diff --git "a/random/val_formal_informal.json" "b/random/val_formal_informal.json" new file mode 100644--- /dev/null +++ "b/random/val_formal_informal.json" @@ -0,0 +1,4082 @@ +[ + { + "formal": "CategoryTheory.Functor.pi'_eval ** I : Type w\u2080 J : Type w\u2081 C : I \u2192 Type u\u2081 inst\u271d\u00b2 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i) D : I \u2192 Type u\u2081 inst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i) A : Type u\u2081 inst\u271d : Category.{u\u2081, u\u2081} A f : (i : I) \u2192 A \u2964 C i i : I \u22a2 pi' f \u22d9 Pi.eval C i = f i ** apply Functor.ext ** case h_map I : Type w\u2080 J : Type w\u2081 C : I \u2192 Type u\u2081 inst\u271d\u00b2 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i) D : I \u2192 Type u\u2081 inst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i) A : Type u\u2081 inst\u271d : Category.{u\u2081, u\u2081} A f : (i : I) \u2192 A \u2964 C i i : I \u22a2 autoParam (\u2200 (X Y : A) (f_1 : X \u27f6 Y), (pi' f \u22d9 Pi.eval C i).map f_1 = eqToHom (_ : ?F.obj X = ?G.obj X) \u226b (f i).map f_1 \u226b eqToHom (_ : (f i).obj Y = (pi' f \u22d9 Pi.eval C i).obj Y)) _auto\u271d case h_obj I : Type w\u2080 J : Type w\u2081 C : I \u2192 Type u\u2081 inst\u271d\u00b2 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i) D : I \u2192 Type u\u2081 inst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i) A : Type u\u2081 inst\u271d : Category.{u\u2081, u\u2081} A f : (i : I) \u2192 A \u2964 C i i : I \u22a2 \u2200 (X : A), (pi' f \u22d9 Pi.eval C i).obj X = (f i).obj X ** intro _ _ _ ** case h_map I : Type w\u2080 J : Type w\u2081 C : I \u2192 Type u\u2081 inst\u271d\u00b2 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i) D : I \u2192 Type u\u2081 inst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i) A : Type u\u2081 inst\u271d : Category.{u\u2081, u\u2081} A f : (i : I) \u2192 A \u2964 C i i : I X\u271d Y\u271d : A f\u271d : X\u271d \u27f6 Y\u271d \u22a2 (pi' f \u22d9 Pi.eval C i).map f\u271d = eqToHom (_ : ?F.obj X\u271d = ?G.obj X\u271d) \u226b (f i).map f\u271d \u226b eqToHom (_ : (f i).obj Y\u271d = (pi' f \u22d9 Pi.eval C i).obj Y\u271d) ** simp ** case h_obj I : Type w\u2080 J : Type w\u2081 C : I \u2192 Type u\u2081 inst\u271d\u00b2 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i) D : I \u2192 Type u\u2081 inst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i) A : Type u\u2081 inst\u271d : Category.{u\u2081, u\u2081} A f : (i : I) \u2192 A \u2964 C i i : I \u22a2 \u2200 (X : A), (pi' f \u22d9 Pi.eval C i).obj X = (f i).obj X ** intro _ ** case h_obj I : Type w\u2080 J : Type w\u2081 C : I \u2192 Type u\u2081 inst\u271d\u00b2 : (i : I) \u2192 Category.{v\u2081, u\u2081} (C i) D : I \u2192 Type u\u2081 inst\u271d\u00b9 : (i : I) \u2192 Category.{v\u2081, u\u2081} (D i) A : Type u\u2081 inst\u271d : Category.{u\u2081, u\u2081} A f : (i : I) \u2192 A \u2964 C i i : I X\u271d : A \u22a2 (pi' f \u22d9 Pi.eval C i).obj X\u271d = (f i).obj X\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.DominatedFinMeasAdditive.add ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' \u22a2 DominatedFinMeasAdditive \u03bc (T + T') (C + C') ** refine' \u27e8hT.1.add hT'.1, fun s hs h\u03bcs => _\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016(T + T') s\u2016 \u2264 (C + C') * ENNReal.toReal (\u2191\u2191\u03bc s) ** rw [Pi.add_apply, add_mul] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : NormedSpace \u211d F' inst\u271d\u00b9 : NormedAddCommGroup G m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b2 : Type u_7 inst\u271d : SeminormedAddCommGroup \u03b2 T T' : Set \u03b1 \u2192 \u03b2 C C' : \u211d hT : DominatedFinMeasAdditive \u03bc T C hT' : DominatedFinMeasAdditive \u03bc T' C' s : Set \u03b1 hs : MeasurableSet s h\u03bcs : \u2191\u2191\u03bc s < \u22a4 \u22a2 \u2016T s + T' s\u2016 \u2264 C * ENNReal.toReal (\u2191\u2191\u03bc s) + C' * ENNReal.toReal (\u2191\u2191\u03bc s) ** exact (norm_add_le _ _).trans (add_le_add (hT.2 s hs h\u03bcs) (hT'.2 s hs h\u03bcs)) ** Qed", + "informal": "" + }, + { + "formal": "EMetric.exists_continuous_real_forall_closedBall_subset ** \u03b9 : Type u_1 X : Type u_2 inst\u271d : EMetricSpace X K U : \u03b9 \u2192 Set X hK : \u2200 (i : \u03b9), IsClosed (K i) hU : \u2200 (i : \u03b9), IsOpen (U i) hKU : \u2200 (i : \u03b9), K i \u2286 U i hfin : LocallyFinite K \u22a2 \u2203 \u03b4, (\u2200 (x : X), 0 < \u2191\u03b4 x) \u2227 \u2200 (i : \u03b9) (x : X), x \u2208 K i \u2192 closedBall x (ENNReal.ofReal (\u2191\u03b4 x)) \u2286 U i ** simpa only [mem_inter_iff, forall_and, mem_preimage, mem_iInter, @forall_swap \u03b9 X] using\n exists_continuous_forall_mem_convex_of_local_const exists_forall_closedBall_subset_aux\u2082\n (exists_forall_closedBall_subset_aux\u2081 hK hU hKU hfin) ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.dist_inversion_mul_dist_center_eq ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c \u22a2 dist (inversion c R x) y * dist x c = dist x (inversion c R y) * dist y c ** rcases eq_or_ne R 0 with rfl | hR ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 \u22a2 dist (inversion c R x) y * dist x c = dist x (inversion c R y) * dist y c ** have hy' : inversion c R y \u2260 c := by simp [*] ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 hy' : inversion c R y \u2260 c \u22a2 dist (inversion c R x) y * dist x c = dist x (inversion c R y) * dist y c ** conv in dist _ y => rw [\u2190 inversion_inversion c hR y] ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 hy' : inversion c R y \u2260 c \u22a2 dist (inversion c R x) (inversion c R (inversion c R y)) * dist x c = dist x (inversion c R y) * dist y c ** rw [dist_inversion_inversion hx hy', dist_inversion_center] ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 hy' : inversion c R y \u2260 c \u22a2 R ^ 2 / (dist x c * (R ^ 2 / dist y c)) * dist x (inversion c R y) * dist x c = dist x (inversion c R y) * dist y c ** have : dist x c \u2260 0 := dist_ne_zero.2 hx ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 hy' : inversion c R y \u2260 c this : dist x c \u2260 0 \u22a2 R ^ 2 / (dist x c * (R ^ 2 / dist y c)) * dist x (inversion c R y) * dist x c = dist x (inversion c R y) * dist y c ** field_simp ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 hy' : inversion c R y \u2260 c this : dist x c \u2260 0 \u22a2 R ^ 2 * dist y c * dist x (inversion c R y) * dist x c = dist x (inversion c R y) * dist y c * (dist x c * R ^ 2) ** ring ** case inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r : \u211d hx : x \u2260 c hy : y \u2260 c \u22a2 dist (inversion c 0 x) y * dist x c = dist x (inversion c 0 y) * dist y c ** simp [dist_comm, mul_comm] ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d x y z : P r R : \u211d hx : x \u2260 c hy : y \u2260 c hR : R \u2260 0 \u22a2 inversion c R y \u2260 c ** simp [*] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.cospanExt_inv_app_one ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D X Y Z X' Y' Z' : C iX : X \u2245 X' iY : Y \u2245 Y' iZ : Z \u2245 Z' f : X \u27f6 Z g : Y \u27f6 Z f' : X' \u27f6 Z' g' : Y' \u27f6 Z' wf : iX.hom \u226b f' = f \u226b iZ.hom wg : iY.hom \u226b g' = g \u226b iZ.hom \u22a2 (cospanExt iX iY iZ wf wg).inv.app WalkingCospan.one = iZ.inv ** dsimp [cospanExt] ** Qed", + "informal": "" + }, + { + "formal": "List.concat_eq_reverse_cons ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 a : \u03b1 l : List \u03b1 \u22a2 concat l a = reverse (a :: reverse l) ** simp only [concat_eq_append, reverse_cons, reverse_reverse] ** Qed", + "informal": "" + }, + { + "formal": "Filter.blimsup_false ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : CompleteLattice \u03b1 f : Filter \u03b2 u : \u03b2 \u2192 \u03b1 \u22a2 (blimsup u f fun x => False) = \u22a5 ** simp [blimsup_eq] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SignedMeasure.of_diff_eq_zero_of_symmDiff_eq_zero_positive ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0 ** rw [restrict_le_restrict_iff] at hsu hsv ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \u2206 v) = 0 \u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0 case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet v case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** have a := hsu (hu.diff hv) (u.diff_subset v) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \u2206 v) = 0 a : \u21910 (u \\ v) \u2264 \u2191s (u \\ v) \u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0 case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet v case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** have b := hsv (hv.diff hu) (v.diff_subset u) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \u2206 v) = 0 a : \u21910 (u \\ v) \u2264 \u2191s (u \\ v) b : \u21910 (v \\ u) \u2264 \u2191s (v \\ u) \u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0 case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet v case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** erw [of_union (Set.disjoint_of_subset_left (u.diff_subset v) disjoint_sdiff_self_right)\n (hu.diff hv) (hv.diff hu)] at hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0 a : \u21910 (u \\ v) \u2264 \u2191s (u \\ v) b : \u21910 (v \\ u) \u2264 \u2191s (v \\ u) \u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0 case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet v case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** rw [zero_apply] at a b ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0 a : 0 \u2264 \u2191s (u \\ v) b : 0 \u2264 \u2191s (v \\ u) \u22a2 \u2191s (u \\ v) = 0 \u2227 \u2191s (v \\ u) = 0 case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet v case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** constructor ** case left \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0 a : 0 \u2264 \u2191s (u \\ v) b : 0 \u2264 \u2191s (v \\ u) \u22a2 \u2191s (u \\ v) = 0 case right \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0 a : 0 \u2264 \u2191s (u \\ v) b : 0 \u2264 \u2191s (v \\ u) \u22a2 \u2191s (v \\ u) = 0 case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet v case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** all_goals first | linarith | infer_instance | assumption ** case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** first | linarith | infer_instance | assumption ** case right \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 u \u2192 \u21910 j \u2264 \u2191s j hsv : \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 v \u2192 \u21910 j \u2264 \u2191s j hs : \u2191s (u \\ v) + \u2191s (v \\ u) = 0 a : 0 \u2264 \u2191s (u \\ v) b : 0 \u2264 \u2191s (v \\ u) \u22a2 \u2191s (v \\ u) = 0 ** linarith ** case hi \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : MeasurableSpace \u03b1 s : SignedMeasure \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd u v w : Set \u03b1 hu : MeasurableSet u hv : MeasurableSet v hsu : VectorMeasure.restrict 0 u \u2264 VectorMeasure.restrict s u hsv : VectorMeasure.restrict 0 v \u2264 VectorMeasure.restrict s v hs : \u2191s (u \u2206 v) = 0 \u22a2 MeasurableSet u ** assumption ** Qed", + "informal": "" + }, + { + "formal": "hasFDerivAt_tsum ** \u03b1 : Type u_1 \u03b2 : Type u_2 \ud835\udd5c : Type u_3 E : Type u_4 F : Type u_5 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : CompleteSpace F u : \u03b1 \u2192 \u211d inst\u271d : NormedSpace \ud835\udd5c F f : \u03b1 \u2192 E \u2192 F f' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F v : \u2115 \u2192 \u03b1 \u2192 \u211d s : Set E x\u2080 x\u271d : E N : \u2115\u221e hu : Summable u hf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x hf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n hf0 : Summable fun n => f n x\u2080 x : E \u22a2 HasFDerivAt (fun y => \u2211' (n : \u03b1), f n y) (\u2211' (n : \u03b1), f' n x) x ** let : NormedSpace \u211d E ** case this \u03b1 : Type u_1 \u03b2 : Type u_2 \ud835\udd5c : Type u_3 E : Type u_4 F : Type u_5 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : CompleteSpace F u : \u03b1 \u2192 \u211d inst\u271d : NormedSpace \ud835\udd5c F f : \u03b1 \u2192 E \u2192 F f' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F v : \u2115 \u2192 \u03b1 \u2192 \u211d s : Set E x\u2080 x\u271d : E N : \u2115\u221e hu : Summable u hf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x hf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n hf0 : Summable fun n => f n x\u2080 x : E \u22a2 NormedSpace \u211d E \u03b1 : Type u_1 \u03b2 : Type u_2 \ud835\udd5c : Type u_3 E : Type u_4 F : Type u_5 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : CompleteSpace F u : \u03b1 \u2192 \u211d inst\u271d : NormedSpace \ud835\udd5c F f : \u03b1 \u2192 E \u2192 F f' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F v : \u2115 \u2192 \u03b1 \u2192 \u211d s : Set E x\u2080 x\u271d : E N : \u2115\u221e hu : Summable u hf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x hf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n hf0 : Summable fun n => f n x\u2080 x : E this : NormedSpace \u211d E := ?this \u22a2 HasFDerivAt (fun y => \u2211' (n : \u03b1), f n y) (\u2211' (n : \u03b1), f' n x) x ** exact NormedSpace.restrictScalars \u211d \ud835\udd5c _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \ud835\udd5c : Type u_3 E : Type u_4 F : Type u_5 inst\u271d\u2075 : IsROrC \ud835\udd5c inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : CompleteSpace F u : \u03b1 \u2192 \u211d inst\u271d : NormedSpace \ud835\udd5c F f : \u03b1 \u2192 E \u2192 F f' : \u03b1 \u2192 E \u2192 E \u2192L[\ud835\udd5c] F v : \u2115 \u2192 \u03b1 \u2192 \u211d s : Set E x\u2080 x\u271d : E N : \u2115\u221e hu : Summable u hf : \u2200 (n : \u03b1) (x : E), HasFDerivAt (f n) (f' n x) x hf' : \u2200 (n : \u03b1) (x : E), \u2016f' n x\u2016 \u2264 u n hf0 : Summable fun n => f n x\u2080 x : E this : NormedSpace \u211d E := NormedSpace.restrictScalars \u211d \ud835\udd5c E \u22a2 HasFDerivAt (fun y => \u2211' (n : \u03b1), f n y) (\u2211' (n : \u03b1), f' n x) x ** exact hasFDerivAt_tsum_of_isPreconnected hu isOpen_univ isPreconnected_univ\n (fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _) ** Qed", + "informal": "" + }, + { + "formal": "Int.neg_mul_comm ** a b : Int \u22a2 -a * b = a * -b ** simp ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.edgeSet_sup ** \u03b9 : Sort u_1 \ud835\udd5c : Type u_2 V : Type u W : Type v X : Type w G : SimpleGraph V G' : SimpleGraph W a b c u v w : V e : Sym2 V G\u2081 G\u2082 : SimpleGraph V \u22a2 edgeSet (G\u2081 \u2294 G\u2082) = edgeSet G\u2081 \u222a edgeSet G\u2082 ** ext \u27e8x, y\u27e9 ** case h.mk.mk \u03b9 : Sort u_1 \ud835\udd5c : Type u_2 V : Type u W : Type v X : Type w G : SimpleGraph V G' : SimpleGraph W a b c u v w : V e : Sym2 V G\u2081 G\u2082 : SimpleGraph V x\u271d : Sym2 V x y : V \u22a2 Quot.mk Setoid.r (x, y) \u2208 edgeSet (G\u2081 \u2294 G\u2082) \u2194 Quot.mk Setoid.r (x, y) \u2208 edgeSet G\u2081 \u222a edgeSet G\u2082 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.principal_one_iff ** op : Ordinal.{u_1} \u2192 Ordinal.{u_1} \u2192 Ordinal.{u_1} \u22a2 Principal op 1 \u2194 op 0 0 = 0 ** refine' \u27e8fun h => _, fun h a b ha hb => _\u27e9 ** case refine'_1 op : Ordinal.{u_1} \u2192 Ordinal.{u_1} \u2192 Ordinal.{u_1} h : Principal op 1 \u22a2 op 0 0 = 0 ** rw [\u2190 lt_one_iff_zero] ** case refine'_1 op : Ordinal.{u_1} \u2192 Ordinal.{u_1} \u2192 Ordinal.{u_1} h : Principal op 1 \u22a2 op 0 0 < 1 ** exact h zero_lt_one zero_lt_one ** case refine'_2 op : Ordinal.{u_1} \u2192 Ordinal.{u_1} \u2192 Ordinal.{u_1} h : op 0 0 = 0 a b : Ordinal.{u_1} ha : a < 1 hb : b < 1 \u22a2 op a b < 1 ** rwa [lt_one_iff_zero, ha, hb] at * ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme ** C : Type u inst\u271d\u00b9 : Category.{v, u} C X\u271d : PresheafedSpace CommRingCat Y\u271d : Scheme f\u271d : X\u271d \u27f6 Y\u271d.toPresheafedSpace H : IsOpenImmersion f\u271d X Y : Scheme f : X \u27f6 Y inst\u271d : AlgebraicGeometry.IsOpenImmersion f \u22a2 toScheme Y f.val = X ** apply scheme_eq_of_locallyRingedSpace_eq ** case H C : Type u inst\u271d\u00b9 : Category.{v, u} C X\u271d : PresheafedSpace CommRingCat Y\u271d : Scheme f\u271d : X\u271d \u27f6 Y\u271d.toPresheafedSpace H : IsOpenImmersion f\u271d X Y : Scheme f : X \u27f6 Y inst\u271d : AlgebraicGeometry.IsOpenImmersion f \u22a2 (toScheme Y f.val).toLocallyRingedSpace = X.toLocallyRingedSpace ** exact locallyRingedSpace_toLocallyRingedSpace f ** Qed", + "informal": "" + }, + { + "formal": "NonemptyFinLinOrd.epi_iff_surjective ** A B : NonemptyFinLinOrd f : A \u27f6 B \u22a2 Epi f \u2194 Function.Surjective \u2191f ** constructor ** case mp A B : NonemptyFinLinOrd f : A \u27f6 B \u22a2 Epi f \u2192 Function.Surjective \u2191f ** intro ** case mp A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f \u22a2 Function.Surjective \u2191f ** dsimp only [Function.Surjective] ** case mp A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f \u22a2 \u2200 (b : \u2191B), \u2203 a, \u2191f a = b ** by_contra' hf' ** case mp A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f hf' : \u2203 b, \u2200 (a : \u2191A), \u2191f a \u2260 b \u22a2 False ** rcases hf' with \u27e8m, hm\u27e9 ** case mp.intro A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m \u22a2 False ** let Y := NonemptyFinLinOrd.of (ULift (Fin 2)) ** case mp.intro A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } h : \u2191p\u2081 m = \u2191p\u2082 m \u22a2 False ** simp [FunLike.coe] at h ** A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 \u22a2 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082 ** simp only ** A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 \u22a2 (if x\u2081 < m then { down := 0 } else { down := 1 }) \u2264 if x\u2082 < m then { down := 0 } else { down := 1 } ** split_ifs with h\u2081 h\u2082 h\u2082 ** case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : x\u2081 < m h\u2082 : x\u2082 < m \u22a2 { down := 0 } \u2264 { down := 0 } case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : x\u2081 < m h\u2082 : \u00acx\u2082 < m \u22a2 { down := 0 } \u2264 { down := 1 } case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 < m h\u2082 : x\u2082 < m \u22a2 { down := 1 } \u2264 { down := 0 } case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 < m h\u2082 : \u00acx\u2082 < m \u22a2 { down := 1 } \u2264 { down := 1 } ** any_goals apply Fin.zero_le ** case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : x\u2081 < m h\u2082 : \u00acx\u2082 < m \u22a2 { down := 0 } \u2264 { down := 1 } ** apply Fin.zero_le ** case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 < m h\u2082 : x\u2082 < m \u22a2 { down := 1 } \u2264 { down := 0 } ** exfalso ** case pos.h A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 < m h\u2082 : x\u2082 < m \u22a2 False ** exact h\u2081 (lt_of_le_of_lt h h\u2082) ** case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 < m h\u2082 : \u00acx\u2082 < m \u22a2 { down := 1 } \u2264 { down := 1 } ** rfl ** A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 \u22a2 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082 ** simp only ** A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 \u22a2 (if x\u2081 \u2264 m then { down := 0 } else { down := 1 }) \u2264 if x\u2082 \u2264 m then { down := 0 } else { down := 1 } ** split_ifs with h\u2081 h\u2082 h\u2082 ** case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : x\u2081 \u2264 m h\u2082 : x\u2082 \u2264 m \u22a2 { down := 0 } \u2264 { down := 0 } case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : x\u2081 \u2264 m h\u2082 : \u00acx\u2082 \u2264 m \u22a2 { down := 0 } \u2264 { down := 1 } case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 \u2264 m h\u2082 : x\u2082 \u2264 m \u22a2 { down := 1 } \u2264 { down := 0 } case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 \u2264 m h\u2082 : \u00acx\u2082 \u2264 m \u22a2 { down := 1 } \u2264 { down := 1 } ** any_goals apply Fin.zero_le ** case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : x\u2081 \u2264 m h\u2082 : \u00acx\u2082 \u2264 m \u22a2 { down := 0 } \u2264 { down := 1 } ** apply Fin.zero_le ** case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 \u2264 m h\u2082 : x\u2082 \u2264 m \u22a2 { down := 1 } \u2264 { down := 0 } ** exfalso ** case pos.h A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 \u2264 m h\u2082 : x\u2082 \u2264 m \u22a2 False ** exact h\u2081 (h.trans h\u2082) ** case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } x\u2081 x\u2082 : \u2191B h : x\u2081 \u2264 x\u2082 h\u2081 : \u00acx\u2081 \u2264 m h\u2082 : \u00acx\u2082 \u2264 m \u22a2 { down := 1 } \u2264 { down := 1 } ** rfl ** A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } \u22a2 \u2191p\u2081 m = \u2191p\u2082 m ** congr ** case e_a A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } \u22a2 p\u2081 = p\u2082 ** rw [\u2190 cancel_epi f] ** case e_a A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } \u22a2 f \u226b p\u2081 = f \u226b p\u2082 ** ext a ** case e_a.w A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A \u22a2 \u2191(f \u226b p\u2081) a = \u2191(f \u226b p\u2082) a ** simp only [coe_of, comp_apply] ** case e_a.w A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A \u22a2 \u2191{ toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } (\u2191f a) = \u2191{ toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } (\u2191f a) ** change ite _ _ _ = ite _ _ _ ** case e_a.w A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A \u22a2 (if \u2191f a < m then { down := 0 } else { down := 1 }) = if \u2191f a \u2264 m then { down := 0 } else { down := 1 } ** split_ifs with h\u2081 h\u2082 h\u2082 ** case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u2191f a < m h\u2082 : \u2191f a \u2264 m \u22a2 { down := 0 } = { down := 0 } case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u2191f a < m h\u2082 : \u00ac\u2191f a \u2264 m \u22a2 { down := 0 } = { down := 1 } case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u00ac\u2191f a < m h\u2082 : \u2191f a \u2264 m \u22a2 { down := 1 } = { down := 0 } case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u00ac\u2191f a < m h\u2082 : \u00ac\u2191f a \u2264 m \u22a2 { down := 1 } = { down := 1 } ** any_goals rfl ** case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u00ac\u2191f a < m h\u2082 : \u00ac\u2191f a \u2264 m \u22a2 { down := 1 } = { down := 1 } ** rfl ** case neg A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u2191f a < m h\u2082 : \u00ac\u2191f a \u2264 m \u22a2 { down := 0 } = { down := 1 } ** exfalso ** case neg.h A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u2191f a < m h\u2082 : \u00ac\u2191f a \u2264 m \u22a2 False ** exact h\u2082 (le_of_lt h\u2081) ** case pos A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u00ac\u2191f a < m h\u2082 : \u2191f a \u2264 m \u22a2 { down := 1 } = { down := 0 } ** exfalso ** case pos.h A B : NonemptyFinLinOrd f : A \u27f6 B a\u271d : Epi f m : \u2191B hm : \u2200 (a : \u2191A), \u2191f a \u2260 m Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2)) p\u2081 : B \u27f6 Y := { toFun := fun b => if b < m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b < m then { down := 0 } else { down := 1 }) x\u2082) } p\u2082 : B \u27f6 Y := { toFun := fun b => if b \u2264 m then { down := 0 } else { down := 1 }, monotone' := (_ : \u2200 (x\u2081 x\u2082 : \u2191B), x\u2081 \u2264 x\u2082 \u2192 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2081 \u2264 (fun b => if b \u2264 m then { down := 0 } else { down := 1 }) x\u2082) } a : (forget NonemptyFinLinOrd).obj A h\u2081 : \u00ac\u2191f a < m h\u2082 : \u2191f a \u2264 m \u22a2 False ** exact hm a (eq_of_le_of_not_lt h\u2082 h\u2081) ** case mpr A B : NonemptyFinLinOrd f : A \u27f6 B \u22a2 Function.Surjective \u2191f \u2192 Epi f ** intro h ** case mpr A B : NonemptyFinLinOrd f : A \u27f6 B h : Function.Surjective \u2191f \u22a2 Epi f ** exact ConcreteCategory.epi_of_surjective f h ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.setLaverage_congr_fun ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 m0 : MeasurableSpace \u03b1 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \u211d E inst\u271d\u00b3 : CompleteSpace E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F \u03bc \u03bd : Measure \u03b1 s t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e hs : MeasurableSet s h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 s \u2192 f x = g x \u22a2 \u2a0d\u207b (x : \u03b1) in s, f x \u2202\u03bc = \u2a0d\u207b (x : \u03b1) in s, g x \u2202\u03bc ** simp only [laverage_eq, set_lintegral_congr_fun hs h] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.length_darts ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' u v : V p : Walk G u v \u22a2 List.length (darts p) = length p ** induction p <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.iterate_derivative_map ** R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d\u00b9 : Semiring R inst\u271d : Semiring S p : R[X] f : R \u2192+* S k : \u2115 \u22a2 (\u2191derivative)^[k] (map f p) = map f ((\u2191derivative)^[k] p) ** induction' k with k ih generalizing p ** case zero R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d\u00b9 : Semiring R inst\u271d : Semiring S p\u271d : R[X] f : R \u2192+* S p : R[X] \u22a2 (\u2191derivative)^[Nat.zero] (map f p) = map f ((\u2191derivative)^[Nat.zero] p) ** simp ** case succ R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n : \u2115 inst\u271d\u00b9 : Semiring R inst\u271d : Semiring S p\u271d : R[X] f : R \u2192+* S k : \u2115 ih : \u2200 (p : R[X]), (\u2191derivative)^[k] (map f p) = map f ((\u2191derivative)^[k] p) p : R[X] \u22a2 (\u2191derivative)^[Nat.succ k] (map f p) = map f ((\u2191derivative)^[Nat.succ k] p) ** simp only [ih, Function.iterate_succ, Polynomial.derivative_map, Function.comp_apply] ** Qed", + "informal": "" + }, + { + "formal": "List.chain_pmap_of_chain ** \u03b1 : Type u \u03b2 : Type v R r : \u03b1 \u2192 \u03b1 \u2192 Prop l\u271d l\u2081 l\u2082 : List \u03b1 a\u271d b : \u03b1 S : \u03b2 \u2192 \u03b2 \u2192 Prop p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 H : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), R a b \u2192 S (f a ha) (f b hb) a : \u03b1 l : List \u03b1 hl\u2081 : Chain R a l ha : p a hl\u2082 : \u2200 (a : \u03b1), a \u2208 l \u2192 p a \u22a2 Chain S (f a ha) (pmap f l hl\u2082) ** induction' l with lh lt l_ih generalizing a ** case nil \u03b1 : Type u \u03b2 : Type v R r : \u03b1 \u2192 \u03b1 \u2192 Prop l\u271d l\u2081 l\u2082 : List \u03b1 a\u271d\u00b9 b : \u03b1 S : \u03b2 \u2192 \u03b2 \u2192 Prop p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 H : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), R a b \u2192 S (f a ha) (f b hb) a\u271d : \u03b1 l : List \u03b1 hl\u2081\u271d : Chain R a\u271d l ha\u271d : p a\u271d hl\u2082\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a a : \u03b1 hl\u2081 : Chain R a [] ha : p a hl\u2082 : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a \u22a2 Chain S (f a ha) (pmap f [] hl\u2082) ** simp ** case cons \u03b1 : Type u \u03b2 : Type v R r : \u03b1 \u2192 \u03b1 \u2192 Prop l\u271d l\u2081 l\u2082 : List \u03b1 a\u271d\u00b9 b : \u03b1 S : \u03b2 \u2192 \u03b2 \u2192 Prop p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 H : \u2200 (a b : \u03b1) (ha : p a) (hb : p b), R a b \u2192 S (f a ha) (f b hb) a\u271d : \u03b1 l : List \u03b1 hl\u2081\u271d : Chain R a\u271d l ha\u271d : p a\u271d hl\u2082\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a lh : \u03b1 lt : List \u03b1 l_ih : \u2200 {a : \u03b1}, Chain R a lt \u2192 \u2200 (ha : p a) (hl\u2082 : \u2200 (a : \u03b1), a \u2208 lt \u2192 p a), Chain S (f a ha) (pmap f lt hl\u2082) a : \u03b1 hl\u2081 : Chain R a (lh :: lt) ha : p a hl\u2082 : \u2200 (a : \u03b1), a \u2208 lh :: lt \u2192 p a \u22a2 Chain S (f a ha) (pmap f (lh :: lt) hl\u2082) ** simp [H _ _ _ _ (rel_of_chain_cons hl\u2081), l_ih (chain_of_chain_cons hl\u2081)] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.HasLiftingProperty.of_arrow_iso_left ** C : Type u_1 inst\u271d : Category.{u_2, u_1} C A\u271d B\u271d B'\u271d X\u271d Y\u271d Y' : C i\u271d : A\u271d \u27f6 B\u271d i'\u271d : B\u271d \u27f6 B'\u271d p\u271d : X\u271d \u27f6 Y\u271d p' : Y\u271d \u27f6 Y' A B A' B' X Y : C i : A \u27f6 B i' : A' \u27f6 B' e : Arrow.mk i \u2245 Arrow.mk i' p : X \u27f6 Y hip : HasLiftingProperty i p \u22a2 HasLiftingProperty i' p ** rw [Arrow.iso_w' e] ** C : Type u_1 inst\u271d : Category.{u_2, u_1} C A\u271d B\u271d B'\u271d X\u271d Y\u271d Y' : C i\u271d : A\u271d \u27f6 B\u271d i'\u271d : B\u271d \u27f6 B'\u271d p\u271d : X\u271d \u27f6 Y\u271d p' : Y\u271d \u27f6 Y' A B A' B' X Y : C i : A \u27f6 B i' : A' \u27f6 B' e : Arrow.mk i \u2245 Arrow.mk i' p : X \u27f6 Y hip : HasLiftingProperty i p \u22a2 HasLiftingProperty (e.inv.left \u226b i \u226b e.hom.right) p ** infer_instance ** Qed", + "informal": "" + }, + { + "formal": "Associates.prod_le_prod_iff_le ** \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a \u22a2 Multiset.prod p \u2264 Multiset.prod q \u2192 p \u2264 q ** rintro \u27e8c, eqc\u27e9 ** case intro \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c \u22a2 p \u2264 q ** refine' Multiset.le_iff_exists_add.2 \u27e8factors c, unique' hq (fun x hx => _) _\u27e9 ** case intro.refine'_1 \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c x : Associates \u03b1 hx : x \u2208 p + factors c \u22a2 Irreducible x ** obtain h | h := Multiset.mem_add.1 hx ** case intro.refine'_1.inl \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c x : Associates \u03b1 hx : x \u2208 p + factors c h : x \u2208 p \u22a2 Irreducible x ** exact hp x h ** case intro.refine'_1.inr \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c x : Associates \u03b1 hx : x \u2208 p + factors c h : x \u2208 factors c \u22a2 Irreducible x ** exact irreducible_of_factor _ h ** case intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c \u22a2 Multiset.prod q = Multiset.prod (p + factors c) ** rw [eqc, Multiset.prod_add] ** case intro.refine'_2 \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c \u22a2 Multiset.prod p * c = Multiset.prod p * Multiset.prod (factors c) ** congr ** case intro.refine'_2.e_a \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c \u22a2 c = Multiset.prod (factors c) ** refine' associated_iff_eq.mp (factors_prod fun hc => _).symm ** case intro.refine'_2.e_a \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c hc : c = 0 \u22a2 False ** refine' not_irreducible_zero (hq _ _) ** case intro.refine'_2.e_a \u03b1 : Type u_1 inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 inst\u271d : Nontrivial \u03b1 p q : Multiset (Associates \u03b1) hp : \u2200 (a : Associates \u03b1), a \u2208 p \u2192 Irreducible a hq : \u2200 (a : Associates \u03b1), a \u2208 q \u2192 Irreducible a c : Associates \u03b1 eqc : Multiset.prod q = Multiset.prod p * c hc : c = 0 \u22a2 0 \u2208 q ** rw [\u2190 prod_eq_zero_iff, eqc, hc, mul_zero] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.absNorm_mem ** S : Type u_1 inst\u271d\u2075 : CommRing S inst\u271d\u2074 : IsDomain S inst\u271d\u00b3 : Infinite S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : Module.Free \u2124 S inst\u271d : Module.Finite \u2124 S I : Ideal S \u22a2 \u2191(\u2191absNorm I) \u2208 I ** rw [absNorm_apply, cardQuot, \u2190 Ideal.Quotient.eq_zero_iff_mem, map_natCast,\n Quotient.index_eq_zero] ** Qed", + "informal": "" + }, + { + "formal": "NNReal.rpow_nat_cast ** x : \u211d\u22650 n : \u2115 \u22a2 \u2191(x ^ \u2191n) = \u2191(x ^ n) ** simpa only [coe_rpow, coe_pow] using Real.rpow_nat_cast x n ** Qed", + "informal": "" + }, + { + "formal": "div_mem ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A M : Type u_5 S : Type u_6 inst\u271d\u00b9 : DivInvMonoid M inst\u271d : SetLike S M hSM : SubgroupClass S M H K : S x y : M hx : x \u2208 H hy : y \u2208 H \u22a2 x / y \u2208 H ** rw [div_eq_mul_inv] ** G : Type u_1 G' : Type u_2 G'' : Type u_3 inst\u271d\u2075 : Group G inst\u271d\u2074 : Group G' inst\u271d\u00b3 : Group G'' A : Type u_4 inst\u271d\u00b2 : AddGroup A M : Type u_5 S : Type u_6 inst\u271d\u00b9 : DivInvMonoid M inst\u271d : SetLike S M hSM : SubgroupClass S M H K : S x y : M hx : x \u2208 H hy : y \u2208 H \u22a2 x * y\u207b\u00b9 \u2208 H ** exact mul_mem hx (inv_mem hy) ** Qed", + "informal": "" + }, + { + "formal": "Int.neg_eq_neg_one_mul ** n : Nat \u22a2 -\u2191(succ n) = -[1 * n+1] ** rw [Nat.one_mul] ** n : Nat \u22a2 -\u2191(succ n) = -[n+1] ** rfl ** n : Nat \u22a2 - -[n+1] = ofNat (succ 0 * succ n) ** rw [Nat.one_mul] ** n : Nat \u22a2 - -[n+1] = ofNat (succ n) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.vanishingIdeal_zeroLocus_eq_radical ** k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I : Ideal (MvPolynomial \u03c3 k) \u22a2 vanishingIdeal (zeroLocus I) = radical I ** rw [I.radical_eq_jacobson] ** k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I : Ideal (MvPolynomial \u03c3 k) \u22a2 vanishingIdeal (zeroLocus I) = jacobson I ** refine' le_antisymm (le_sInf _) fun p hp x hx => _ ** case refine'_1 k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I : Ideal (MvPolynomial \u03c3 k) \u22a2 \u2200 (b : Ideal (MvPolynomial \u03c3 k)), b \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 vanishingIdeal (zeroLocus I) \u2264 b ** rintro J \u27e8hJI, hJ\u27e9 ** case refine'_1.intro k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I J : Ideal (MvPolynomial \u03c3 k) hJI : I \u2264 J hJ : IsMaximal J \u22a2 vanishingIdeal (zeroLocus I) \u2264 J ** obtain \u27e8x, hx\u27e9 := (isMaximal_iff_eq_vanishingIdeal_singleton J).1 hJ ** case refine'_1.intro.intro k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I J : Ideal (MvPolynomial \u03c3 k) hJI : I \u2264 J hJ : IsMaximal J x : \u03c3 \u2192 k hx : J = vanishingIdeal {x} \u22a2 vanishingIdeal (zeroLocus I) \u2264 J ** refine' hx.symm \u25b8 vanishingIdeal_anti_mono fun y hy p hp => _ ** case refine'_1.intro.intro k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I J : Ideal (MvPolynomial \u03c3 k) hJI : I \u2264 J hJ : IsMaximal J x : \u03c3 \u2192 k hx : J = vanishingIdeal {x} y : \u03c3 \u2192 k hy : y \u2208 {x} p : MvPolynomial \u03c3 k hp : p \u2208 I \u22a2 \u2191(eval y) p = 0 ** rw [\u2190 mem_vanishingIdeal_singleton_iff, Set.mem_singleton_iff.1 hy, \u2190 hx] ** case refine'_1.intro.intro k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I J : Ideal (MvPolynomial \u03c3 k) hJI : I \u2264 J hJ : IsMaximal J x : \u03c3 \u2192 k hx : J = vanishingIdeal {x} y : \u03c3 \u2192 k hy : y \u2208 {x} p : MvPolynomial \u03c3 k hp : p \u2208 I \u22a2 p \u2208 J ** refine' hJI hp ** case refine'_2 k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I : Ideal (MvPolynomial \u03c3 k) p : MvPolynomial \u03c3 k hp : p \u2208 jacobson I x : \u03c3 \u2192 k hx : x \u2208 zeroLocus I \u22a2 \u2191(eval x) p = 0 ** rw [\u2190 mem_vanishingIdeal_singleton_iff x p] ** case refine'_2 k : Type u_1 inst\u271d\u00b2 : Field k \u03c3 : Type u_2 inst\u271d\u00b9 : IsAlgClosed k inst\u271d : Finite \u03c3 I : Ideal (MvPolynomial \u03c3 k) p : MvPolynomial \u03c3 k hp : p \u2208 jacobson I x : \u03c3 \u2192 k hx : x \u2208 zeroLocus I \u22a2 p \u2208 vanishingIdeal {x} ** refine' (mem_sInf.mp hp)\n \u27e8le_trans (le_vanishingIdeal_zeroLocus I) (vanishingIdeal_anti_mono fun y hy => hy.symm \u25b8 hx),\n MvPolynomial.vanishingIdeal_singleton_isMaximal\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMap.continuousMap_mem_subalgebra_closure_of_separatesPoints ** X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : CompactSpace X A : Subalgebra \u211d C(X, \u211d) w : Subalgebra.SeparatesPoints A f : C(X, \u211d) \u22a2 f \u2208 Subalgebra.topologicalClosure A ** rw [subalgebra_topologicalClosure_eq_top_of_separatesPoints A w] ** X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : CompactSpace X A : Subalgebra \u211d C(X, \u211d) w : Subalgebra.SeparatesPoints A f : C(X, \u211d) \u22a2 f \u2208 \u22a4 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Fin.isCycle_cycleRange ** n : \u2115 i : Fin (n + 1) h0 : i \u2260 0 \u22a2 IsCycle (cycleRange i) ** cases' i with i hi ** case mk n i : \u2115 hi : i < n + 1 h0 : { val := i, isLt := hi } \u2260 0 \u22a2 IsCycle (cycleRange { val := i, isLt := hi }) ** cases i ** case mk.succ n n\u271d : \u2115 hi : Nat.succ n\u271d < n + 1 h0 : { val := Nat.succ n\u271d, isLt := hi } \u2260 0 \u22a2 IsCycle (cycleRange { val := Nat.succ n\u271d, isLt := hi }) ** exact isCycle_finRotate.extendDomain _ ** case mk.zero n : \u2115 hi : Nat.zero < n + 1 h0 : { val := Nat.zero, isLt := hi } \u2260 0 \u22a2 IsCycle (cycleRange { val := Nat.zero, isLt := hi }) ** exact (h0 rfl).elim ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.CNF_foldr ** b o : Ordinal.{u_1} \u22a2 foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b 0) = 0 ** rw [CNF_zero] ** b o : Ordinal.{u_1} \u22a2 foldr (fun p r => b ^ p.1 * p.2 + r) 0 [] = 0 ** rfl ** b o\u271d o : Ordinal.{u_1} ho : o \u2260 0 IH : foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b (o % b ^ log b o)) = o % b ^ log b o \u22a2 foldr (fun p r => b ^ p.1 * p.2 + r) 0 (CNF b o) = o ** rw [CNF_ne_zero ho, foldr_cons, IH, div_add_mod] ** Qed", + "informal": "" + }, + { + "formal": "IsOpen.inter_closure ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a : \u03b1 s\u271d s\u2081 s\u2082 t\u271d : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 s t : Set \u03b1 h : IsOpen s \u22a2 (closure (s \u2229 t))\u1d9c \u2286 (s \u2229 closure t)\u1d9c ** simpa only [\u2190 interior_compl, compl_inter] using IsClosed.interior_union_left h.isClosed_compl ** Qed", + "informal": "" + }, + { + "formal": "biSup_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9\u271d : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9\u271d \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f g s\u271d t : \u03b9\u271d \u2192 \u03b1 a\u271d b : \u03b1 \u03b9 : Type u_9 a : \u03b1 s : Set \u03b9 hs : Set.Nonempty s \u22a2 \u2a06 i \u2208 s, a = a ** haveI : Nonempty s := Set.nonempty_coe_sort.mpr hs ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9\u271d : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9\u271d \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f g s\u271d t : \u03b9\u271d \u2192 \u03b1 a\u271d b : \u03b1 \u03b9 : Type u_9 a : \u03b1 s : Set \u03b9 hs : Set.Nonempty s this : Nonempty \u2191s \u22a2 \u2a06 i \u2208 s, a = a ** rw [\u2190 iSup_subtype'', iSup_const] ** Qed", + "informal": "" + }, + { + "formal": "BoundedContinuousFunction.exists_forall_mem_restrict_eq_of_closed ** X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : NormalSpace Y s : Set Y f : \u2191s \u2192\u1d47 \u211d hs : IsClosed s t : Set \u211d inst\u271d : OrdConnected t hf : \u2200 (x : \u2191s), \u2191f x \u2208 t hne : Set.Nonempty t \u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 restrict g s = f ** rcases exists_extension_forall_mem_of_closedEmbedding f hf hne\n (closedEmbedding_subtype_val hs) with\n \u27e8g, hg, hgf\u27e9 ** case intro.intro X : Type u_1 Y : Type u_2 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace Y inst\u271d\u00b9 : NormalSpace Y s : Set Y f : \u2191s \u2192\u1d47 \u211d hs : IsClosed s t : Set \u211d inst\u271d : OrdConnected t hf : \u2200 (x : \u2191s), \u2191f x \u2208 t hne : Set.Nonempty t g : Y \u2192\u1d47 \u211d hg : \u2200 (y : Y), \u2191g y \u2208 t hgf : \u2191g \u2218 Subtype.val = \u2191f \u22a2 \u2203 g, (\u2200 (y : Y), \u2191g y \u2208 t) \u2227 restrict g s = f ** exact \u27e8g, hg, FunLike.coe_injective hgf\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "contDiff_top ** \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F \u22a2 ContDiff \ud835\udd5c \u22a4 f \u2194 \u2200 (n : \u2115), ContDiff \ud835\udd5c (\u2191n) f ** simp [contDiffOn_univ.symm, contDiffOn_top] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.exists_circumsphere_eq_of_cospherical_subset ** V : Type u_1 P : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : InnerProductSpace \u211d V inst\u271d\u00b3 : MetricSpace P inst\u271d\u00b2 : NormedAddTorsor V P s : AffineSubspace \u211d P ps : Set P h : ps \u2286 \u2191s inst\u271d\u00b9 : Nonempty { x // x \u2208 s } n : \u2115 inst\u271d : FiniteDimensional \u211d { x // x \u2208 direction s } hd : finrank \u211d { x // x \u2208 direction s } = n hc : Cospherical ps \u22a2 \u2203 c, \u2200 (sx : Simplex \u211d P n), Set.range sx.points \u2286 ps \u2192 Simplex.circumsphere sx = c ** obtain \u27e8r, hr\u27e9 := exists_circumradius_eq_of_cospherical_subset h hd hc ** case intro V : Type u_1 P : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : InnerProductSpace \u211d V inst\u271d\u00b3 : MetricSpace P inst\u271d\u00b2 : NormedAddTorsor V P s : AffineSubspace \u211d P ps : Set P h : ps \u2286 \u2191s inst\u271d\u00b9 : Nonempty { x // x \u2208 s } n : \u2115 inst\u271d : FiniteDimensional \u211d { x // x \u2208 direction s } hd : finrank \u211d { x // x \u2208 direction s } = n hc : Cospherical ps r : \u211d hr : \u2200 (sx : Simplex \u211d P n), Set.range sx.points \u2286 ps \u2192 Simplex.circumradius sx = r \u22a2 \u2203 c, \u2200 (sx : Simplex \u211d P n), Set.range sx.points \u2286 ps \u2192 Simplex.circumsphere sx = c ** obtain \u27e8c, hc\u27e9 := exists_circumcenter_eq_of_cospherical_subset h hd hc ** case intro.intro V : Type u_1 P : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : InnerProductSpace \u211d V inst\u271d\u00b3 : MetricSpace P inst\u271d\u00b2 : NormedAddTorsor V P s : AffineSubspace \u211d P ps : Set P h : ps \u2286 \u2191s inst\u271d\u00b9 : Nonempty { x // x \u2208 s } n : \u2115 inst\u271d : FiniteDimensional \u211d { x // x \u2208 direction s } hd : finrank \u211d { x // x \u2208 direction s } = n hc\u271d : Cospherical ps r : \u211d hr : \u2200 (sx : Simplex \u211d P n), Set.range sx.points \u2286 ps \u2192 Simplex.circumradius sx = r c : P hc : \u2200 (sx : Simplex \u211d P n), Set.range sx.points \u2286 ps \u2192 Simplex.circumcenter sx = c \u22a2 \u2203 c, \u2200 (sx : Simplex \u211d P n), Set.range sx.points \u2286 ps \u2192 Simplex.circumsphere sx = c ** exact \u27e8\u27e8c, r\u27e9, fun sx hsx => Sphere.ext _ _ (hc sx hsx) (hr sx hsx)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "QuadraticForm.polar_smul_right ** S : Type u_1 T : Type u_2 R : Type u_3 M : Type u_4 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M Q : QuadraticForm R M a : R x y : M \u22a2 polar (\u2191Q) x (a \u2022 y) = a * polar (\u2191Q) x y ** rw [polar_comm Q x, polar_comm Q x, polar_smul_left] ** Qed", + "informal": "" + }, + { + "formal": "FixedPoints.toAlgHom_bijective ** M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F \u22a2 Function.Bijective (MulSemiringAction.toAlgHom { x // x \u2208 subfield G F } F) ** cases nonempty_fintype G ** case intro M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Function.Bijective (MulSemiringAction.toAlgHom { x // x \u2208 subfield G F } F) ** rw [Fintype.bijective_iff_injective_and_card] ** case intro M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Function.Injective (MulSemiringAction.toAlgHom { x // x \u2208 subfield G F } F) \u2227 Fintype.card G = Fintype.card (F \u2192\u2090[{ x // x \u2208 subfield G F }] F) ** constructor ** case intro.left M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Function.Injective (MulSemiringAction.toAlgHom { x // x \u2208 subfield G F } F) ** exact MulSemiringAction.toAlgHom_injective _ F ** case intro.right M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Fintype.card G = Fintype.card (F \u2192\u2090[{ x // x \u2208 subfield G F }] F) ** apply le_antisymm ** case intro.right.a M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Fintype.card G \u2264 Fintype.card (F \u2192\u2090[{ x // x \u2208 subfield G F }] F) ** exact Fintype.card_le_of_injective _ (MulSemiringAction.toAlgHom_injective _ F) ** case intro.right.a M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Fintype.card (F \u2192\u2090[{ x // x \u2208 subfield G F }] F) \u2264 Fintype.card G ** rw [\u2190 finrank_eq_card G F] ** case intro.right.a M : Type u inst\u271d\u2079 : Monoid M G\u271d : Type u inst\u271d\u2078 : Group G\u271d F\u271d : Type v inst\u271d\u2077 : Field F\u271d inst\u271d\u2076 : MulSemiringAction M F\u271d inst\u271d\u2075 : MulSemiringAction G\u271d F\u271d m : M G : Type u F : Type v inst\u271d\u2074 : Group G inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Finite G inst\u271d\u00b9 : MulSemiringAction G F inst\u271d : FaithfulSMul G F val\u271d : Fintype G \u22a2 Fintype.card (F \u2192\u2090[{ x // x \u2208 subfield G F }] F) \u2264 finrank { x // x \u2208 subfield G F } F ** exact LE.le.trans_eq (finrank_algHom _ F) (finrank_linear_map' _ _ _) ** Qed", + "informal": "" + }, + { + "formal": "tangentMapWithin_id ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u2075 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M inst\u271d\u00b9\u00b3 : ChartedSpace H M inst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2079 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2078 : TopologicalSpace M' inst\u271d\u2077 : ChartedSpace H' M' inst\u271d\u2076 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b3 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace M'' inst\u271d\u00b9 : ChartedSpace H'' M'' inst\u271d : SmoothManifoldWithCorners I'' M'' s : Set M x : M p : TangentBundle I M hs : UniqueMDiffWithinAt I s p.proj \u22a2 tangentMapWithin I I id s p = p ** simp only [tangentMapWithin, id.def] ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u2075 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M inst\u271d\u00b9\u00b3 : ChartedSpace H M inst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2079 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2078 : TopologicalSpace M' inst\u271d\u2077 : ChartedSpace H' M' inst\u271d\u2076 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b3 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace M'' inst\u271d\u00b9 : ChartedSpace H'' M'' inst\u271d : SmoothManifoldWithCorners I'' M'' s : Set M x : M p : TangentBundle I M hs : UniqueMDiffWithinAt I s p.proj \u22a2 { proj := p.proj, snd := \u2191(mfderivWithin I I id s p.proj) p.snd } = p ** rw [mfderivWithin_id] ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u2075 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M inst\u271d\u00b9\u00b3 : ChartedSpace H M inst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2079 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2078 : TopologicalSpace M' inst\u271d\u2077 : ChartedSpace H' M' inst\u271d\u2076 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b3 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace M'' inst\u271d\u00b9 : ChartedSpace H'' M'' inst\u271d : SmoothManifoldWithCorners I'' M'' s : Set M x : M p : TangentBundle I M hs : UniqueMDiffWithinAt I s p.proj \u22a2 { proj := p.proj, snd := \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I p.proj)) p.snd } = p ** rcases p with \u27e8\u27e9 ** case mk \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u2075 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M inst\u271d\u00b9\u00b3 : ChartedSpace H M inst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2079 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2078 : TopologicalSpace M' inst\u271d\u2077 : ChartedSpace H' M' inst\u271d\u2076 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b3 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace M'' inst\u271d\u00b9 : ChartedSpace H'' M'' inst\u271d : SmoothManifoldWithCorners I'' M'' s : Set M x proj\u271d : M snd\u271d : TangentSpace I proj\u271d hs : UniqueMDiffWithinAt I s { proj := proj\u271d, snd := snd\u271d }.proj \u22a2 { proj := { proj := proj\u271d, snd := snd\u271d }.proj, snd := \u2191(ContinuousLinearMap.id \ud835\udd5c (TangentSpace I { proj := proj\u271d, snd := snd\u271d }.proj)) { proj := proj\u271d, snd := snd\u271d }.snd } = { proj := proj\u271d, snd := snd\u271d } ** rfl ** case hxs \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u2075 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M inst\u271d\u00b9\u00b3 : ChartedSpace H M inst\u271d\u00b9\u00b2 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2079 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2078 : TopologicalSpace M' inst\u271d\u2077 : ChartedSpace H' M' inst\u271d\u2076 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u2075 : NormedAddCommGroup E'' inst\u271d\u2074 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b3 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2 : TopologicalSpace M'' inst\u271d\u00b9 : ChartedSpace H'' M'' inst\u271d : SmoothManifoldWithCorners I'' M'' s : Set M x : M p : TangentBundle I M hs : UniqueMDiffWithinAt I s p.proj \u22a2 UniqueMDiffWithinAt I s p.proj ** exact hs ** Qed", + "informal": "" + }, + { + "formal": "Stream'.Seq.destruct_eq_nil ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 \u22a2 destruct s = none \u2192 s = nil ** dsimp [destruct] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 \u22a2 Option.map (fun a' => (a', tail s)) (get? s 0) = none \u2192 s = nil ** induction' f0 : get? s 0 <;> intro h ** case none \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 x\u271d : Option \u03b1 f0\u271d : get? s 0 = x\u271d f0 : get? s 0 = none h : Option.map (fun a' => (a', tail s)) none = none \u22a2 s = nil ** apply Subtype.eq ** case none.a \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 x\u271d : Option \u03b1 f0\u271d : get? s 0 = x\u271d f0 : get? s 0 = none h : Option.map (fun a' => (a', tail s)) none = none \u22a2 \u2191s = \u2191nil ** funext n ** case none.a.h \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 x\u271d : Option \u03b1 f0\u271d : get? s 0 = x\u271d f0 : get? s 0 = none h : Option.map (fun a' => (a', tail s)) none = none n : \u2115 \u22a2 \u2191s n = \u2191nil n ** induction' n with n IH ** case none.a.h.zero \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 x\u271d : Option \u03b1 f0\u271d : get? s 0 = x\u271d f0 : get? s 0 = none h : Option.map (fun a' => (a', tail s)) none = none \u22a2 \u2191s Nat.zero = \u2191nil Nat.zero case none.a.h.succ \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 x\u271d : Option \u03b1 f0\u271d : get? s 0 = x\u271d f0 : get? s 0 = none h : Option.map (fun a' => (a', tail s)) none = none n : \u2115 IH : \u2191s n = \u2191nil n \u22a2 \u2191s (Nat.succ n) = \u2191nil (Nat.succ n) ** exacts [f0, s.2 IH] ** case some \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w s : Seq \u03b1 x\u271d : Option \u03b1 f0\u271d : get? s 0 = x\u271d val\u271d : \u03b1 f0 : get? s 0 = some val\u271d h : Option.map (fun a' => (a', tail s)) (some val\u271d) = none \u22a2 s = nil ** contradiction ** Qed", + "informal": "" + }, + { + "formal": "affineSpan_pair_le_of_mem_of_mem ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P \u03b9 : Type u_4 p\u2081 p\u2082 : P s : AffineSubspace k P hp\u2081 : p\u2081 \u2208 s hp\u2082 : p\u2082 \u2208 s \u22a2 affineSpan k {p\u2081, p\u2082} \u2264 s ** rw [affineSpan_le, Set.insert_subset_iff, Set.singleton_subset_iff] ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P \u03b9 : Type u_4 p\u2081 p\u2082 : P s : AffineSubspace k P hp\u2081 : p\u2081 \u2208 s hp\u2082 : p\u2082 \u2208 s \u22a2 p\u2081 \u2208 \u2191s \u2227 p\u2082 \u2208 \u2191s ** exact \u27e8hp\u2081, hp\u2082\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Real.deriv_tan_sub_id ** x : \u211d h : cos x \u2260 0 \u22a2 HasDerivAt (fun y => tan y - y) (\u21911 / cos x ^ 2 - 1) x ** simpa using (hasDerivAt_tan h).add (hasDerivAt_id x).neg ** Qed", + "informal": "" + }, + { + "formal": "Pell.xy_modEq_yn ** a : \u2115 a1 : 1 < a n : \u2115 \u22a2 xn a1 (n * 0) \u2261 xn a1 n ^ 0 [MOD yn a1 n ^ 2] \u2227 yn a1 (n * 0) \u2261 0 * xn a1 n ^ (0 - 1) * yn a1 n [MOD yn a1 n ^ 3] ** constructor <;> simp <;> exact Nat.ModEq.refl _ ** a : \u2115 a1 : 1 < a n k : \u2115 \u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227 yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3] ** let \u27e8hx, hy\u27e9 := xy_modEq_yn n k ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] \u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227 yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3] ** have L : xn a1 (n * k) * xn a1 n + d a1 * yn a1 (n * k) * yn a1 n \u2261\n xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] :=\n (hx.mul_right _).add <|\n modEq_zero_iff_dvd.2 <| by\n rw [_root_.pow_succ']\n exact\n mul_dvd_mul_right\n (dvd_mul_of_dvd_right\n (modEq_zero_iff_dvd.1 <|\n (hy.of_dvd <| by simp [_root_.pow_succ']).trans <|\n modEq_zero_iff_dvd.2 <| by simp)\n _) _ ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] \u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227 yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3] ** have R : xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261\n xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] :=\n ModEq.add\n (by\n rw [_root_.pow_succ']\n exact hx.mul_right' _) <| by\n have : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n := by\n cases' k with k <;> simp [_root_.pow_succ']; ring_nf\n rw [\u2190 this]\n exact hy.mul_right _ ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] R : xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] \u22a2 xn a1 (n * (k + 1)) \u2261 xn a1 n ^ (k + 1) [MOD yn a1 n ^ 2] \u2227 yn a1 (n * (k + 1)) \u2261 (k + 1) * xn a1 n ^ (k + 1 - 1) * yn a1 n [MOD yn a1 n ^ 3] ** rw [add_tsub_cancel_right, Nat.mul_succ, xn_add, yn_add, pow_succ' (xn _ n), Nat.succ_mul,\n add_comm (k * xn _ n ^ k) (xn _ n ^ k), right_distrib] ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] R : xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] \u22a2 xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n [MOD yn a1 n ^ 2] \u2227 xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n \u2261 xn a1 n ^ k * yn a1 n + k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] ** exact \u27e8L, R\u27e9 ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] \u22a2 yn a1 n ^ 2 \u2223 Pell.d a1 * yn a1 (n * k) * yn a1 n ** rw [_root_.pow_succ'] ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] \u22a2 yn a1 n ^ 1 * yn a1 n \u2223 Pell.d a1 * yn a1 (n * k) * yn a1 n ** exact\n mul_dvd_mul_right\n (dvd_mul_of_dvd_right\n (modEq_zero_iff_dvd.1 <|\n (hy.of_dvd <| by simp [_root_.pow_succ']).trans <|\n modEq_zero_iff_dvd.2 <| by simp)\n _) _ ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] \u22a2 yn a1 n ^ 1 \u2223 yn a1 n ^ 3 ** simp [_root_.pow_succ'] ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] \u22a2 yn a1 n ^ 1 \u2223 k * xn a1 n ^ (k - 1) * yn a1 n ** simp ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] \u22a2 xn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] ** rw [_root_.pow_succ'] ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] \u22a2 xn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 2 * yn a1 n] ** exact hx.mul_right' _ ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] \u22a2 yn a1 (n * k) * xn a1 n \u2261 k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] ** have : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n := by\n cases' k with k <;> simp [_root_.pow_succ']; ring_nf ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] this : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n \u22a2 yn a1 (n * k) * xn a1 n \u2261 k * xn a1 n ^ k * yn a1 n [MOD yn a1 n ^ 3] ** rw [\u2190 this] ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] this : k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n \u22a2 yn a1 (n * k) * xn a1 n \u2261 k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n [MOD yn a1 n ^ 3] ** exact hy.mul_right _ ** a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * k) \u2261 xn a1 n ^ k [MOD yn a1 n ^ 2] hy : yn a1 (n * k) \u2261 k * xn a1 n ^ (k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * k) * xn a1 n + Pell.d a1 * yn a1 (n * k) * yn a1 n \u2261 xn a1 n ^ k * xn a1 n + 0 [MOD yn a1 n ^ 2] \u22a2 k * xn a1 n ^ (k - 1) * yn a1 n * xn a1 n = k * xn a1 n ^ k * yn a1 n ** cases' k with k <;> simp [_root_.pow_succ'] ** case succ a : \u2115 a1 : 1 < a n k : \u2115 hx : xn a1 (n * succ k) \u2261 xn a1 n ^ succ k [MOD yn a1 n ^ 2] hy : yn a1 (n * succ k) \u2261 succ k * xn a1 n ^ (succ k - 1) * yn a1 n [MOD yn a1 n ^ 3] L : xn a1 (n * succ k) * xn a1 n + Pell.d a1 * yn a1 (n * succ k) * yn a1 n \u2261 xn a1 n ^ succ k * xn a1 n + 0 [MOD yn a1 n ^ 2] \u22a2 succ k * xn a1 n ^ k * yn a1 n * xn a1 n = succ k * (xn a1 n ^ k * xn a1 n) * yn a1 n ** ring_nf ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Endofunctor.Algebra.iso_of_iso ** C : Type u inst\u271d\u00b9 : Category.{v, u} C F : C \u2964 C A A\u2080 A\u2081 A\u2082 : Algebra F f\u271d : A\u2080 \u27f6 A\u2081 g : A\u2081 \u27f6 A\u2082 f : A\u2080 \u27f6 A\u2081 inst\u271d : IsIso f.f \u22a2 F.map (inv f.f) \u226b A\u2080.str = A\u2081.str \u226b inv f.f ** rw [IsIso.eq_comp_inv f.1, Category.assoc, \u2190 f.h] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C F : C \u2964 C A A\u2080 A\u2081 A\u2082 : Algebra F f\u271d : A\u2080 \u27f6 A\u2081 g : A\u2081 \u27f6 A\u2082 f : A\u2080 \u27f6 A\u2081 inst\u271d : IsIso f.f \u22a2 F.map (inv f.f) \u226b F.map f.f \u226b A\u2081.str = A\u2081.str ** simp ** C : Type u inst\u271d\u00b9 : Category.{v, u} C F : C \u2964 C A A\u2080 A\u2081 A\u2082 : Algebra F f\u271d : A\u2080 \u27f6 A\u2081 g : A\u2081 \u27f6 A\u2082 f : A\u2080 \u27f6 A\u2081 inst\u271d : IsIso f.f \u22a2 f \u226b Hom.mk (inv f.f) = \ud835\udfd9 A\u2080 ** aesop_cat ** C : Type u inst\u271d\u00b9 : Category.{v, u} C F : C \u2964 C A A\u2080 A\u2081 A\u2082 : Algebra F f\u271d : A\u2080 \u27f6 A\u2081 g : A\u2081 \u27f6 A\u2082 f : A\u2080 \u27f6 A\u2081 inst\u271d : IsIso f.f \u22a2 Hom.mk (inv f.f) \u226b f = \ud835\udfd9 A\u2081 ** aesop_cat ** Qed", + "informal": "" + }, + { + "formal": "integrableOn_Iic_iff_integrableOn_Iio' ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 hb : \u2191\u2191\u03bc {b} \u2260 \u22a4 \u22a2 IntegrableOn f (Iic b) \u2194 IntegrableOn f (Iio b) ** rw [\u2190 Iio_union_right, integrableOn_union,\n eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.nodup_powerset ** \u03b1 : Type u_1 s : Multiset \u03b1 l : List \u03b1 h : Nodup (Quotient.mk (isSetoid \u03b1) l) \u22a2 Nodup (powerset (Quotient.mk (isSetoid \u03b1) l)) ** simp only [quot_mk_to_coe, powerset_coe', coe_nodup] ** \u03b1 : Type u_1 s : Multiset \u03b1 l : List \u03b1 h : Nodup (Quotient.mk (isSetoid \u03b1) l) \u22a2 List.Nodup (List.map ofList (sublists' l)) ** refine' (nodup_sublists'.2 h).map_on _ ** \u03b1 : Type u_1 s : Multiset \u03b1 l : List \u03b1 h : Nodup (Quotient.mk (isSetoid \u03b1) l) \u22a2 \u2200 (x : List \u03b1), x \u2208 sublists' l \u2192 \u2200 (y : List \u03b1), y \u2208 sublists' l \u2192 \u2191x = \u2191y \u2192 x = y ** exact fun x sx y sy e =>\n (h.sublist_ext (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1 (Quotient.exact e) ** Qed", + "informal": "" + }, + { + "formal": "Array.map_data ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 \u22a2 (map f arr).data = List.map f arr.data ** rw [map, mapM_eq_foldlM] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 \u22a2 (Id.run (foldlM (fun bs a => push bs <$> f a) #[] arr 0 (size arr))).data = List.map f arr.data ** apply congrArg data (foldl_eq_foldl_data (fun bs a => push bs (f a)) #[] arr) |>.trans ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 \u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data ** have H (l arr) : List.foldl (fun bs a => push bs (f a)) arr l = \u27e8arr.data ++ l.map f\u27e9 := by\n induction l generalizing arr <;> simp [*] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr : Array \u03b1 H : \u2200 (l : List \u03b1) (arr : Array \u03b2), List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l } \u22a2 (List.foldl (fun bs a => push bs (f a)) #[] arr.data).data = List.map f arr.data ** simp [H] ** \u03b1 : Type u_1 \u03b2 : Type u_2 f : \u03b1 \u2192 \u03b2 arr\u271d : Array \u03b1 l : List \u03b1 arr : Array \u03b2 \u22a2 List.foldl (fun bs a => push bs (f a)) arr l = { data := arr.data ++ List.map f l } ** induction l generalizing arr <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.nonsing_inv_nonsing_inv ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A B : Matrix n n \u03b1 h : IsUnit (det A) \u22a2 A\u207b\u00b9\u207b\u00b9 = 1 * A\u207b\u00b9\u207b\u00b9 ** rw [Matrix.one_mul] ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A B : Matrix n n \u03b1 h : IsUnit (det A) \u22a2 1 * A\u207b\u00b9\u207b\u00b9 = A * A\u207b\u00b9 * A\u207b\u00b9\u207b\u00b9 ** rw [A.mul_nonsing_inv h] ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A B : Matrix n n \u03b1 h : IsUnit (det A) \u22a2 A * A\u207b\u00b9 * A\u207b\u00b9\u207b\u00b9 = A ** rw [Matrix.mul_assoc, A\u207b\u00b9.mul_nonsing_inv (A.isUnit_nonsing_inv_det h), Matrix.mul_one] ** Qed", + "informal": "" + }, + { + "formal": "Nat.floor_div_eq_div ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedSemifield \u03b1 inst\u271d : FloorSemiring \u03b1 m n : \u2115 \u22a2 \u230a\u2191m / \u2191n\u230b\u208a = m / n ** convert floor_div_nat (m : \u03b1) n ** case h.e'_3.h.e'_5 F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : LinearOrderedSemifield \u03b1 inst\u271d : FloorSemiring \u03b1 m n : \u2115 \u22a2 m = \u230a\u2191m\u230b\u208a ** rw [m.floor_coe] ** Qed", + "informal": "" + }, + { + "formal": "neg_div_neg_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 K : Type u_3 inst\u271d\u00b9 : DivisionMonoid K inst\u271d : HasDistribNeg K a\u271d b\u271d a b : K \u22a2 -a / -b = a / b ** rw [div_neg_eq_neg_div, neg_div, neg_neg] ** Qed", + "informal": "" + }, + { + "formal": "UniqueFactorizationMonoid.multiplicative_prime_power ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p is_coprime : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y \u22a2 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) ** letI := Classical.decEq \u03b1 ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p is_coprime : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 \u22a2 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) ** induction' s using Finset.induction_on with p s hps ih ** case insert \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s\u271d : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p is_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 p : \u03b1 s : Finset \u03b1 hps : \u00acp \u2208 s ih : (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) is_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1 is_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q \u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p) ** have hpr_p := is_prime _ (Finset.mem_insert_self _ _) ** case insert \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s\u271d : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p is_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 p : \u03b1 s : Finset \u03b1 hps : \u00acp \u2208 s ih : (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) is_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1 is_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q hpr_p : Prime p \u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p) ** have hpr_s : \u2200 p \u2208 s, Prime p := fun p hp => is_prime _ (Finset.mem_insert_of_mem hp) ** case insert \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s\u271d : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p is_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 p : \u03b1 s : Finset \u03b1 hps : \u00acp \u2208 s ih : (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) is_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1 is_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q hpr_p : Prime p hpr_s : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p \u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p) ** have hcp_p := fun i => prime_pow_coprime_prod_of_coprime_insert i p hps is_prime is_coprime ** case insert \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s\u271d : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p is_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 p : \u03b1 s : Finset \u03b1 hps : \u00acp \u2208 s ih : (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) is_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1 is_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q hpr_p : Prime p hpr_s : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p hcp_p : \u2200 (i : \u03b1 \u2192 \u2115) (q : \u03b1), q \u2223 p ^ i p \u2192 q \u2223 \u220f p' in s, p' ^ i p' \u2192 IsUnit q \u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p) ** have hcp_s : \u2200 (p) (_ : p \u2208 s) (q) (_ : q \u2208 s), p \u2223 q \u2192 p = q := fun p hp q hq =>\n is_coprime p (Finset.mem_insert_of_mem hp) q (Finset.mem_insert_of_mem hq) ** case insert \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s\u271d : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 Prime p is_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s\u271d \u2192 \u2200 (q : \u03b1), q \u2208 s\u271d \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 p : \u03b1 s : Finset \u03b1 hps : \u00acp \u2208 s ih : (\u2200 (p : \u03b1), p \u2208 s \u2192 Prime p) \u2192 (\u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q) \u2192 f (\u220f p in s, p ^ (i p + j p)) = f (\u220f p in s, p ^ i p) * f (\u220f p in s, p ^ j p) is_prime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 Prime p_1 is_coprime : \u2200 (p_1 : \u03b1), p_1 \u2208 insert p s \u2192 \u2200 (q : \u03b1), q \u2208 insert p s \u2192 p_1 \u2223 q \u2192 p_1 = q hpr_p : Prime p hpr_s : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p hcp_p : \u2200 (i : \u03b1 \u2192 \u2115) (q : \u03b1), q \u2223 p ^ i p \u2192 q \u2223 \u220f p' in s, p' ^ i p' \u2192 IsUnit q hcp_s : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q \u22a2 f (\u220f p in insert p s, p ^ (i p + j p)) = f (\u220f p in insert p s, p ^ i p) * f (\u220f p in insert p s, p ^ j p) ** rw [Finset.prod_insert hps, Finset.prod_insert hps, Finset.prod_insert hps, hcp (hcp_p _),\n hpr _ hpr_p, hcp (hcp_p _), hpr _ hpr_p, hcp (hcp_p (fun p => i p + j p)), hpr _ hpr_p,\n ih hpr_s hcp_s, pow_add, mul_assoc, mul_left_comm (f p ^ j p), mul_assoc] ** case empty \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : CancelCommMonoidWithZero \u03b1 inst\u271d\u00b9 : UniqueFactorizationMonoid \u03b1 \u03b2 : Type u_3 inst\u271d : CancelCommMonoidWithZero \u03b2 f : \u03b1 \u2192 \u03b2 s : Finset \u03b1 i j : \u03b1 \u2192 \u2115 is_prime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 Prime p is_coprime\u271d : \u2200 (p : \u03b1), p \u2208 s \u2192 \u2200 (q : \u03b1), q \u2208 s \u2192 p \u2223 q \u2192 p = q h1 : \u2200 {x y : \u03b1}, IsUnit y \u2192 f (x * y) = f x * f y hpr : \u2200 {p : \u03b1} (i : \u2115), Prime p \u2192 f (p ^ i) = f p ^ i hcp : \u2200 {x y : \u03b1}, (\u2200 (p : \u03b1), p \u2223 x \u2192 p \u2223 y \u2192 IsUnit p) \u2192 f (x * y) = f x * f y this : DecidableEq \u03b1 := Classical.decEq \u03b1 is_prime : \u2200 (p : \u03b1), p \u2208 \u2205 \u2192 Prime p is_coprime : \u2200 (p : \u03b1), p \u2208 \u2205 \u2192 \u2200 (q : \u03b1), q \u2208 \u2205 \u2192 p \u2223 q \u2192 p = q \u22a2 f (\u220f p in \u2205, p ^ (i p + j p)) = f (\u220f p in \u2205, p ^ i p) * f (\u220f p in \u2205, p ^ j p) ** simpa using h1 isUnit_one ** Qed", + "informal": "" + }, + { + "formal": "WithTop.mul_top ** \u03b1 : Type u_1 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : Zero \u03b1 inst\u271d : Mul \u03b1 a : WithTop \u03b1 h : a \u2260 0 \u22a2 a * \u22a4 = \u22a4 ** rw [mul_top', if_neg h] ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.exists_fundamental_sequence ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u22a2 \u2203 f, IsFundamentalSequence a (ord (cof a)) f ** suffices h : \u2203 o f, IsFundamentalSequence a o f ** case h \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u22a2 \u2203 o f, IsFundamentalSequence a o f ** rcases exists_lsub_cof a with \u27e8\u03b9, f, hf, h\u03b9\u27e9 ** case h.intro.intro.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a \u22a2 \u2203 o f, IsFundamentalSequence a o f ** rcases ord_eq \u03b9 with \u27e8r, wo, hr\u27e9 ** case h.intro.intro.intro.intro.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r \u22a2 \u2203 o f, IsFundamentalSequence a o f ** haveI := wo ** case h.intro.intro.intro.intro.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this : IsWellOrder \u03b9 r \u22a2 \u2203 o f, IsFundamentalSequence a o f ** let r' := Subrel r { i | \u2200 j, r j i \u2192 f j < f i } ** case h.intro.intro.intro.intro.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u22a2 \u2203 o f, IsFundamentalSequence a o f ** let hrr' : r' \u21aar r := Subrel.relEmbedding _ _ ** case h.intro.intro.intro.intro.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u22a2 \u2203 o f, IsFundamentalSequence a o f ** haveI := hrr'.isWellOrder ** case h.intro.intro.intro.intro.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' \u22a2 \u2203 o f, IsFundamentalSequence a o f ** refine'\n \u27e8_, _, hrr'.ordinal_type_le.trans _, @fun i j _ h _ => (enum r' j h).prop _ _,\n le_antisymm (blsub_le fun i hi => lsub_le_iff.1 hf.le _) _\u27e9 ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} h : \u2203 o f, IsFundamentalSequence a o f \u22a2 \u2203 f, IsFundamentalSequence a (ord (cof a)) f ** rcases h with \u27e8o, f, hf\u27e9 ** case intro.intro \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop a o : Ordinal.{u} f : (b : Ordinal.{u}) \u2192 b < o \u2192 Ordinal.{u} hf : IsFundamentalSequence a o f \u22a2 \u2203 f, IsFundamentalSequence a (ord (cof a)) f ** exact \u27e8_, hf.ord_cof\u27e9 ** case h.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' \u22a2 type r \u2264 ord (cof a) ** rw [\u2190 h\u03b9, hr] ** case h.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i j : Ordinal.{u} x\u271d\u00b9 : i < type r' h : j < type r' x\u271d : i < j \u22a2 r \u2191(enum r' i x\u271d\u00b9) \u2191(enum r' j h) ** change r (hrr'.1 _) (hrr'.1 _) ** case h.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i j : Ordinal.{u} x\u271d\u00b9 : i < type r' h : j < type r' x\u271d : i < j \u22a2 r (\u2191hrr'.toEmbedding (enum r' i x\u271d\u00b9)) (\u2191hrr'.toEmbedding (enum r' j h)) ** rwa [hrr'.2, @enum_lt_enum _ r'] ** case h.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' \u22a2 a \u2264 blsub (type r') fun j h => f \u2191(enum r' j h) ** rw [\u2190 hf, lsub_le_iff] ** case h.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' \u22a2 \u2200 (i : \u03b9), f i < blsub (type r') fun j h => f \u2191(enum r' j h) ** intro i ** case h.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 \u22a2 f i < blsub (type r') fun j h => f \u2191(enum r' j h) ** suffices h : \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f i) i' hi' ** case h \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 \u22a2 \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' ** by_cases h : \u2200 j, r j i \u2192 f j < f i ** case h.intro.intro.intro.intro.intro.refine'_3 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' \u22a2 f i < blsub (type r') fun j h => f \u2191(enum r' j h) ** rcases h with \u27e8i', hi', hfg\u27e9 ** case h.intro.intro.intro.intro.intro.refine'_3.intro.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 i' : Ordinal.{u} hi' : i' < type r' hfg : f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' \u22a2 f i < blsub (type r') fun j h => f \u2191(enum r' j h) ** exact hfg.trans_lt (lt_blsub _ _ _) ** case pos \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2200 (j : \u03b9), r j i \u2192 f j < f i \u22a2 \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' ** refine' \u27e8typein r' \u27e8i, h\u27e9, typein_lt_type _ _, _\u27e9 ** case pos \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2200 (j : \u03b9), r j i \u2192 f j < f i \u22a2 f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) (typein r' { val := i, property := h }) (_ : typein r' { val := i, property := h } < type r') ** rw [bfamilyOfFamily'_typein] ** case neg \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u00ac\u2200 (j : \u03b9), r j i \u2192 f j < f i \u22a2 \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' ** push_neg at h ** case neg \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2203 j, r j i \u2227 f i \u2264 f j \u22a2 \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' ** cases' wo.wf.min_mem _ h with hji hij ** case neg.intro \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2203 j, r j i \u2227 f i \u2264 f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) i hij : f i \u2264 f (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) \u22a2 \u2203 i' hi', f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) i' hi' ** refine' \u27e8typein r' \u27e8_, fun k hkj => lt_of_lt_of_le _ hij\u27e9, typein_lt_type _ _, _\u27e9 ** case neg.intro.refine'_1 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2203 j, r j i \u2227 f i \u2264 f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) i hij : f i \u2264 f (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) k : \u03b9 hkj : r k (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) \u22a2 f k < f i ** by_contra' H ** case neg.intro.refine'_1 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2203 j, r j i \u2227 f i \u2264 f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) i hij : f i \u2264 f (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) k : \u03b9 hkj : r k (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) H : f i \u2264 f k \u22a2 False ** exact (wo.wf.not_lt_min _ h \u27e8IsTrans.trans _ _ _ hkj hji, H\u27e9) hkj ** case neg.intro.refine'_2 \u03b1 : Type u_1 r\u271d : \u03b1 \u2192 \u03b1 \u2192 Prop a : Ordinal.{u} \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{u} hf : lsub f = a h\u03b9 : #\u03b9 = cof a r : \u03b9 \u2192 \u03b9 \u2192 Prop wo : IsWellOrder \u03b9 r hr : ord #\u03b9 = type r this\u271d : IsWellOrder \u03b9 r r' : \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 \u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} \u2192 Prop := Subrel r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} hrr' : r' \u21aar r := Subrel.relEmbedding r {i | \u2200 (j : \u03b9), r j i \u2192 f j < f i} this : IsWellOrder (\u2191{i | \u2200 (j : \u03b9), r j i \u2192 f j < f i}) r' i : \u03b9 h : \u2203 j, r j i \u2227 f i \u2264 f j hji : r (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) i hij : f i \u2264 f (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) \u22a2 f i \u2264 bfamilyOfFamily' r' (fun i => f \u2191i) (typein r' { val := WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h, property := (_ : \u2200 (k : \u03b9), r k (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) \u2192 f k < f (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h)) }) (_ : typein r' { val := WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h, property := (_ : \u2200 (k : \u03b9), r k (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h) \u2192 f k < f (WellFounded.min (_ : WellFounded r) (fun x => r x i \u2227 f i \u2264 f x) h)) } < type r') ** rwa [bfamilyOfFamily'_typein] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.mem_tail_support_append_iff ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' t u v w : V p : Walk G u v p' : Walk G v w \u22a2 t \u2208 List.tail (support (append p p')) \u2194 t \u2208 List.tail (support p) \u2228 t \u2208 List.tail (support p') ** rw [tail_support_append, List.mem_append] ** Qed", + "informal": "" + }, + { + "formal": "Basis.coe_mkFinCons ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 v : \u03b9 \u2192 M inst\u271d\u2078 : Ring R inst\u271d\u2077 : CommRing R\u2082 inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : AddCommGroup M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R\u2082 M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' c d : R x y\u271d : M b\u271d : Basis \u03b9 R M n : \u2115 N : Submodule R M y : M b : Basis (Fin n) R { x // x \u2208 N } hli : \u2200 (c : R) (x : M), x \u2208 N \u2192 c \u2022 y + x = 0 \u2192 c = 0 hsp : \u2200 (z : M), \u2203 c, z + c \u2022 y \u2208 N \u22a2 \u2191(mkFinCons y b hli hsp) = Fin.cons y (Subtype.val \u2218 \u2191b) ** unfold mkFinCons ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 v : \u03b9 \u2192 M inst\u271d\u2078 : Ring R inst\u271d\u2077 : CommRing R\u2082 inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : AddCommGroup M' inst\u271d\u2074 : AddCommGroup M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R\u2082 M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' c d : R x y\u271d : M b\u271d : Basis \u03b9 R M n : \u2115 N : Submodule R M y : M b : Basis (Fin n) R { x // x \u2208 N } hli : \u2200 (c : R) (x : M), x \u2208 N \u2192 c \u2022 y + x = 0 \u2192 c = 0 hsp : \u2200 (z : M), \u2203 c, z + c \u2022 y \u2208 N \u22a2 \u2191(let_fun span_b := (_ : span R (Set.range (\u2191(Submodule.subtype N) \u2218 \u2191b)) = N); Basis.mk (_ : LinearIndependent R (Fin.cons y (\u2191(Submodule.subtype N) \u2218 \u2191b))) (_ : \u2200 (x : M), x \u2208 \u22a4 \u2192 x \u2208 span R (Set.range (Fin.cons y (\u2191(Submodule.subtype N) \u2218 \u2191b))))) = Fin.cons y (Subtype.val \u2218 \u2191b) ** exact coe_mk (v := Fin.cons y (N.subtype \u2218 b)) _ _ ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.radical_le_vanishingIdeal_zeroLocus ** k : Type u_1 inst\u271d : Field k \u03c3 : Type u_2 I : Ideal (MvPolynomial \u03c3 k) \u22a2 radical I \u2264 vanishingIdeal (zeroLocus I) ** intro p hp x hx ** k : Type u_1 inst\u271d : Field k \u03c3 : Type u_2 I : Ideal (MvPolynomial \u03c3 k) p : MvPolynomial \u03c3 k hp : p \u2208 radical I x : \u03c3 \u2192 k hx : x \u2208 zeroLocus I \u22a2 \u2191(eval x) p = 0 ** rw [\u2190 mem_vanishingIdeal_singleton_iff] ** k : Type u_1 inst\u271d : Field k \u03c3 : Type u_2 I : Ideal (MvPolynomial \u03c3 k) p : MvPolynomial \u03c3 k hp : p \u2208 radical I x : \u03c3 \u2192 k hx : x \u2208 zeroLocus I \u22a2 p \u2208 vanishingIdeal {x} ** rw [radical_eq_sInf] at hp ** k : Type u_1 inst\u271d : Field k \u03c3 : Type u_2 I : Ideal (MvPolynomial \u03c3 k) p : MvPolynomial \u03c3 k hp : p \u2208 sInf {J | I \u2264 J \u2227 IsPrime J} x : \u03c3 \u2192 k hx : x \u2208 zeroLocus I \u22a2 p \u2208 vanishingIdeal {x} ** refine'\n (mem_sInf.mp hp)\n \u27e8le_trans (le_vanishingIdeal_zeroLocus I)\n (vanishingIdeal_anti_mono fun y hy => hy.symm \u25b8 hx),\n IsMaximal.isPrime' _\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "LinearIndependent.mono ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M t s : Set M h : t \u2286 s \u22a2 (LinearIndependent R fun x => \u2191x) \u2192 LinearIndependent R fun x => \u2191x ** simp only [linearIndependent_subtype_disjoint] ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M t s : Set M h : t \u2286 s \u22a2 Disjoint (Finsupp.supported R R s) (LinearMap.ker (Finsupp.total M M R id)) \u2192 Disjoint (Finsupp.supported R R t) (LinearMap.ker (Finsupp.total M M R id)) ** exact Disjoint.mono_left (Finsupp.supported_mono h) ** Qed", + "informal": "" + }, + { + "formal": "Function.Periodic.nat_mul_sub_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f g : \u03b1 \u2192 \u03b2 c c\u2081 c\u2082 x : \u03b1 inst\u271d : Ring \u03b1 h : Periodic f c n : \u2115 \u22a2 f (\u2191n * c - x) = f (-x) ** simpa only [sub_eq_neg_add] using h.nat_mul n (-x) ** Qed", + "informal": "" + }, + { + "formal": "PrimeSpectrum.basicOpen_zero ** R : Type u S : Type v inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S \u22a2 \u2191(basicOpen 0) = \u2191\u22a5 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.bitCasesOn_bit ** n\u271d : \u2115 C : \u2115 \u2192 Sort u H : (b : Bool) \u2192 (n : \u2115) \u2192 C (bit b n) b : Bool n : \u2115 \u22a2 HEq (H (bodd (bit b n)) (div2 (bit b n))) (H b n) ** rw [bodd_bit, div2_bit] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_le_of_forall_fin_meas_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m m0 : MeasurableSpace \u03b1 E : Type u_5 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : MeasurableSpace E inst\u271d\u00b2 : OpensMeasurableSpace E inst\u271d\u00b9 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : SigmaFinite \u03bc C : \u211d\u22650\u221e f : \u03b1 \u2192 \u211d\u22650\u221e hf_meas : AEMeasurable f hf : \u2200 (s : Set \u03b1), MeasurableSet s \u2192 \u2191\u2191\u03bc s \u2260 \u22a4 \u2192 \u222b\u207b (x : \u03b1) in s, f x \u2202\u03bc \u2264 C \u22a2 SigmaFinite (Measure.trim \u03bc (_ : inst\u271d\u00b9 \u2264 inst\u271d\u00b9)) ** rwa [trim_eq_self] ** Qed", + "informal": "" + }, + { + "formal": "Set.iInter_eq_compl_iUnion_compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 s : \u03b9 \u2192 Set \u03b2 \u22a2 \u22c2 i, s i = (\u22c3 i, (s i)\u1d9c)\u1d9c ** simp only [compl_iUnion, compl_compl] ** Qed", + "informal": "" + }, + { + "formal": "Nat.bit0_div_two ** m n : \u2115 \u22a2 bit0 n / 2 = n ** rw [\u2190 Nat.bit0_eq_bit0, bit0_eq_two_mul, two_mul_div_two_of_even (even_bit0 n)] ** Qed", + "informal": "" + }, + { + "formal": "Nat.Primrec.casesOn' ** f g : \u2115 \u2192 \u2115 hf : Nat.Primrec f hg : Nat.Primrec g n : \u2115 \u22a2 unpaired (fun z n => Nat.rec (f z) (fun y IH => g (Nat.pair (unpair (Nat.pair z (Nat.pair y IH))).1 (unpair (unpair (Nat.pair z (Nat.pair y IH))).2).1)) n) n = unpaired (fun z n => Nat.casesOn n (f z) fun y => g (Nat.pair z y)) n ** simp ** Qed", + "informal": "" + }, + { + "formal": "IsPiSystem.insert_empty ** \u03b1 : Type u_1 S : Set (Set \u03b1) h_pi : IsPiSystem S \u22a2 IsPiSystem (insert \u2205 S) ** intro s hs t ht hst ** \u03b1 : Type u_1 S : Set (Set \u03b1) h_pi : IsPiSystem S s : Set \u03b1 hs : s \u2208 insert \u2205 S t : Set \u03b1 ht : t \u2208 insert \u2205 S hst : Set.Nonempty (s \u2229 t) \u22a2 s \u2229 t \u2208 insert \u2205 S ** cases' hs with hs hs ** case inl \u03b1 : Type u_1 S : Set (Set \u03b1) h_pi : IsPiSystem S s t : Set \u03b1 ht : t \u2208 insert \u2205 S hst : Set.Nonempty (s \u2229 t) hs : s = \u2205 \u22a2 s \u2229 t \u2208 insert \u2205 S ** simp [hs] ** case inr \u03b1 : Type u_1 S : Set (Set \u03b1) h_pi : IsPiSystem S s t : Set \u03b1 ht : t \u2208 insert \u2205 S hst : Set.Nonempty (s \u2229 t) hs : s \u2208 S \u22a2 s \u2229 t \u2208 insert \u2205 S ** cases' ht with ht ht ** case inr.inl \u03b1 : Type u_1 S : Set (Set \u03b1) h_pi : IsPiSystem S s t : Set \u03b1 hst : Set.Nonempty (s \u2229 t) hs : s \u2208 S ht : t = \u2205 \u22a2 s \u2229 t \u2208 insert \u2205 S ** simp [ht] ** case inr.inr \u03b1 : Type u_1 S : Set (Set \u03b1) h_pi : IsPiSystem S s t : Set \u03b1 hst : Set.Nonempty (s \u2229 t) hs : s \u2208 S ht : t \u2208 S \u22a2 s \u2229 t \u2208 insert \u2205 S ** exact Set.mem_insert_of_mem _ (h_pi s hs t ht hst) ** Qed", + "informal": "" + }, + { + "formal": "TopCat.Presheaf.pushforward_eq' ** C : Type u inst\u271d : Category.{v, u} C X Y : TopCat f g : X \u27f6 Y h : f = g \u2131 : Presheaf C X \u22a2 f _* \u2131 = g _* \u2131 ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMul.of_nhds_one ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) \u22a2 Continuous fun p => p.1 * p.2 ** rw [continuous_iff_continuousAt] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) \u22a2 \u2200 (x : M \u00d7 M), ContinuousAt (fun p => p.1 * p.2) x ** rintro \u27e8x\u2080, y\u2080\u27e9 ** case mk \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 \u22a2 ContinuousAt (fun p => p.1 * p.2) (x\u2080, y\u2080) ** have key\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x := by\n ext x\n simp [mul_assoc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M \u22a2 (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 ** ext p ** case h \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M p : M \u00d7 M \u22a2 x\u2080 * p.1 * (p.2 * y\u2080) = (((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1) p ** simp [uncurry, mul_assoc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 \u22a2 ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x ** ext x ** case h \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 x : M \u22a2 ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) x = x\u2080 * y\u2080 * x ** simp [mul_assoc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 key\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x \u22a2 map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd (x\u2080, y\u2080)) = map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080) ** rw [nhds_prod_eq] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 key\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x \u22a2 map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd x\u2080 \u00d7\u02e2 \ud835\udcdd y\u2080) = map (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) ** simp_rw [uncurry, hleft x\u2080, hright y\u2080, prod_map_map_eq, Filter.map_map, Function.comp] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 key\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x \u22a2 map (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) = map ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) (map (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1)) ** rw [key, \u2190 Filter.map_map] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M\u271d : Type u_4 N : Type u_5 inst\u271d\u2075 : TopologicalSpace X inst\u271d\u2074 : TopologicalSpace M\u271d inst\u271d\u00b3 : Mul M\u271d inst\u271d\u00b2 : ContinuousMul M\u271d M : Type u inst\u271d\u00b9 : Monoid M inst\u271d : TopologicalSpace M hmul : Tendsto (uncurry fun x x_1 => x * x_1) (\ud835\udcdd 1 \u00d7\u02e2 \ud835\udcdd 1) (\ud835\udcdd 1) hleft : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x\u2080 * x) (\ud835\udcdd 1) hright : \u2200 (x\u2080 : M), \ud835\udcdd x\u2080 = map (fun x => x * x\u2080) (\ud835\udcdd 1) x\u2080 y\u2080 : M key : (fun p => x\u2080 * p.1 * (p.2 * y\u2080)) = ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) \u2218 uncurry fun x x_1 => x * x_1 key\u2082 : ((fun x => x\u2080 * x) \u2218 fun x => y\u2080 * x) = fun x => x\u2080 * y\u2080 * x \u22a2 map ((fun x => x\u2080 * x) \u2218 fun x => x * y\u2080) (\ud835\udcdd 1) = \ud835\udcdd (x\u2080 * y\u2080) ** rw [\u2190 Filter.map_map, \u2190 hright, hleft y\u2080, Filter.map_map, key\u2082, \u2190 hleft] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Bicategory.inv_hom_whiskerLeft ** B : Type u inst\u271d : Bicategory B a b c d e : B f : a \u27f6 b g h : b \u27f6 c \u03b7 : g \u2245 h \u22a2 f \u25c1 \u03b7.inv \u226b f \u25c1 \u03b7.hom = \ud835\udfd9 (f \u226b h) ** rw [\u2190 whiskerLeft_comp, inv_hom_id, whiskerLeft_id] ** Qed", + "informal": "" + }, + { + "formal": "pow_dvd_pow_of_dvd ** \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 a\u271d b\u271d a b : \u03b1 h : a \u2223 b \u22a2 a ^ 0 \u2223 b ^ 0 ** rw [pow_zero, pow_zero] ** \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 a\u271d b\u271d a b : \u03b1 h : a \u2223 b n : \u2115 \u22a2 a ^ (n + 1) \u2223 b ^ (n + 1) ** rw [pow_succ, pow_succ] ** \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 a\u271d b\u271d a b : \u03b1 h : a \u2223 b n : \u2115 \u22a2 a * a ^ n \u2223 b * b ^ n ** exact mul_dvd_mul h (pow_dvd_pow_of_dvd h n) ** Qed", + "informal": "" + }, + { + "formal": "multiplicity_eq_multiplicity_span ** R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R \u22a2 multiplicity (span {a}) (span {b}) = multiplicity a b ** by_cases h : Finite a b ** case pos R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : multiplicity.Finite a b \u22a2 multiplicity (span {a}) (span {b}) = multiplicity a b ** rw [\u2190 PartENat.natCast_get (finite_iff_dom.mp h)] ** case pos R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : multiplicity.Finite a b \u22a2 multiplicity (span {a}) (span {b}) = \u2191(Part.get (multiplicity a b) (_ : (multiplicity a b).Dom)) ** refine (multiplicity.unique\n (show Ideal.span {a} ^ (multiplicity a b).get h \u2223 Ideal.span {b} from ?_) ?_).symm <;>\n rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd] ** case pos.refine_1 R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : multiplicity.Finite a b \u22a2 a ^ Part.get (multiplicity a b) h \u2223 b case pos.refine_2 R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : multiplicity.Finite a b \u22a2 \u00aca ^ (Part.get (multiplicity a b) h + 1) \u2223 b ** exact pow_multiplicity_dvd h ** case pos.refine_2 R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : multiplicity.Finite a b \u22a2 \u00aca ^ (Part.get (multiplicity a b) h + 1) \u2223 b ** exact multiplicity.is_greatest\n ((PartENat.lt_coe_iff _ _).mpr (Exists.intro (finite_iff_dom.mp h) (Nat.lt_succ_self _))) ** case neg R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : \u00acmultiplicity.Finite a b \u22a2 multiplicity (span {a}) (span {b}) = multiplicity a b ** suffices \u00acFinite (Ideal.span ({a} : Set R)) (Ideal.span ({b} : Set R)) by\n rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this\n rw [h, this] ** case neg R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : \u00acmultiplicity.Finite a b \u22a2 \u00acmultiplicity.Finite (span {a}) (span {b}) ** exact not_finite_iff_forall.mpr fun n => by\n rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd]\n exact not_finite_iff_forall.mp h n ** R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : \u00acmultiplicity.Finite a b this : \u00acmultiplicity.Finite (span {a}) (span {b}) \u22a2 multiplicity (span {a}) (span {b}) = multiplicity a b ** rw [finite_iff_dom, PartENat.not_dom_iff_eq_top] at h this ** R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h\u271d : \u00acmultiplicity.Finite a b h : multiplicity a b = \u22a4 this\u271d : \u00acmultiplicity.Finite (span {a}) (span {b}) this : multiplicity (span {a}) (span {b}) = \u22a4 \u22a2 multiplicity (span {a}) (span {b}) = multiplicity a b ** rw [h, this] ** R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : \u00acmultiplicity.Finite a b n : \u2115 \u22a2 span {a} ^ n \u2223 span {b} ** rw [Ideal.span_singleton_pow, span_singleton_dvd_span_singleton_iff_dvd] ** R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u2077 : CommRing R inst\u271d\u2076 : CommRing A inst\u271d\u2075 : Field K inst\u271d\u2074 : IsDomain A inst\u271d\u00b3 : IsDomain R inst\u271d\u00b2 : IsPrincipalIdealRing R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : R h : \u00acmultiplicity.Finite a b n : \u2115 \u22a2 a ^ n \u2223 b ** exact not_finite_iff_forall.mp h n ** Qed", + "informal": "" + }, + { + "formal": "List.splitOn_intercalate ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls : ls \u2260 [] \u22a2 splitOn x (intercalate [x] ls) = ls ** simp only [intercalate] ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls : ls \u2260 [] \u22a2 splitOn x (join (intersperse [x] ls)) = ls ** induction' ls with hd tl ih ** case cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 tl : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 tl \u2192 \u00acx \u2208 l) \u2192 tl \u2260 [] \u2192 splitOn x (join (intersperse [x] tl)) = tl hx : \u2200 (l : List \u03b1), l \u2208 hd :: tl \u2192 \u00acx \u2208 l hls : hd :: tl \u2260 [] \u22a2 splitOn x (join (intersperse [x] (hd :: tl))) = hd :: tl ** cases tl ** case nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hx : \u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l hls : [] \u2260 [] \u22a2 splitOn x (join (intersperse [x] [])) = [] ** contradiction ** case cons.nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] \u22a2 splitOn x (join (intersperse [x] [hd])) = [hd] ** suffices hd.splitOn x = [hd] by simpa [join] ** case cons.nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] \u22a2 splitOn x hd = [hd] ** refine' splitOnP_eq_single _ _ _ ** case cons.nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] \u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 hd \u2192 \u00ac(x_1 == x) = true ** intro y hy H ** case cons.nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] y : \u03b1 hy : y \u2208 hd H : (y == x) = true \u22a2 False ** rw [eq_of_beq H] at hy ** case cons.nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] y : \u03b1 hy : x \u2208 hd H : (y == x) = true \u22a2 False ** refine' hx hd _ hy ** case cons.nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] y : \u03b1 hy : x \u2208 hd H : (y == x) = true \u22a2 hd \u2208 [hd] ** simp ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd : List \u03b1 ih : (\u2200 (l : List \u03b1), l \u2208 [] \u2192 \u00acx \u2208 l) \u2192 [] \u2260 [] \u2192 splitOn x (join (intersperse [x] [])) = [] hx : \u2200 (l : List \u03b1), l \u2208 [hd] \u2192 \u00acx \u2208 l hls : [hd] \u2260 [] this : splitOn x hd = [hd] \u22a2 splitOn x (join (intersperse [x] [hd])) = [hd] ** simpa [join] ** case cons.cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] \u22a2 splitOn x (join (intersperse [x] (hd :: head\u271d :: tail\u271d))) = hd :: head\u271d :: tail\u271d ** simp only [intersperse_cons_cons, singleton_append, join] ** case cons.cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] \u22a2 splitOn x (hd ++ x :: join (intersperse [x] (head\u271d :: tail\u271d))) = hd :: head\u271d :: tail\u271d ** specialize ih _ _ ** case cons.cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d this : \u2200 (as : List \u03b1), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as \u22a2 splitOn x (hd ++ x :: join (intersperse [x] (head\u271d :: tail\u271d))) = hd :: head\u271d :: tail\u271d case h \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d \u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 hd \u2192 \u00ac(fun x_2 => x_2 == x) x_1 = true ** case h =>\n intro y hy H\n rw [eq_of_beq H] at hy\n exact hx hd (.head _) hy ** case cons.cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d this : \u2200 (as : List \u03b1), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as \u22a2 splitOn x (hd ++ x :: join (intersperse [x] (head\u271d :: tail\u271d))) = hd :: head\u271d :: tail\u271d ** simp only [splitOn] at ih \u22a2 ** case cons.cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOnP (fun x_1 => x_1 == x) (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d this : \u2200 (as : List \u03b1), splitOnP (fun x_1 => x_1 == x) (hd ++ x :: as) = hd :: splitOnP (fun x_1 => x_1 == x) as \u22a2 splitOnP (fun x_1 => x_1 == x) (hd ++ x :: join (intersperse [x] (head\u271d :: tail\u271d))) = hd :: head\u271d :: tail\u271d ** rw [this, ih] ** case cons.cons.specialize_1 \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] \u22a2 \u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l ** intro l hl ** case cons.cons.specialize_1 \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] l : List \u03b1 hl : l \u2208 head\u271d :: tail\u271d \u22a2 \u00acx \u2208 l ** apply hx l ** case cons.cons.specialize_1 \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] l : List \u03b1 hl : l \u2208 head\u271d :: tail\u271d \u22a2 l \u2208 hd :: head\u271d :: tail\u271d ** simp at hl \u22a2 ** case cons.cons.specialize_1 \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] l : List \u03b1 hl : l = head\u271d \u2228 l \u2208 tail\u271d \u22a2 l = hd \u2228 l = head\u271d \u2228 l \u2208 tail\u271d ** exact Or.inr hl ** case cons.cons.specialize_2 \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) ih : (\u2200 (l : List \u03b1), l \u2208 head\u271d :: tail\u271d \u2192 \u00acx \u2208 l) \u2192 head\u271d :: tail\u271d \u2260 [] \u2192 splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] \u22a2 head\u271d :: tail\u271d \u2260 [] ** exact List.noConfusion ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d \u22a2 \u2200 (x_1 : \u03b1), x_1 \u2208 hd \u2192 \u00ac(fun x_2 => x_2 == x) x_1 = true ** intro y hy H ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d y : \u03b1 hy : y \u2208 hd H : (fun x_1 => x_1 == x) y = true \u22a2 False ** rw [eq_of_beq H] at hy ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool xs ys : List \u03b1 ls : List (List \u03b1) f : List \u03b1 \u2192 List \u03b1 inst\u271d : DecidableEq \u03b1 x : \u03b1 hx\u271d : \u2200 (l : List \u03b1), l \u2208 ls \u2192 \u00acx \u2208 l hls\u271d : ls \u2260 [] hd head\u271d : List \u03b1 tail\u271d : List (List \u03b1) hx : \u2200 (l : List \u03b1), l \u2208 hd :: head\u271d :: tail\u271d \u2192 \u00acx \u2208 l hls : hd :: head\u271d :: tail\u271d \u2260 [] ih : splitOn x (join (intersperse [x] (head\u271d :: tail\u271d))) = head\u271d :: tail\u271d y : \u03b1 hy : x \u2208 hd H : (fun x_1 => x_1 == x) y = true \u22a2 False ** exact hx hd (.head _) hy ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.sum_modByMonic_coeff ** R : Type u S : Type v T : Type w A : Type z a b : R n\u271d : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q n : \u2115 hn : degree q \u2264 \u2191n \u22a2 \u2211 i : Fin n, \u2191(monomial \u2191i) (coeff (p %\u2098 q) \u2191i) = p %\u2098 q ** nontriviality R ** R : Type u S : Type v T : Type w A : Type z a b : R n\u271d : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q n : \u2115 hn : degree q \u2264 \u2191n \u271d : Nontrivial R \u22a2 \u2211 i : Fin n, \u2191(monomial \u2191i) (coeff (p %\u2098 q) \u2191i) = p %\u2098 q ** exact\n (sum_fin (fun i c => monomial i c) (by simp) ((degree_modByMonic_lt _ hq).trans_le hn)).trans\n (sum_monomial_eq _) ** R : Type u S : Type v T : Type w A : Type z a b : R n\u271d : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q n : \u2115 hn : degree q \u2264 \u2191n \u271d : Nontrivial R \u22a2 \u2200 (i : \u2115), (fun i c => \u2191(monomial i) c) i 0 = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Subtype.range_coe ** \u03b1 : Type u_1 s : Set \u03b1 \u22a2 range val = s ** rw [\u2190 Set.image_univ] ** \u03b1 : Type u_1 s : Set \u03b1 \u22a2 val '' univ = s ** simp [-Set.image_univ, coe_image] ** Qed", + "informal": "" + }, + { + "formal": "Finset.weightedVSubOfPoint_indicator_subset ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b2 : Ring k inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082\u271d : Finset \u03b9\u2082 w : \u03b9 \u2192 k p : \u03b9 \u2192 P b : P s\u2081 s\u2082 : Finset \u03b9 h : s\u2081 \u2286 s\u2082 \u22a2 \u2191(weightedVSubOfPoint s\u2081 p b) w = \u2191(weightedVSubOfPoint s\u2082 p b) (Set.indicator (\u2191s\u2081) w) ** rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply] ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b2 : Ring k inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082\u271d : Finset \u03b9\u2082 w : \u03b9 \u2192 k p : \u03b9 \u2192 P b : P s\u2081 s\u2082 : Finset \u03b9 h : s\u2081 \u2286 s\u2082 \u22a2 \u2211 i in s\u2081, w i \u2022 (p i -\u1d65 b) = \u2211 i in s\u2082, Set.indicator (\u2191s\u2081) w i \u2022 (p i -\u1d65 b) ** exact\n Set.sum_indicator_subset_of_eq_zero w (fun i wi => wi \u2022 (p i -\u1d65 b : V)) h fun i => zero_smul k _ ** Qed", + "informal": "" + }, + { + "formal": "Orientation.inner_rightAngleRotation_right ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E \u22a2 inner x (\u2191(rightAngleRotation o) y) = -\u2191(\u2191(areaForm o) x) y ** rw [rightAngleRotation] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E \u22a2 inner x (\u2191(LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o) (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id) (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id)) y) = -\u2191(\u2191(areaForm o) x) y ** exact o.inner_rightAngleRotationAux\u2081_right x y ** Qed", + "informal": "" + }, + { + "formal": "polynomialFunctions.starClosure_eq_adjoin_X ** R : Type u_1 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : TopologicalSpace R inst\u271d\u00b2 : TopologicalSemiring R inst\u271d\u00b9 : StarRing R inst\u271d : ContinuousStar R s : Set R \u22a2 Subalgebra.starClosure (polynomialFunctions s) = adjoin R {\u2191(toContinuousMapOnAlgHom s) X} ** rw [polynomialFunctions.eq_adjoin_X s, adjoin_eq_starClosure_adjoin] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.leadingCoeff_linear ** R : Type u S : Type v a b c d : R n m : \u2115 inst\u271d : Semiring R p q : R[X] \u03b9 : Type u_1 ha : a \u2260 0 \u22a2 leadingCoeff (\u2191C a * X + \u2191C b) = a ** rw [add_comm, leadingCoeff_add_of_degree_lt (degree_C_lt_degree_C_mul_X ha),\n leadingCoeff_C_mul_X] ** Qed", + "informal": "" + }, + { + "formal": "Nat.snd_mem_divisors_of_mem_antidiagonal ** n : \u2115 x : \u2115 \u00d7 \u2115 h : x \u2208 divisorsAntidiagonal n \u22a2 x.2 \u2208 divisors n ** rw [mem_divisorsAntidiagonal] at h ** n : \u2115 x : \u2115 \u00d7 \u2115 h : x.1 * x.2 = n \u2227 n \u2260 0 \u22a2 x.2 \u2208 divisors n ** simp [Dvd.intro_left _ h.1, h.2] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.OuterMeasure.isometry_comap_mkMetric ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u22a2 \u2191(comap f) (mkMetric m) = mkMetric m ** simp only [mkMetric, mkMetric', mkMetric'.pre, inducedOuterMeasure, comap_iSup] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u22a2 \u2a06 i, \u2a06 (_ : i > 0), \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) = \u2a06 r, \u2a06 (_ : r > 0), boundedBy (extend fun s x => m (diam s)) ** refine' surjective_id.iSup_congr id fun \u03b5 => surjective_id.iSup_congr id fun h\u03b5 => _ ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 \u22a2 \u2191(comap f) (boundedBy (extend fun s x => m (diam s))) = boundedBy (extend fun s x => m (diam s)) ** rw [comap_boundedBy _ (H.imp _ id)] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 \u22a2 (boundedBy fun s => extend (fun s x => m (diam s)) (f '' s)) = boundedBy (extend fun s x => m (diam s)) ** congr with s : 1 ** case e_m.h \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X \u22a2 extend (fun s x => m (diam s)) (f '' s) = extend (fun s x => m (diam s)) s ** apply extend_congr ** case e_m.h.hP \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X \u22a2 diam (f '' s) \u2264 \u03b5 \u2194 diam s \u2264 id \u03b5 ** simp [hf.ediam_image] ** case e_m.h.hm \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X \u22a2 diam (f '' s) \u2264 \u03b5 \u2192 diam s \u2264 id \u03b5 \u2192 m (diam (f '' s)) = m (diam s) ** intros ** case e_m.h.hm \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 s : Set X ha\u271d : diam (f '' s) \u2264 \u03b5 hb\u271d : diam s \u2264 id \u03b5 \u22a2 m (diam (f '' s)) = m (diam s) ** simp [hf.injective.subsingleton_image_iff, hf.ediam_image] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 \u22a2 Monotone m \u2192 Monotone fun s => extend (fun s x => m (diam s)) \u2191s ** intro h_mono s t hst ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t \u22a2 (fun s => extend (fun s x => m (diam s)) \u2191s) s \u2264 (fun s => extend (fun s x => m (diam s)) \u2191s) t ** simp only [extend, le_iInf_iff] ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t \u22a2 diam \u2191t \u2264 \u03b5 \u2192 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t) ** intro ht ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t ht : diam \u2191t \u2264 \u03b5 \u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam \u2191t) ** apply le_trans _ (h_mono (diam_mono hst)) ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b9 : EMetricSpace X inst\u271d : EMetricSpace Y m : \u211d\u22650\u221e \u2192 \u211d\u22650\u221e f : X \u2192 Y hf : Isometry f H : Monotone m \u2228 Surjective f \u03b5 : \u211d\u22650\u221e h\u03b5 : id \u03b5 > 0 h_mono : Monotone m s t : { s // Set.Nonempty s } hst : s \u2264 t ht : diam \u2191t \u2264 \u03b5 \u22a2 \u2a05 (_ : diam \u2191s \u2264 \u03b5), m (diam \u2191s) \u2264 m (diam ((fun a => \u2191a) s)) ** simp only [(diam_mono hst).trans ht, le_refl, ciInf_pos] ** Qed", + "informal": "" + }, + { + "formal": "NNReal.exists_pos_sum_of_countable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 \u03b9 : Type u_4 inst\u271d : Countable \u03b9 \u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5 ** cases nonempty_encodable \u03b9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 \u03b9 : Type u_4 inst\u271d : Countable \u03b9 val\u271d : Encodable \u03b9 \u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5 ** obtain \u27e8a, a0, a\u03b5\u27e9 := exists_between (pos_iff_ne_zero.2 h\u03b5) ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 \u03b9 : Type u_4 inst\u271d : Countable \u03b9 val\u271d : Encodable \u03b9 a : \u211d\u22650 a0 : 0 < a a\u03b5 : a < \u03b5 \u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5 ** obtain \u27e8\u03b5', h\u03b5', c, hc, hc\u03b5\u27e9 := posSumOfEncodable a0 \u03b9 ** case intro.intro.intro.mk.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9\u271d : Type u_3 \u03b5 : \u211d\u22650 h\u03b5 : \u03b5 \u2260 0 \u03b9 : Type u_4 inst\u271d : Countable \u03b9 val\u271d : Encodable \u03b9 a : \u211d\u22650 a0 : 0 < a a\u03b5 : a < \u03b5 \u03b5' : \u03b9 \u2192 \u211d h\u03b5' : \u2200 (i : \u03b9), 0 < \u03b5' i c : \u211d hc : HasSum \u03b5' c hc\u03b5 : c \u2264 (fun a => \u2191a) a \u22a2 \u2203 \u03b5', (\u2200 (i : \u03b9), 0 < \u03b5' i) \u2227 \u2203 c, HasSum \u03b5' c \u2227 c < \u03b5 ** exact\n \u27e8fun i => \u27e8\u03b5' i, (h\u03b5' i).le\u27e9, fun i => NNReal.coe_lt_coe.1 <| h\u03b5' i,\n \u27e8c, hasSum_le (fun i => (h\u03b5' i).le) hasSum_zero hc\u27e9, NNReal.hasSum_coe.1 hc,\n a\u03b5.trans_le' <| NNReal.coe_le_coe.1 hc\u03b5\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "dite_eq_left_iff ** \u03b1 : Sort u_2 \u03b2 : Sort ?u.31578 \u03c3 : \u03b1 \u2192 Sort u_1 f : \u03b1 \u2192 \u03b2 P Q : Prop inst\u271d\u00b9 : Decidable P inst\u271d : Decidable Q a b c : \u03b1 A : P \u2192 \u03b1 B : \u00acP \u2192 \u03b1 \u22a2 dite P (fun x => a) B = a \u2194 \u2200 (h : \u00acP), B h = a ** by_cases P <;> simp [*, forall_prop_of_true, forall_prop_of_false] ** Qed", + "informal": "" + }, + { + "formal": "Finset.Ico_filter_le ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d b\u271d a b c : \u03b1 \u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b ** cases le_total a c with\n| inl h => rw [Ico_filter_le_of_left_le h, max_eq_right h]\n| inr h => rw [Ico_filter_le_of_le_left h, max_eq_left h] ** case inl \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d b\u271d a b c : \u03b1 h : a \u2264 c \u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b ** rw [Ico_filter_le_of_left_le h, max_eq_right h] ** case inr \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a\u271d b\u271d a b c : \u03b1 h : c \u2264 a \u22a2 filter (fun x => c \u2264 x) (Ico a b) = Ico (max a c) b ** rw [Ico_filter_le_of_le_left h, max_eq_left h] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.rightDistributor_inv_comp_biproduct_\u03c0 ** C : Type u_1 inst\u271d\u2075 : Category.{u_2, u_1} C inst\u271d\u2074 : Preadditive C inst\u271d\u00b3 : MonoidalCategory C inst\u271d\u00b2 : MonoidalPreadditive C inst\u271d\u00b9 : HasFiniteBiproducts C J : Type inst\u271d : Fintype J f : J \u2192 C X : C j : J \u22a2 (rightDistributor f X).inv \u226b (biproduct.\u03c0 f j \u2297 \ud835\udfd9 X) = biproduct.\u03c0 (fun j => f j \u2297 X) j ** simp [rightDistributor_inv, Preadditive.sum_comp, \u2190 comp_tensor_id, biproduct.\u03b9_\u03c0, dite_tensor,\n comp_dite] ** Qed", + "informal": "" + }, + { + "formal": "Finset.forall_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b2 f g : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a : \u03b1 b c : \u03b2 p : \u03b2 \u2192 Prop \u22a2 (\u2200 (b : \u03b2), b \u2208 image f s \u2192 p b) \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 p (f a) ** simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082] ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.multiplicity_pow_self ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 p : \u03b1 h0 : p \u2260 0 hu : \u00acIsUnit p n : \u2115 \u22a2 multiplicity p (p ^ n) = \u2191n ** rw [eq_coe_iff] ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 p : \u03b1 h0 : p \u2260 0 hu : \u00acIsUnit p n : \u2115 \u22a2 p ^ n \u2223 p ^ n \u2227 \u00acp ^ (n + 1) \u2223 p ^ n ** use dvd_rfl ** case right \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 p : \u03b1 h0 : p \u2260 0 hu : \u00acIsUnit p n : \u2115 \u22a2 \u00acp ^ (n + 1) \u2223 p ^ n ** rw [pow_dvd_pow_iff h0 hu] ** case right \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 p : \u03b1 h0 : p \u2260 0 hu : \u00acIsUnit p n : \u2115 \u22a2 \u00acn + 1 \u2264 n ** apply Nat.not_succ_le_self ** Qed", + "informal": "" + }, + { + "formal": "List.zipWith_swap_prod_support' ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l l' : List \u03b1 \u22a2 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2264 \u2191(toFinset l \u2294 toFinset l') ** simp only [Set.sup_eq_union, Set.le_eq_subset] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l l' : List \u03b1 \u22a2 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') ** induction' l with y l hl generalizing l' ** case nil \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x : \u03b1 l'\u271d l' : List \u03b1 \u22a2 {x | \u2191(prod (zipWith swap [] l')) x \u2260 x} \u2286 \u2191(toFinset [] \u2294 toFinset l') ** simp ** case cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') l' : List \u03b1 \u22a2 {x | \u2191(prod (zipWith swap (y :: l) l')) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset l') ** cases' l' with z l' ** case cons.nil \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l' : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') \u22a2 {x | \u2191(prod (zipWith swap (y :: l) [])) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset []) ** simp ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 \u22a2 {x | \u2191(prod (zipWith swap (y :: l) (z :: l'))) x \u2260 x} \u2286 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** intro x ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 \u22a2 x \u2208 {x | \u2191(prod (zipWith swap (y :: l) (z :: l'))) x \u2260 x} \u2192 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** simp only [Set.union_subset_iff, mem_cons, zipWith_cons_cons, foldr, prod_cons,\n mul_apply] ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 \u22a2 x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} \u2192 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** intro hx ** case cons.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** by_cases h : x \u2208 { x | (zipWith swap l l').prod x \u2260 x } ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} h : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** specialize hl l' h ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} h : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} hl : x \u2208 \u2191(toFinset l \u2294 toFinset l') \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** simp only [ge_iff_le, Finset.le_eq_subset, Finset.sup_eq_union, Finset.coe_union,\n coe_toFinset, Set.mem_union, Set.mem_setOf_eq] at hl ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} h : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} hl : x \u2208 l \u2228 x \u2208 l' hm : x \u2208 l' \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** simp only [Finset.coe_insert, Set.mem_insert_iff, Finset.mem_coe, toFinset_cons,\n mem_toFinset] at hm \u22a2 ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} h : x \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} hl : x \u2208 l \u2228 x \u2208 l' hm : x \u2208 l' \u22a2 x \u2208 insert y (toFinset l) \u2294 insert z (toFinset l') ** simp [hm] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} h : \u00acx \u2208 {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** simp only [not_not, Set.mem_setOf_eq] at h ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 hx : x \u2208 {x | \u2191(swap y z) (\u2191(prod (zipWith swap l l')) x) \u2260 x} h : \u2191(prod (zipWith swap l l')) x = x \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** simp only [h, Set.mem_setOf_eq] at hx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 h : \u2191(prod (zipWith swap l l')) x = x hx : \u2191(swap y z) x \u2260 x \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** rw [swap_apply_ne_self_iff] at hx ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l'\u271d : List \u03b1 y : \u03b1 l : List \u03b1 hl : \u2200 (l' : List \u03b1), {x | \u2191(prod (zipWith swap l l')) x \u2260 x} \u2286 \u2191(toFinset l \u2294 toFinset l') z : \u03b1 l' : List \u03b1 x : \u03b1 h : \u2191(prod (zipWith swap l l')) x = x hx : y \u2260 z \u2227 (x = y \u2228 x = z) \u22a2 x \u2208 \u2191(toFinset (y :: l) \u2294 toFinset (z :: l')) ** rcases hx with \u27e8hyz, rfl | rfl\u27e9 <;> simp ** Qed", + "informal": "" + }, + { + "formal": "Int.bit1_ne_zero ** m : \u2124 \u22a2 bit1 m \u2260 0 ** simpa only [bit0_zero] using bit1_ne_bit0 m 0 ** Qed", + "informal": "" + }, + { + "formal": "parallelogram_law ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \u22a2 inner (x + y) (x + y) + inner (x - y) (x - y) = 2 * (inner x x + inner y y) ** simp only [inner_add_add_self, inner_sub_sub_self] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E x y : E \u22a2 inner x x + inner x y + inner y x + inner y y + (inner x x - inner x y - inner y x + inner y y) = 2 * (inner x x + inner y y) ** ring ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.sum_taylor_eq ** R\u271d : Type u_1 inst\u271d\u00b9 : Semiring R\u271d r\u271d : R\u271d f\u271d : R\u271d[X] R : Type u_2 inst\u271d : CommRing R f : R[X] r : R \u22a2 (sum (\u2191(taylor r) f) fun i a => \u2191C a * (X - \u2191C r) ^ i) = f ** rw [\u2190 comp_eq_sum_left, sub_eq_add_neg, \u2190 C_neg, \u2190 taylor_apply, taylor_taylor, neg_add_self,\n taylor_zero] ** Qed", + "informal": "" + }, + { + "formal": "Associates.eq_of_factors_eq_factors ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d : UniqueFactorizationMonoid \u03b1 dec : DecidableEq \u03b1 dec' : DecidableEq (Associates \u03b1) a b : Associates \u03b1 h : factors a = factors b \u22a2 a = b ** have : a.factors.prod = b.factors.prod := by rw [h] ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d : UniqueFactorizationMonoid \u03b1 dec : DecidableEq \u03b1 dec' : DecidableEq (Associates \u03b1) a b : Associates \u03b1 h : factors a = factors b this : FactorSet.prod (factors a) = FactorSet.prod (factors b) \u22a2 a = b ** rwa [factors_prod, factors_prod] at this ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 dec_irr : (p : Associates \u03b1) \u2192 Decidable (Irreducible p) inst\u271d : UniqueFactorizationMonoid \u03b1 dec : DecidableEq \u03b1 dec' : DecidableEq (Associates \u03b1) a b : Associates \u03b1 h : factors a = factors b \u22a2 FactorSet.prod (factors a) = FactorSet.prod (factors b) ** rw [h] ** Qed", + "informal": "" + }, + { + "formal": "Nat.factorial_mul_pow_sub_le_factorial ** m\u271d n\u271d n m : \u2115 hnm : n \u2264 m \u22a2 n ! * n ^ (m - n) \u2264 m ! ** suffices n ! * (n + 1) ^ (m - n) \u2264 m ! from by\n apply LE.le.trans _ this\n apply mul_le_mul_left\n apply pow_le_pow_of_le_left (le_succ n) ** m\u271d n\u271d n m : \u2115 hnm : n \u2264 m \u22a2 n ! * (n + 1) ^ (m - n) \u2264 m ! ** have := @Nat.factorial_mul_pow_le_factorial n (m - n) ** m\u271d n\u271d n m : \u2115 hnm : n \u2264 m this : n ! * succ n ^ (m - n) \u2264 (n + (m - n))! \u22a2 n ! * (n + 1) ^ (m - n) \u2264 m ! ** simp [hnm] at this ** m\u271d n\u271d n m : \u2115 hnm : n \u2264 m this : n ! * succ n ^ (m - n) \u2264 m ! \u22a2 n ! * (n + 1) ^ (m - n) \u2264 m ! ** exact this ** m\u271d n\u271d n m : \u2115 hnm : n \u2264 m this : n ! * (n + 1) ^ (m - n) \u2264 m ! \u22a2 n ! * n ^ (m - n) \u2264 m ! ** apply LE.le.trans _ this ** m\u271d n\u271d n m : \u2115 hnm : n \u2264 m this : n ! * (n + 1) ^ (m - n) \u2264 m ! \u22a2 n ! * n ^ (m - n) \u2264 n ! * (n + 1) ^ (m - n) ** apply mul_le_mul_left ** case h m\u271d n\u271d n m : \u2115 hnm : n \u2264 m this : n ! * (n + 1) ^ (m - n) \u2264 m ! \u22a2 n ^ (m - n) \u2264 (n + 1) ^ (m - n) ** apply pow_le_pow_of_le_left (le_succ n) ** Qed", + "informal": "" + }, + { + "formal": "Finset.pred_card_le_card_erase ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 \u22a2 card s - 1 \u2264 card (erase s a) ** by_cases h : a \u2208 s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : a \u2208 s \u22a2 card s - 1 \u2264 card (erase s a) ** exact (card_erase_of_mem h).ge ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : \u00aca \u2208 s \u22a2 card s - 1 \u2264 card (erase s a) ** rw [erase_eq_of_not_mem h] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 a b : \u03b1 inst\u271d : DecidableEq \u03b1 h : \u00aca \u2208 s \u22a2 card s - 1 \u2264 card s ** exact Nat.sub_le _ _ ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.cyclotomic_expand_eq_cyclotomic ** p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R \u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R ** rcases n.eq_zero_or_pos with (rfl | hzero) ** case inr p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 \u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R ** haveI := NeZero.of_pos hzero ** case inr p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n \u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R ** suffices expand \u2124 p (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 by\n rw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, map_cyclotomic_int, map_cyclotomic] ** case inr p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n \u22a2 \u2191(expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 ** refine' eq_of_monic_of_dvd_of_natDegree_le (cyclotomic.monic _ _)\n ((cyclotomic.monic n \u2124).expand hp.pos) _ _ ** case inl p : \u2115 hp : Nat.Prime p R : Type u_1 inst\u271d : CommRing R hdiv : p \u2223 0 \u22a2 \u2191(expand R p) (cyclotomic 0 R) = cyclotomic (0 * p) R ** simp ** p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this\u271d : NeZero n this : \u2191(expand \u2124 p) (cyclotomic n \u2124) = cyclotomic (n * p) \u2124 \u22a2 \u2191(expand R p) (cyclotomic n R) = cyclotomic (n * p) R ** rw [\u2190 map_cyclotomic_int, \u2190 map_expand, this, map_cyclotomic_int, map_cyclotomic] ** case inr.refine'_1 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n \u22a2 cyclotomic (n * p) \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124) ** have hpos := Nat.mul_pos hzero hp.pos ** case inr.refine'_1 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n hpos : n * p > 0 \u22a2 cyclotomic (n * p) \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124) ** have hprim := Complex.isPrimitiveRoot_exp _ hpos.ne.symm ** case inr.refine'_1 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n hpos : n * p > 0 hprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p) \u22a2 cyclotomic (n * p) \u2124 \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124) ** rw [cyclotomic_eq_minpoly hprim hpos] ** case inr.refine'_1 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n hpos : n * p > 0 hprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p) \u22a2 minpoly \u2124 (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) \u2223 \u2191(expand \u2124 p) (cyclotomic n \u2124) ** refine' minpoly.isIntegrallyClosed_dvd (hprim.isIntegral hpos) _ ** case inr.refine'_1 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n hpos : n * p > 0 hprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p) \u22a2 \u2191(aeval (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)))) (\u2191(expand \u2124 p) (cyclotomic n \u2124)) = 0 ** rw [aeval_def, \u2190 eval_map, map_expand, map_cyclotomic, expand_eval, \u2190 IsRoot.def,\n @isRoot_cyclotomic_iff] ** case inr.refine'_1 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n hpos : n * p > 0 hprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p) \u22a2 IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p)) ^ p) n ** convert IsPrimitiveRoot.pow_of_dvd hprim hp.ne_zero (dvd_mul_left p n) ** case h.e'_4 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n hpos : n * p > 0 hprim : IsPrimitiveRoot (Complex.exp (2 * \u2191Real.pi * Complex.I / \u2191(n * p))) (n * p) \u22a2 n = n * p / p ** rw [Nat.mul_div_cancel _ hp.pos] ** case inr.refine'_2 p n : \u2115 hp : Nat.Prime p hdiv : p \u2223 n R : Type u_1 inst\u271d : CommRing R hzero : n > 0 this : NeZero n \u22a2 natDegree (\u2191(expand \u2124 p) (cyclotomic n \u2124)) \u2264 natDegree (cyclotomic (n * p) \u2124) ** rw [natDegree_expand, natDegree_cyclotomic, natDegree_cyclotomic, mul_comm n,\n Nat.totient_mul_of_prime_of_dvd hp hdiv, mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "SimplexCategory.\u03b4_comp_\u03c3_of_le ** n : \u2115 i : Fin (n + 2) j : Fin (n + 1) H : i \u2264 Fin.castSucc j \u22a2 \u03b4 (Fin.castSucc i) \u226b \u03c3 (Fin.succ j) = \u03c3 j \u226b \u03b4 i ** rcases i with \u27e8i, hi\u27e9 ** case mk n : \u2115 j : Fin (n + 1) i : \u2115 hi : i < n + 2 H : { val := i, isLt := hi } \u2264 Fin.castSucc j \u22a2 \u03b4 (Fin.castSucc { val := i, isLt := hi }) \u226b \u03c3 (Fin.succ j) = \u03c3 j \u226b \u03b4 { val := i, isLt := hi } ** rcases j with \u27e8j, hj\u27e9 ** case mk.mk n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 H : { val := i, isLt := hi } \u2264 Fin.castSucc { val := j, isLt := hj } \u22a2 \u03b4 (Fin.castSucc { val := i, isLt := hi }) \u226b \u03c3 (Fin.succ { val := j, isLt := hj }) = \u03c3 { val := j, isLt := hj } \u226b \u03b4 { val := i, isLt := hi } ** ext \u27e8k, hk\u27e9 ** case mk.mk.a.h.h.mk.h n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 H : { val := i, isLt := hi } \u2264 Fin.castSucc { val := j, isLt := hj } k : \u2115 hk : k < len [n + 1] + 1 \u22a2 \u2191(\u2191(Hom.toOrderHom (\u03b4 (Fin.castSucc { val := i, isLt := hi }) \u226b \u03c3 (Fin.succ { val := j, isLt := hj }))) { val := k, isLt := hk }) = \u2191(\u2191(Hom.toOrderHom (\u03c3 { val := j, isLt := hj } \u226b \u03b4 { val := i, isLt := hi })) { val := k, isLt := hk }) ** simp at H hk ** case mk.mk.a.h.h.mk.h n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j \u22a2 \u2191(\u2191(Hom.toOrderHom (\u03b4 (Fin.castSucc { val := i, isLt := hi }) \u226b \u03c3 (Fin.succ { val := j, isLt := hj }))) { val := k, isLt := hk }) = \u2191(\u2191(Hom.toOrderHom (\u03c3 { val := j, isLt := hj } \u226b \u03b4 { val := i, isLt := hi })) { val := k, isLt := hk }) ** dsimp [\u03c3, \u03b4, Fin.predAbove, Fin.succAbove] ** case mk.mk.a.h.h.mk.h n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j \u22a2 \u2191(if h : { val := j + 1, isLt := (_ : j + 1 < Nat.succ (n + 1 + 1)) } < if k < i then { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) } then Fin.pred (if k < i then { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) (_ : (if k < i then { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) \u2260 0) else Fin.castLT (if k < i then { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) (_ : \u2191(if k < i then { val := k, isLt := (_ : k < Nat.succ (n + 1 + 1)) } else { val := k + 1, isLt := (_ : Nat.succ k < Nat.succ (n + 1 + 1)) }) < n + 1 + 1)) = \u2191(if \u2191(if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } \u2260 0) else { val := k, isLt := (_ : \u2191{ val := k, isLt := hk } < n + 1) }) < i then Fin.castSucc (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } \u2260 0) else { val := k, isLt := (_ : \u2191{ val := k, isLt := hk } < n + 1) }) else Fin.succ (if h : { val := j, isLt := (_ : j < Nat.succ (n + 1)) } < { val := k, isLt := hk } then Fin.pred { val := k, isLt := hk } (_ : { val := k, isLt := hk } \u2260 0) else { val := k, isLt := (_ : \u2191{ val := k, isLt := hk } < n + 1) })) ** simp only [Fin.lt_iff_val_lt_val, Fin.dite_val, Fin.ite_val, Fin.coe_pred, ge_iff_le,\n Fin.coe_castLT, dite_eq_ite, Fin.coe_castSucc, Fin.val_succ] ** case mk.mk.a.h.h.mk.h n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j \u22a2 (if j + 1 < if k < i then k else k + 1 then (if k < i then k else k + 1) - 1 else if k < i then k else k + 1) = if (if j < k then k - 1 else k) < i then if j < k then k - 1 else k else (if j < k then k - 1 else k) + 1 ** split_ifs ** case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : k < i h\u271d\u00b2 : j + 1 < k h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k - 1 = k - 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : k < i h\u271d\u00b2 : j + 1 < k h\u271d\u00b9 : j < k h\u271d : \u00ack - 1 < i \u22a2 k - 1 = k - 1 + 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : k < i h\u271d\u00b9 : j + 1 < k h\u271d : \u00acj < k \u22a2 k - 1 = k case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : k < i h\u271d\u00b2 : \u00acj + 1 < k h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k = k - 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : k < i h\u271d\u00b2 : \u00acj + 1 < k h\u271d\u00b9 : j < k h\u271d : \u00ack - 1 < i \u22a2 k = k - 1 + 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : k < i h\u271d\u00b9 : \u00acj + 1 < k h\u271d : \u00acj < k \u22a2 k = k case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : j + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k + 1 - 1 = k - 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : j + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : \u00ack - 1 < i \u22a2 k + 1 - 1 = k - 1 + 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : \u00ack < i h\u271d\u00b9 : j + 1 < k + 1 h\u271d : \u00acj < k \u22a2 k + 1 - 1 = k + 1 case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : \u00acj + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k + 1 = k - 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : \u00acj + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : \u00ack - 1 < i \u22a2 k + 1 = k - 1 + 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : \u00ack < i h\u271d\u00b9 : \u00acj + 1 < k + 1 h\u271d : \u00acj < k \u22a2 k + 1 = k + 1 ** all_goals try simp <;> linarith ** case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : k < i h\u271d\u00b9 : j + 1 < k h\u271d : \u00acj < k \u22a2 k - 1 = k case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : k < i h\u271d\u00b2 : \u00acj + 1 < k h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k = k - 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : k < i h\u271d\u00b2 : \u00acj + 1 < k h\u271d\u00b9 : j < k h\u271d : \u00ack - 1 < i \u22a2 k = k - 1 + 1 case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : j + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k + 1 - 1 = k - 1 case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : j + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : \u00ack - 1 < i \u22a2 k + 1 - 1 = k - 1 + 1 case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : \u00acj + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k + 1 = k - 1 ** all_goals cases k <;> simp at * <;> linarith ** case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : \u00ack < i h\u271d\u00b9 : \u00acj + 1 < k + 1 h\u271d : \u00acj < k \u22a2 k + 1 = k + 1 ** try simp <;> linarith ** case neg n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b2 : \u00ack < i h\u271d\u00b9 : \u00acj + 1 < k + 1 h\u271d : \u00acj < k \u22a2 k + 1 = k + 1 ** simp <;> linarith ** case pos n i : \u2115 hi : i < n + 2 j : \u2115 hj : j < n + 1 k : \u2115 hk : k < n + 1 + 1 H : i \u2264 j h\u271d\u00b3 : \u00ack < i h\u271d\u00b2 : \u00acj + 1 < k + 1 h\u271d\u00b9 : j < k h\u271d : k - 1 < i \u22a2 k + 1 = k - 1 ** cases k <;> simp at * <;> linarith ** Qed", + "informal": "" + }, + { + "formal": "SetTheory.PGame.neg_fuzzy_neg_iff ** x y : PGame \u22a2 -x \u2016 -y \u2194 x \u2016 y ** rw [Fuzzy, Fuzzy, neg_lf_neg_iff, neg_lf_neg_iff, and_comm] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_map_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hg : Measurable g \u22a2 \u222b\u207b (a : \u03b2), f a \u2202Measure.map g \u03bc \u2264 \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc ** rw [\u2190 iSup_lintegral_measurable_le_eq_lintegral, \u2190 iSup_lintegral_measurable_le_eq_lintegral] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hg : Measurable g \u22a2 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => f a), \u222b\u207b (a : \u03b2), g_1 a \u2202Measure.map g \u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => f (g a)), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc ** refine' iSup\u2082_le fun i hi => iSup_le fun h'i => _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hg : Measurable g i : \u03b2 \u2192 \u211d\u22650\u221e hi : Measurable i h'i : i \u2264 fun a => f a \u22a2 \u222b\u207b (a : \u03b2), i a \u2202Measure.map g \u03bc \u2264 \u2a06 g_1, \u2a06 (_ : Measurable g_1), \u2a06 (_ : g_1 \u2264 fun a => f (g a)), \u222b\u207b (a : \u03b1), g_1 a \u2202\u03bc ** refine' le_iSup\u2082_of_le (i \u2218 g) (hi.comp hg) _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 m\u03b2 : MeasurableSpace \u03b2 f : \u03b2 \u2192 \u211d\u22650\u221e g : \u03b1 \u2192 \u03b2 hg : Measurable g i : \u03b2 \u2192 \u211d\u22650\u221e hi : Measurable i h'i : i \u2264 fun a => f a \u22a2 \u222b\u207b (a : \u03b2), i a \u2202Measure.map g \u03bc \u2264 \u2a06 (_ : i \u2218 g \u2264 fun a => f (g a)), \u222b\u207b (a : \u03b1), (i \u2218 g) a \u2202\u03bc ** exact le_iSup_of_le (fun x => h'i (g x)) (le_of_eq (lintegral_map hi hg)) ** Qed", + "informal": "" + }, + { + "formal": "Nat.shiftRight_eq_div_pow ** m k : Nat \u22a2 m >>> (k + 1) = m / 2 ^ (k + 1) ** rw [shiftRight_add, shiftRight_eq_div_pow m k] ** m k : Nat \u22a2 (m / 2 ^ k) >>> 1 = m / 2 ^ (k + 1) ** simp [Nat.div_div_eq_div_mul, \u2190 Nat.pow_succ] ** Qed", + "informal": "" + }, + { + "formal": "Function.Embedding.invFun_restrict ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 f : \u03b1 \u21aa \u03b2 b : \u2191(Set.range \u2191f) inst\u271d : Nonempty \u03b1 \u22a2 Set.restrict (Set.range \u2191f) (invFun \u2191f) = invOfMemRange f ** ext \u27e8b, h\u27e9 ** case h.mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 f : \u03b1 \u21aa \u03b2 b\u271d : \u2191(Set.range \u2191f) inst\u271d : Nonempty \u03b1 b : \u03b2 h : b \u2208 Set.range \u2191f \u22a2 Set.restrict (Set.range \u2191f) (invFun \u2191f) { val := b, property := h } = invOfMemRange f { val := b, property := h } ** apply f.injective ** case h.mk.a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : Fintype \u03b1 inst\u271d\u00b9 : DecidableEq \u03b2 f : \u03b1 \u21aa \u03b2 b\u271d : \u2191(Set.range \u2191f) inst\u271d : Nonempty \u03b1 b : \u03b2 h : b \u2208 Set.range \u2191f \u22a2 \u2191f (Set.restrict (Set.range \u2191f) (invFun \u2191f) { val := b, property := h }) = \u2191f (invOfMemRange f { val := b, property := h }) ** simp [f.left_inv_of_invOfMemRange, @invFun_eq _ _ _ f b (Set.mem_range.mp h)] ** Qed", + "informal": "" + }, + { + "formal": "IsCoprime.mul_add_left_left ** R : Type u inst\u271d : CommRing R x y : R h : IsCoprime x y z : R \u22a2 IsCoprime (y * z + x) y ** rw [add_comm] ** R : Type u inst\u271d : CommRing R x y : R h : IsCoprime x y z : R \u22a2 IsCoprime (x + y * z) y ** exact h.add_mul_left_left z ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Supermartingale.smul_nonneg ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc \u22a2 Supermartingale (c \u2022 f) \u2131 \u03bc ** refine' \u27e8hf.1.smul c, fun i j hij => _, fun i => (hf.2.2 i).smul c\u27e9 ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 \u03bc[(c \u2022 f) j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i ** refine' (condexp_smul c (f j)).le.trans _ ** \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j \u22a2 c \u2022 \u03bc[f j|\u2191\u2131 i] \u2264\u1d50[\u03bc] (c \u2022 f) i ** filter_upwards [hf.2.1 i j hij] with _ hle ** case h \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j a\u271d : \u03a9 hle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d \u22a2 (c \u2022 \u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 (c \u2022 f) i a\u271d ** simp_rw [Pi.smul_apply] ** case h \u03a9 : Type u_1 E : Type u_2 \u03b9 : Type u_3 inst\u271d\u2077 : Preorder \u03b9 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : CompleteSpace E f\u271d g : \u03b9 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u03b9 m0 F : Type u_4 inst\u271d\u00b3 : NormedLatticeAddCommGroup F inst\u271d\u00b2 : NormedSpace \u211d F inst\u271d\u00b9 : CompleteSpace F inst\u271d : OrderedSMul \u211d F f : \u03b9 \u2192 \u03a9 \u2192 F c : \u211d hc : 0 \u2264 c hf : Supermartingale f \u2131 \u03bc i j : \u03b9 hij : i \u2264 j a\u271d : \u03a9 hle : (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 f i a\u271d \u22a2 c \u2022 (\u03bc[f j|\u2191\u2131 i]) a\u271d \u2264 c \u2022 f i a\u271d ** exact smul_le_smul_of_nonneg hle hc ** Qed", + "informal": "" + }, + { + "formal": "Set.sigma_insert ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2 : \u03b9 \u2192 Type u_4 s s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x : (i : \u03b9) \u00d7 \u03b1 i i j : \u03b9 a\u271d : \u03b1 i a : (i : \u03b9) \u2192 \u03b1 i \u22a2 (Set.Sigma s fun i => insert (a i) (t i)) = (fun i => { fst := i, snd := a i }) '' s \u222a Set.Sigma s t ** simp_rw [insert_eq, sigma_union, sigma_singleton] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.snorm_eq_integral_rpow_norm ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p \u22a2 snorm f p \u03bc = ENNReal.ofReal ((\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9) ** have A : \u222b\u207b a : \u03b1, ENNReal.ofReal (\u2016f a\u2016 ^ p.toReal) \u2202\u03bc = \u222b\u207b a : \u03b1, \u2016f a\u2016\u208a ^ p.toReal \u2202\u03bc := by\n apply lintegral_congr\n intro x\n rw [\u2190 ofReal_rpow_of_nonneg (norm_nonneg _) toReal_nonneg, ofReal_norm_eq_coe_nnnorm,\n \u2190 ENNReal.coe_rpow_of_nonneg _ toReal_nonneg] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 snorm f p \u03bc = ENNReal.ofReal ((\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9) ** simp only [snorm_eq_lintegral_rpow_nnnorm hp1 hp2, one_div] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 = ENNReal.ofReal ((\u222b (a : \u03b1), \u2016f a\u2016 ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9) ** rw [integral_eq_lintegral_of_nonneg_ae] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 = ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9) case hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2016f a\u2016 ^ ENNReal.toReal p case hfm \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 AEStronglyMeasurable (fun a => \u2016f a\u2016 ^ ENNReal.toReal p) \u03bc ** rotate_left ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 = ENNReal.ofReal (ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9) ** rw [A, \u2190 ofReal_rpow_of_nonneg toReal_nonneg (inv_nonneg.2 toReal_nonneg), ofReal_toReal] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc ** apply lintegral_congr ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p \u22a2 \u2200 (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) = \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) ** intro x ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p x : \u03b1 \u22a2 ENNReal.ofReal (\u2016f x\u2016 ^ ENNReal.toReal p) = \u2191(\u2016f x\u2016\u208a ^ ENNReal.toReal p) ** rw [\u2190 ofReal_rpow_of_nonneg (norm_nonneg _) toReal_nonneg, ofReal_norm_eq_coe_nnnorm,\n \u2190 ENNReal.coe_rpow_of_nonneg _ toReal_nonneg] ** case hf \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 0 \u2264\u1d50[\u03bc] fun a => \u2016f a\u2016 ^ ENNReal.toReal p ** exact eventually_of_forall fun x => Real.rpow_nonneg_of_nonneg (norm_nonneg _) _ ** case hfm \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 AEStronglyMeasurable (fun a => \u2016f a\u2016 ^ ENNReal.toReal p) \u03bc ** exact (hf.aestronglyMeasurable.norm.aemeasurable.pow_const _).aestronglyMeasurable ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 (\u222b\u207b (x : \u03b1), \u2191\u2016f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 = (\u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc) ^ (ENNReal.toReal p)\u207b\u00b9 ** simp_rw [\u2190 ENNReal.coe_rpow_of_nonneg _ toReal_nonneg] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u2260 \u22a4 ** simp_rw [\u2190 ENNReal.coe_rpow_of_nonneg _ toReal_nonneg] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b2 : NormedAddCommGroup E inst\u271d\u00b9\u00b9 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2077 : NormedAddCommGroup F inst\u271d\u2076 : NormedSpace \u211d F inst\u271d\u2075 : CompleteSpace F G : Type u_5 inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : FirstCountableTopology X H : Type u_7 inst\u271d : NormedAddCommGroup H f : \u03b1 \u2192 H p : \u211d\u22650\u221e hp1 : p \u2260 0 hp2 : p \u2260 \u22a4 hf : Mem\u2112p f p A : \u222b\u207b (a : \u03b1), ENNReal.ofReal (\u2016f a\u2016 ^ ENNReal.toReal p) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191(\u2016f a\u2016\u208a ^ ENNReal.toReal p) \u2202\u03bc \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 ** exact (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp1 hp2 hf.2).ne ** Qed", + "informal": "" + }, + { + "formal": "MultilinearMap.mkPiRing_zero ** R : Type uR S : Type uS \u03b9 : Type u\u03b9 n : \u2115 M : Fin (Nat.succ n) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u2077 : CommSemiring R inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 AddCommMonoid (M i) inst\u271d\u2074 : AddCommMonoid M\u2082 inst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 Module R (M i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u00b9 : Module R M\u2082 f f' : MultilinearMap R M\u2081 M\u2082 inst\u271d : Fintype \u03b9 \u22a2 MultilinearMap.mkPiRing R \u03b9 0 = 0 ** ext ** case H R : Type uR S : Type uS \u03b9 : Type u\u03b9 n : \u2115 M : Fin (Nat.succ n) \u2192 Type v M\u2081 : \u03b9 \u2192 Type v\u2081 M\u2082 : Type v\u2082 M\u2083 : Type v\u2083 M' : Type v' inst\u271d\u2077 : CommSemiring R inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u2081 i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 AddCommMonoid (M i) inst\u271d\u2074 : AddCommMonoid M\u2082 inst\u271d\u00b3 : (i : Fin (Nat.succ n)) \u2192 Module R (M i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (M\u2081 i) inst\u271d\u00b9 : Module R M\u2082 f f' : MultilinearMap R M\u2081 M\u2082 inst\u271d : Fintype \u03b9 x\u271d : \u03b9 \u2192 R \u22a2 \u2191(MultilinearMap.mkPiRing R \u03b9 0) x\u271d = \u21910 x\u271d ** rw [mkPiRing_apply, smul_zero, MultilinearMap.zero_apply] ** Qed", + "informal": "" + }, + { + "formal": "IsCoprime.sq_add_sq_ne_zero ** R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b \u22a2 a ^ 2 + b ^ 2 \u2260 0 ** intro h' ** R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b h' : a ^ 2 + b ^ 2 = 0 \u22a2 False ** obtain \u27e8ha, hb\u27e9 := (add_eq_zero_iff'\n(by rw [pow_two]; exact mul_self_nonneg _)\n (by rw [pow_two]; exact mul_self_nonneg _)).mp h' ** case intro R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b h' : a ^ 2 + b ^ 2 = 0 ha : a ^ 2 = 0 hb : b ^ 2 = 0 \u22a2 False ** obtain rfl := pow_eq_zero ha ** case intro R : Type u_1 inst\u271d : LinearOrderedCommRing R b : R hb : b ^ 2 = 0 h : IsCoprime 0 b h' : 0 ^ 2 + b ^ 2 = 0 ha : 0 ^ 2 = 0 \u22a2 False ** obtain rfl := pow_eq_zero hb ** case intro R : Type u_1 inst\u271d : LinearOrderedCommRing R ha hb : 0 ^ 2 = 0 h : IsCoprime 0 0 h' : 0 ^ 2 + 0 ^ 2 = 0 \u22a2 False ** exact not_isCoprime_zero_zero h ** R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b h' : a ^ 2 + b ^ 2 = 0 \u22a2 0 \u2264 a ^ 2 ** rw [pow_two] ** R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b h' : a ^ 2 + b ^ 2 = 0 \u22a2 0 \u2264 a * a ** exact mul_self_nonneg _ ** R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b h' : a ^ 2 + b ^ 2 = 0 \u22a2 0 \u2264 b ^ 2 ** rw [pow_two] ** R : Type u_1 inst\u271d : LinearOrderedCommRing R a b : R h : IsCoprime a b h' : a ^ 2 + b ^ 2 = 0 \u22a2 0 \u2264 b * b ** exact mul_self_nonneg _ ** Qed", + "informal": "" + }, + { + "formal": "ProperCone.pointed_zero ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : TopologicalSpace E inst\u271d\u00b9 : T1Space E inst\u271d : Module \ud835\udd5c E \u22a2 ConvexCone.Pointed 0.toConvexCone ** simp [ConvexCone.pointed_zero] ** Qed", + "informal": "" + }, + { + "formal": "IsAdjoinRootMonic.coeff_root_pow ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f \u22a2 \u2191(coeff h) (root h.toIsAdjoinRoot ^ n) = Pi.single n 1 ** ext i ** case h R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 \u22a2 \u2191(coeff h) (root h.toIsAdjoinRoot ^ n) i = Pi.single n 1 i ** rw [coeff_apply] ** case h R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 \u22a2 (if hi : i < natDegree f then \u2191(\u2191(basis h).repr (root h.toIsAdjoinRoot ^ n)) { val := i, isLt := hi } else 0) = Pi.single n 1 i ** split_ifs with hi ** case pos R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 hi : i < natDegree f \u22a2 \u2191(\u2191(basis h).repr (root h.toIsAdjoinRoot ^ n)) { val := i, isLt := hi } = Pi.single n 1 i ** calc\n h.basis.repr (h.root ^ n) \u27e8i, _\u27e9 = h.basis.repr (h.basis \u27e8n, hn\u27e9) \u27e8i, hi\u27e9 := by\n rw [h.basis_apply, Fin.val_mk]\n _ = Pi.single (f := fun _ => R) ((\u27e8n, hn\u27e9 : Fin _) : \u2115) (1 : (fun _ => R) n)\n \u2191(\u27e8i, _\u27e9 : Fin _) := by\n rw [h.basis.repr_self, \u2190 Finsupp.single_eq_pi_single,\n Finsupp.single_apply_left Fin.val_injective]\n _ = Pi.single (f := fun _ => R) n 1 i := by rw [Fin.val_mk, Fin.val_mk] ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 hi : i < natDegree f \u22a2 \u2191(\u2191(basis h).repr (root h.toIsAdjoinRoot ^ n)) { val := i, isLt := hi } = \u2191(\u2191(basis h).repr (\u2191(basis h) { val := n, isLt := hn })) { val := i, isLt := hi } ** rw [h.basis_apply, Fin.val_mk] ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 hi : i < natDegree f \u22a2 \u2191(\u2191(basis h).repr (\u2191(basis h) { val := n, isLt := hn })) { val := i, isLt := hi } = Pi.single (\u2191{ val := n, isLt := hn }) 1 \u2191{ val := i, isLt := ?m.1386915 } ** rw [h.basis.repr_self, \u2190 Finsupp.single_eq_pi_single,\n Finsupp.single_apply_left Fin.val_injective] ** R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 hi : i < natDegree f \u22a2 Pi.single (\u2191{ val := n, isLt := hn }) 1 \u2191{ val := i, isLt := hi } = Pi.single n 1 i ** rw [Fin.val_mk, Fin.val_mk] ** case neg R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 hi : \u00aci < natDegree f \u22a2 0 = Pi.single n 1 i ** refine (Pi.single_eq_of_ne (f := fun _ => R) ?_ (1 : (fun _ => R) n)).symm ** case neg R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f n : \u2115 hn : n < natDegree f i : \u2115 hi : \u00aci < natDegree f \u22a2 i \u2260 n ** rintro rfl ** case neg R : Type u S : Type v inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Ring S f : R[X] inst\u271d : Algebra R S h : IsAdjoinRootMonic S f i : \u2115 hi : \u00aci < natDegree f hn : i < natDegree f \u22a2 False ** simp [hi] at hn ** Qed", + "informal": "" + }, + { + "formal": "Set.preimage_const_add_Ioo ** \u03b1 : Type u_1 inst\u271d : OrderedAddCommGroup \u03b1 a b c : \u03b1 \u22a2 (fun x => a + x) \u207b\u00b9' Ioo b c = Ioo (b - a) (c - a) ** simp [\u2190 Ioi_inter_Iio] ** Qed", + "informal": "" + }, + { + "formal": "pi_le_borel_pi ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 \u03b9 : Type u_6 \u03c0 : \u03b9 \u2192 Type u_7 inst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i) inst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i) \u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i) ** have : \u2039\u2200 i, MeasurableSpace (\u03c0 i)\u203a = fun i => borel (\u03c0 i) :=\n funext fun i => BorelSpace.measurable_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 \u03b9 : Type u_6 \u03c0 : \u03b9 \u2192 Type u_7 inst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c0 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 MeasurableSpace (\u03c0 i) inst\u271d : \u2200 (i : \u03b9), BorelSpace (\u03c0 i) this : inst\u271d\u00b9 = fun i => borel (\u03c0 i) \u22a2 MeasurableSpace.pi \u2264 borel ((i : \u03b9) \u2192 \u03c0 i) ** exact iSup_le fun i => comap_le_iff_le_map.2 <| (continuous_apply i).borel_measurable ** Qed", + "informal": "" + }, + { + "formal": "isLUB_biSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s\u271d t : \u03b9 \u2192 \u03b1 a b : \u03b1 s : Set \u03b2 f : \u03b2 \u2192 \u03b1 \u22a2 IsLUB (f '' s) (\u2a06 x \u2208 s, f x) ** simpa only [range_comp, Subtype.range_coe, iSup_subtype'] using\n @isLUB_iSup \u03b1 s _ (f \u2218 fun x => (x : \u03b2)) ** Qed", + "informal": "" + }, + { + "formal": "Finset.mem_dfinsupp_iff ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 f : \u03a0\u2080 (i : \u03b9), \u03b1 i t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) \u22a2 f \u2208 dfinsupp s t \u2194 support f \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i ** refine' mem_map.trans \u27e8_, _\u27e9 ** case refine'_1 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 f : \u03a0\u2080 (i : \u03b9), \u03b1 i t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) \u22a2 (\u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := fun f => DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s), inj' := (_ : Function.Injective (DFinsupp.mk s \u2218 fun f i => f \u2191i (_ : \u2191i \u2208 \u2191s))) } a = f) \u2192 support f \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i ** rintro \u27e8f, hf, rfl\u27e9 ** case refine'_1.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) f : (a : \u03b9) \u2192 a \u2208 s \u2192 (fun i => \u03b1 i) a hf : f \u2208 pi s t \u22a2 support (\u2191{ toFun := fun f => DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s), inj' := (_ : Function.Injective (DFinsupp.mk s \u2218 fun f i => f \u2191i (_ : \u2191i \u2208 \u2191s))) } f) \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(\u2191{ toFun := fun f => DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s), inj' := (_ : Function.Injective (DFinsupp.mk s \u2218 fun f i => f \u2191i (_ : \u2191i \u2208 \u2191s))) } f) i \u2208 t i ** rw [Function.Embedding.coeFn_mk] ** case refine'_1.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) f : (a : \u03b9) \u2192 a \u2208 s \u2192 (fun i => \u03b1 i) a hf : f \u2208 pi s t \u22a2 support (DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s)) \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s)) i \u2208 t i ** refine' \u27e8support_mk_subset, fun i hi => _\u27e9 ** case refine'_1.intro.intro \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) f : (a : \u03b9) \u2192 a \u2208 s \u2192 (fun i => \u03b1 i) a hf : f \u2208 pi s t i : \u03b9 hi : i \u2208 s \u22a2 \u2191(DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s)) i \u2208 t i ** convert mem_pi.1 hf i hi ** case h.e'_4 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) f : (a : \u03b9) \u2192 a \u2208 s \u2192 (fun i => \u03b1 i) a hf : f \u2208 pi s t i : \u03b9 hi : i \u2208 s \u22a2 \u2191(DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s)) i = f i hi ** exact mk_of_mem hi ** case refine'_2 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 f : \u03a0\u2080 (i : \u03b9), \u03b1 i t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) \u22a2 (support f \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i) \u2192 \u2203 a, a \u2208 pi s t \u2227 \u2191{ toFun := fun f => DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s), inj' := (_ : Function.Injective (DFinsupp.mk s \u2218 fun f i => f \u2191i (_ : \u2191i \u2208 \u2191s))) } a = f ** refine' fun h => \u27e8fun i _ => f i, mem_pi.2 h.2, _\u27e9 ** case refine'_2 \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 f : \u03a0\u2080 (i : \u03b9), \u03b1 i t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) h : support f \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i \u22a2 (\u2191{ toFun := fun f => DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s), inj' := (_ : Function.Injective (DFinsupp.mk s \u2218 fun f i => f \u2191i (_ : \u2191i \u2208 \u2191s))) } fun i x => \u2191f i) = f ** ext i ** case refine'_2.h \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 f : \u03a0\u2080 (i : \u03b9), \u03b1 i t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) h : support f \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i i : \u03b9 \u22a2 \u2191(\u2191{ toFun := fun f => DFinsupp.mk s fun i => f \u2191i (_ : \u2191i \u2208 \u2191s), inj' := (_ : Function.Injective (DFinsupp.mk s \u2218 fun f i => f \u2191i (_ : \u2191i \u2208 \u2191s))) } fun i x => \u2191f i) i = \u2191f i ** dsimp ** case refine'_2.h \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : DecidableEq \u03b9 inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) s : Finset \u03b9 f : \u03a0\u2080 (i : \u03b9), \u03b1 i t : (i : \u03b9) \u2192 Finset (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) h : support f \u2286 s \u2227 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191f i \u2208 t i i : \u03b9 \u22a2 (if i \u2208 s then \u2191f i else 0) = \u2191f i ** exact ite_eq_left_iff.2 fun hi => (not_mem_support_iff.1 fun H => hi <| h.1 H).symm ** Qed", + "informal": "" + }, + { + "formal": "aemeasurable_add_measure_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 \u22a2 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f ** rw [\u2190 sum_cond, aemeasurable_sum_measure_iff, Bool.forall_bool, and_comm] ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 \u22a2 AEMeasurable f \u2227 AEMeasurable f \u2194 AEMeasurable f \u2227 AEMeasurable f ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Wbtw.sameRay_vsub_left ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : StrictOrderedCommRing R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P h : Wbtw R x y z \u22a2 SameRay R (y -\u1d65 x) (z -\u1d65 x) ** rcases h with \u27e8t, \u27e8ht0, _\u27e9, rfl\u27e9 ** case intro.intro.intro R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : StrictOrderedCommRing R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x z : P t : R ht0 : 0 \u2264 t right\u271d : t \u2264 1 \u22a2 SameRay R (\u2191(lineMap x z) t -\u1d65 x) (z -\u1d65 x) ** simpa [lineMap_apply] using SameRay.sameRay_nonneg_smul_left (z -\u1d65 x) ht0 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.tendsto_lintegral_of_dominated_convergence' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)) ** have : \u2200 n, \u222b\u207b a, F n a \u2202\u03bc = \u222b\u207b a, (hF_meas n).mk (F n) a \u2202\u03bc := fun n =>\n lintegral_congr_ae (hF_meas n).ae_eq_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), F n a \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)) ** simp_rw [this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc \u22a2 Tendsto (fun n => \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc) atTop (\ud835\udcdd (\u222b\u207b (a : \u03b1), f a \u2202\u03bc)) ** apply\n tendsto_lintegral_of_dominated_convergence bound (fun n => (hF_meas n).measurable_mk) _ h_fin ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a)) ** have : \u2200 n, \u2200\u1d50 a \u2202\u03bc, (hF_meas n).mk (F n) a = F n a := fun n => (hF_meas n).ae_eq_mk.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this\u271d : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc this : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a)) ** have : \u2200\u1d50 a \u2202\u03bc, \u2200 n, (hF_meas n).mk (F n) a = F n a := ae_all_iff.mpr this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this\u271d\u00b9 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc this\u271d : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a this : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a \u22a2 \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a)) ** filter_upwards [this, h_lim] with a H H' ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this\u271d\u00b9 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc this\u271d : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a this : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a a : \u03b1 H : \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a H' : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) \u22a2 Tendsto (fun n => AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a) atTop (\ud835\udcdd (f a)) ** simp_rw [H] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this\u271d\u00b9 : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc this\u271d : \u2200 (n : \u2115), \u2200\u1d50 (a : \u03b1) \u2202\u03bc, AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a this : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a a : \u03b1 H : \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a = F n a H' : Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) \u22a2 Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) ** exact H' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc \u22a2 \u2200 (n : \u2115), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) \u2264\u1d50[\u03bc] bound ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc n : \u2115 \u22a2 AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) \u2264\u1d50[\u03bc] bound ** filter_upwards [h_bound n, (hF_meas n).ae_eq_mk] with a H H' ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 F : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e f bound : \u03b1 \u2192 \u211d\u22650\u221e hF_meas : \u2200 (n : \u2115), AEMeasurable (F n) h_bound : \u2200 (n : \u2115), F n \u2264\u1d50[\u03bc] bound h_fin : \u222b\u207b (a : \u03b1), bound a \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b1) \u2202\u03bc, Tendsto (fun n => F n a) atTop (\ud835\udcdd (f a)) this : \u2200 (n : \u2115), \u222b\u207b (a : \u03b1), F n a \u2202\u03bc = \u222b\u207b (a : \u03b1), AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2202\u03bc n : \u2115 a : \u03b1 H : F n a \u2264 bound a H' : F n a = AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u22a2 AEMeasurable.mk (F n) (_ : AEMeasurable (F n)) a \u2264 bound a ** rwa [H'] at H ** Qed", + "informal": "" + }, + { + "formal": "Nat.frequently_odd ** \u22a2 \u2203\u1da0 (m : \u2115) in atTop, Odd m ** simpa only [odd_iff] using frequently_mod_eq one_lt_two ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.integral_eq_lintegral_pos_part_sub_lintegral_neg_part ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** let f\u2081 := hf.toL1 f ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** have eq\u2081 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 := by\n rw [L1.norm_def]\n congr 1\n apply lintegral_congr_ae\n filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n rw [h\u2081, h\u2082, ENNReal.ofReal]\n congr 1\n apply NNReal.eq\n rw [Real.nnnorm_of_nonneg (le_max_right _ _)]\n rw [Real.coe_toNNReal', NNReal.coe_mk] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** have eq\u2082 : ENNReal.toReal (\u222b\u207b a, ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 := by\n rw [L1.norm_def]\n congr 1\n apply lintegral_congr_ae\n filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082\n rw [h\u2081, h\u2082, ENNReal.ofReal]\n congr 1\n apply NNReal.eq\n simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg]\n rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) ** rw [eq\u2081, eq\u2082, integral, dif_pos, dif_pos] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 L1.integral (Integrable.toL1 (fun a => f a) ?hc) = \u2016Lp.posPart f\u2081\u2016 - \u2016Lp.negPart f\u2081\u2016 case hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 Integrable fun a => f a case hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 CompleteSpace \u211d case hc \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 eq\u2082 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 \u22a2 CompleteSpace \u211d ** exact L1.integral_eq_norm_posPart_sub _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 ** rw [L1.norm_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc) ** congr 1 ** case e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a \u2202\u03bc ** apply lintegral_congr_ae ** case e_a.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf \u22a2 (fun a => ENNReal.ofReal (f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u2016\u208a ** filter_upwards [Lp.coeFn_posPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 ENNReal.ofReal (f a\u271d) = \u2191\u2016\u2191\u2191(Lp.posPart f\u2081) a\u271d\u2016\u208a ** rw [h\u2081, h\u2082, ENNReal.ofReal] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a ** congr 1 ** case h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 Real.toNNReal (f a\u271d) = \u2016max (f a\u271d) 0\u2016\u208a ** apply NNReal.eq ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191\u2016max (f a\u271d) 0\u2016\u208a ** rw [Real.nnnorm_of_nonneg (le_max_right _ _)] ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.posPart f\u2081) a\u271d = max (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (f a\u271d)) = \u2191{ val := max (f a\u271d) 0, property := (_ : 0 \u2264 max (f a\u271d) 0) } ** rw [Real.coe_toNNReal', NNReal.coe_mk] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = \u2016Lp.negPart f\u2081\u2016 ** rw [L1.norm_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc) ** congr 1 ** case e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a \u2202\u03bc ** apply lintegral_congr_ae ** case e_a.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 \u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1d50[\u03bc] fun a => \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u2016\u208a ** filter_upwards [Lp.coeFn_negPart f\u2081, hf.coeFn_toL1] with _ h\u2081 h\u2082 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 ENNReal.ofReal (-f a\u271d) = \u2191\u2016\u2191\u2191(Lp.negPart f\u2081) a\u271d\u2016\u208a ** rw [h\u2081, h\u2082, ENNReal.ofReal] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a ** congr 1 ** case h.e_a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 Real.toNNReal (-f a\u271d) = \u2016-min (f a\u271d) 0\u2016\u208a ** apply NNReal.eq ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 \u2191(Real.toNNReal (-f a\u271d)) = \u2191\u2016-min (f a\u271d) 0\u2016\u208a ** simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg] ** case h.e_a.a \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : Integrable f f\u2081 : { x // x \u2208 Lp \u211d 1 } := Integrable.toL1 f hf eq\u2081 : ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) = \u2016Lp.posPart f\u2081\u2016 a\u271d : \u03b1 h\u2081 : \u2191\u2191(Lp.negPart f\u2081) a\u271d = -min (\u2191\u2191f\u2081 a\u271d) 0 h\u2082 : \u2191\u2191(Integrable.toL1 f hf) a\u271d = f a\u271d \u22a2 max (-f a\u271d) 0 = \u2016min (f a\u271d) 0\u2016 ** rw [Real.norm_of_nonpos (min_le_right _ _), \u2190 max_neg_neg, neg_zero] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.GlueData.t'_iji ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v, u\u2081} C C' : Type u\u2082 inst\u271d : Category.{v, u\u2082} C' D : GlueData C i j : D.J \u22a2 t' D i j i = pullback.fst \u226b t D i j \u226b inv pullback.snd ** rw [\u2190 Category.assoc, \u2190 D.t_fac] ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v, u\u2081} C C' : Type u\u2082 inst\u271d : Category.{v, u\u2082} C' D : GlueData C i j : D.J \u22a2 t' D i j i = (t' D i j i \u226b pullback.snd) \u226b inv pullback.snd ** simp ** Qed", + "informal": "" + }, + { + "formal": "Tuple.eq_sort_iff' ** n : \u2115 \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f : Fin n \u2192 \u03b1 \u03c3 : Equiv.Perm (Fin n) \u22a2 \u03c3 = sort f \u2194 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) ** constructor <;> intro h ** case mp n : \u2115 \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f : Fin n \u2192 \u03b1 \u03c3 : Equiv.Perm (Fin n) h : \u03c3 = sort f \u22a2 StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) ** rw [h, sort, Equiv.trans_assoc, Equiv.symm_trans_self] ** case mp n : \u2115 \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f : Fin n \u2192 \u03b1 \u03c3 : Equiv.Perm (Fin n) h : \u03c3 = sort f \u22a2 StrictMono \u2191((graphEquiv\u2082 f).trans (Equiv.refl { x // x \u2208 graph f })) ** exact (graphEquiv\u2082 f).strictMono ** case mpr n : \u2115 \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f : Fin n \u2192 \u03b1 \u03c3 : Equiv.Perm (Fin n) h : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) \u22a2 \u03c3 = sort f ** have := Subsingleton.elim (graphEquiv\u2082 f) (h.orderIsoOfSurjective _ <| Equiv.surjective _) ** case mpr n : \u2115 \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f : Fin n \u2192 \u03b1 \u03c3 : Equiv.Perm (Fin n) h : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) this : graphEquiv\u2082 f = StrictMono.orderIsoOfSurjective (\u2191(\u03c3.trans (graphEquiv\u2081 f))) h (_ : Function.Surjective \u2191(\u03c3.trans (graphEquiv\u2081 f))) \u22a2 \u03c3 = sort f ** ext1 x ** case mpr.H n : \u2115 \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 f : Fin n \u2192 \u03b1 \u03c3 : Equiv.Perm (Fin n) h : StrictMono \u2191(\u03c3.trans (graphEquiv\u2081 f)) this : graphEquiv\u2082 f = StrictMono.orderIsoOfSurjective (\u2191(\u03c3.trans (graphEquiv\u2081 f))) h (_ : Function.Surjective \u2191(\u03c3.trans (graphEquiv\u2081 f))) x : Fin n \u22a2 \u2191\u03c3 x = \u2191(sort f) x ** exact (graphEquiv\u2081 f).apply_eq_iff_eq_symm_apply.1 (FunLike.congr_fun this x).symm ** Qed", + "informal": "" + }, + { + "formal": "Nat.prod_div_divisors ** n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 \u22a2 \u220f d in divisors n, f (n / d) = Finset.prod (divisors n) f ** by_cases hn : n = 0 ** case neg n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : \u00acn = 0 \u22a2 \u220f d in divisors n, f (n / d) = Finset.prod (divisors n) f ** rw [\u2190 prod_image] ** case pos n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : n = 0 \u22a2 \u220f d in divisors n, f (n / d) = Finset.prod (divisors n) f ** simp [hn] ** case neg n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : \u00acn = 0 \u22a2 \u220f x in image (fun d => n / d) (divisors n), f x = Finset.prod (divisors n) f ** exact prod_congr (image_div_divisors_eq_divisors n) (by simp) ** n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : \u00acn = 0 \u22a2 \u2200 (x : \u2115), x \u2208 divisors n \u2192 f x = f x ** simp ** case neg n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : \u00acn = 0 \u22a2 \u2200 (x : \u2115), x \u2208 divisors n \u2192 \u2200 (y : \u2115), y \u2208 divisors n \u2192 n / x = n / y \u2192 x = y ** intro x hx y hy h ** case neg n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : \u00acn = 0 x : \u2115 hx : x \u2208 divisors n y : \u2115 hy : y \u2208 divisors n h : n / x = n / y \u22a2 x = y ** rw [mem_divisors] at hx hy ** case neg n\u271d : \u2115 \u03b1 : Type u_1 inst\u271d : CommMonoid \u03b1 n : \u2115 f : \u2115 \u2192 \u03b1 hn : \u00acn = 0 x : \u2115 hx : x \u2223 n \u2227 n \u2260 0 y : \u2115 hy : y \u2223 n \u2227 n \u2260 0 h : n / x = n / y \u22a2 x = y ** exact (div_eq_iff_eq_of_dvd_dvd hn hx.1 hy.1).mp h ** Qed", + "informal": "" + }, + { + "formal": "Function.fixedPoints_id ** \u03b1 : Type u \u03b2 : Type v f fa g : \u03b1 \u2192 \u03b1 x y : \u03b1 fb : \u03b2 \u2192 \u03b2 m n k : \u2115 e : Perm \u03b1 x\u271d : \u03b1 \u22a2 x\u271d \u2208 fixedPoints id \u2194 x\u271d \u2208 Set.univ ** simpa using isFixedPt_id _ ** Qed", + "informal": "" + }, + { + "formal": "EMetric.diam_pos_iff ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b1 \u03b3 : Type w inst\u271d : EMetricSpace \u03b3 s : Set \u03b3 \u22a2 0 < diam s \u2194 Set.Nontrivial s ** simp only [pos_iff_ne_zero, Ne.def, diam_eq_zero_iff, Set.not_subsingleton_iff] ** Qed", + "informal": "" + }, + { + "formal": "UniqueFactorizationMonoid.count_normalizedFactors_eq' ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : Nontrivial R inst\u271d\u00b9 : NormalizationMonoid R dec_dvd : DecidableRel Dvd.dvd inst\u271d : DecidableEq R p x : R hp : p = 0 \u2228 Irreducible p hnorm : \u2191normalize p = p n : \u2115 hle : p ^ n \u2223 x hlt : \u00acp ^ (n + 1) \u2223 x \u22a2 count p (normalizedFactors x) = n ** rcases hp with (rfl | hp) ** case inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : Nontrivial R inst\u271d\u00b9 : NormalizationMonoid R dec_dvd : DecidableRel Dvd.dvd inst\u271d : DecidableEq R x : R n : \u2115 hnorm : \u2191normalize 0 = 0 hle : 0 ^ n \u2223 x hlt : \u00ac0 ^ (n + 1) \u2223 x \u22a2 count 0 (normalizedFactors x) = n ** cases n ** case inl.zero \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : Nontrivial R inst\u271d\u00b9 : NormalizationMonoid R dec_dvd : DecidableRel Dvd.dvd inst\u271d : DecidableEq R x : R hnorm : \u2191normalize 0 = 0 hle : 0 ^ Nat.zero \u2223 x hlt : \u00ac0 ^ (Nat.zero + 1) \u2223 x \u22a2 count 0 (normalizedFactors x) = Nat.zero ** exact count_eq_zero.2 (zero_not_mem_normalizedFactors _) ** case inl.succ \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : Nontrivial R inst\u271d\u00b9 : NormalizationMonoid R dec_dvd : DecidableRel Dvd.dvd inst\u271d : DecidableEq R x : R hnorm : \u2191normalize 0 = 0 n\u271d : \u2115 hle : 0 ^ Nat.succ n\u271d \u2223 x hlt : \u00ac0 ^ (Nat.succ n\u271d + 1) \u2223 x \u22a2 count 0 (normalizedFactors x) = Nat.succ n\u271d ** rw [zero_pow (Nat.succ_pos _)] at hle hlt ** case inl.succ \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : Nontrivial R inst\u271d\u00b9 : NormalizationMonoid R dec_dvd : DecidableRel Dvd.dvd inst\u271d : DecidableEq R x : R hnorm : \u2191normalize 0 = 0 n\u271d : \u2115 hle : 0 \u2223 x hlt : \u00ac0 \u2223 x \u22a2 count 0 (normalizedFactors x) = Nat.succ n\u271d ** exact absurd hle hlt ** case inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u2074 : CancelCommMonoidWithZero R inst\u271d\u00b3 : UniqueFactorizationMonoid R inst\u271d\u00b2 : Nontrivial R inst\u271d\u00b9 : NormalizationMonoid R dec_dvd : DecidableRel Dvd.dvd inst\u271d : DecidableEq R p x : R hnorm : \u2191normalize p = p n : \u2115 hle : p ^ n \u2223 x hlt : \u00acp ^ (n + 1) \u2223 x hp : Irreducible p \u22a2 count p (normalizedFactors x) = n ** exact count_normalizedFactors_eq hp hnorm hle hlt ** Qed", + "informal": "" + }, + { + "formal": "Set.Finite.lowerClosure ** \u03b1 : Type u_1 inst\u271d\u00b9 : Preorder \u03b1 s : Set \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 hs : Set.Finite s \u22a2 Set.Finite \u2191(lowerClosure s) ** rw [coe_lowerClosure] ** \u03b1 : Type u_1 inst\u271d\u00b9 : Preorder \u03b1 s : Set \u03b1 inst\u271d : LocallyFiniteOrderBot \u03b1 hs : Set.Finite s \u22a2 Set.Finite (\u22c3 a \u2208 s, Iic a) ** exact hs.biUnion fun _ _ => finite_Iic _ ** Qed", + "informal": "" + }, + { + "formal": "CauchyFilter.comp_gen ** \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 (Filter.lift' (Filter.lift' (\ud835\udce4 \u03b1) gen) fun s => s \u25cb s) = Filter.lift' (\ud835\udce4 \u03b1) fun s => gen s \u25cb gen s ** rw [lift'_lift'_assoc] ** case hg \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 Monotone gen ** exact monotone_gen ** case hh \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 Monotone fun s => s \u25cb s ** exact monotone_id.compRel monotone_id ** \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 (Filter.lift' (\ud835\udce4 \u03b1) fun s => gen (s \u25cb s)) = Filter.lift' (Filter.lift' (\ud835\udce4 \u03b1) fun s => s \u25cb s) gen ** rw [lift'_lift'_assoc] ** case hg \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 Monotone fun s => s \u25cb s ** exact monotone_id.compRel monotone_id ** case hh \u03b1 : Type u inst\u271d\u00b2 : UniformSpace \u03b1 \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : UniformSpace \u03b2 inst\u271d : UniformSpace \u03b3 \u22a2 Monotone gen ** exact monotone_gen ** Qed", + "informal": "" + }, + { + "formal": "SmoothBumpFunction.isClosed_image_of_isClosed ** E : Type uE inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : FiniteDimensional \u211d E H : Type uH inst\u271d\u00b3 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M s : Set M hsc : IsClosed s hs : s \u2286 support \u2191f \u22a2 IsClosed (\u2191(extChartAt I c) '' s) ** rw [f.image_eq_inter_preimage_of_subset_support hs] ** E : Type uE inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : FiniteDimensional \u211d E H : Type uH inst\u271d\u00b3 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M s : Set M hsc : IsClosed s hs : s \u2286 support \u2191f \u22a2 IsClosed (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I \u2229 \u2191(LocalEquiv.symm (extChartAt I c)) \u207b\u00b9' s) ** refine' ContinuousOn.preimage_closed_of_closed\n ((continuousOn_extChartAt_symm _ _).mono f.closedBall_subset) _ hsc ** E : Type uE inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : FiniteDimensional \u211d E H : Type uH inst\u271d\u00b3 : TopologicalSpace H I : ModelWithCorners \u211d E H M : Type uM inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : SmoothManifoldWithCorners I M c : M f : SmoothBumpFunction I c x : M s : Set M hsc : IsClosed s hs : s \u2286 support \u2191f \u22a2 IsClosed (closedBall (\u2191(extChartAt I c) c) f.rOut \u2229 range \u2191I) ** exact IsClosed.inter isClosed_ball I.closed_range ** Qed", + "informal": "" + }, + { + "formal": "Profinite.exists_locallyConstant_finite_nonempty ** J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 \u22a2 \u2203 j g, f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) g ** inhabit \u03b1 ** J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 \u22a2 \u2203 j g, f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) g ** obtain \u27e8j, gg, h\u27e9 := exists_locallyConstant_finite_aux _ hC f ** case intro.intro J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u22a2 \u2203 j g, f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) g ** let \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 ** case intro.intro J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u22a2 \u2203 j g, f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) g ** let \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a : \u03b1, \u03b9 a = f then h.choose else default ** case intro.intro J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default \u22a2 \u2203 j g, f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) g ** refine' \u27e8j, gg.map \u03c3, _\u27e9 ** case intro.intro J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default \u22a2 f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) (LocallyConstant.map \u03c3 gg) ** ext x ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop \u22a2 \u2191f x = \u2191(LocallyConstant.comap (\u2191(C.\u03c0.app j)) (LocallyConstant.map \u03c3 gg)) x ** erw [LocallyConstant.coe_comap _ _ (C.\u03c0.app j).continuous] ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop \u22a2 \u2191f x = (\u2191(LocallyConstant.map \u03c3 gg) \u2218 \u2191(C.\u03c0.app j)) x ** dsimp ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop \u22a2 \u2191f x = if h : \u2203 a, (fun b => if a = b then 0 else 1) = \u2191gg (\u2191(C.\u03c0.app j) x) then Exists.choose h else default ** have h1 : \u03b9 (f x) = gg (C.\u03c0.app j x) := by\n change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _\n erw [h, LocallyConstant.coe_comap _ _ (C.\u03c0.app j).continuous]\n rfl ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) \u22a2 \u2191f x = if h : \u2203 a, (fun b => if a = b then 0 else 1) = \u2191gg (\u2191(C.\u03c0.app j) x) then Exists.choose h else default ** have h2 : \u2203 a : \u03b1, \u03b9 a = gg (C.\u03c0.app j x) := \u27e8f x, h1\u27e9 ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) \u22a2 \u2191f x = if h : \u2203 a, (fun b => if a = b then 0 else 1) = \u2191gg (\u2191(C.\u03c0.app j) x) then Exists.choose h else default ** erw [dif_pos h2] ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) \u22a2 \u2191f x = Exists.choose h2 ** apply_fun \u03b9 ** J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop \u22a2 \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) ** change f.map (fun a b => if a = b then (0 : Fin 2) else 1) x = _ ** J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop \u22a2 \u2191(LocallyConstant.map (fun a b => if a = b then 0 else 1) f) x = \u2191gg (\u2191(C.\u03c0.app j) x) ** erw [h, LocallyConstant.coe_comap _ _ (C.\u03c0.app j).continuous] ** J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop \u22a2 (\u2191gg \u2218 \u2191(C.\u03c0.app j)) x = \u2191gg (\u2191(C.\u03c0.app j) x) ** rfl ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) \u22a2 \u03b9 (\u2191f x) = \u03b9 (Exists.choose h2) ** rw [h2.choose_spec] ** case intro.intro.h J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) \u22a2 \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) ** exact h1 ** case intro.intro.h.inj J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) \u22a2 Function.Injective \u03b9 ** intro a b hh ** case intro.intro.h.inj J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) a b : \u03b1 hh : \u03b9 a = \u03b9 b \u22a2 a = b ** have hhh := congr_fun hh a ** case intro.intro.h.inj J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) a b : \u03b1 hh : \u03b9 a = \u03b9 b hhh : \u03b9 a a = \u03b9 b a \u22a2 a = b ** dsimp at hhh ** case intro.intro.h.inj J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) a b : \u03b1 hh : \u03b9 a = \u03b9 b hhh : (if a = a then 0 else 1) = if b = a then 0 else 1 \u22a2 a = b ** rw [if_pos rfl] at hhh ** case intro.intro.h.inj J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) a b : \u03b1 hh : \u03b9 a = \u03b9 b hhh : 0 = if b = a then 0 else 1 \u22a2 a = b ** split_ifs at hhh with hh1 ** case pos J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) a b : \u03b1 hh : \u03b9 a = \u03b9 b hh1 : b = a hhh : 0 = 0 \u22a2 a = b ** exact hh1.symm ** case neg J : Type u inst\u271d\u00b3 : SmallCategory J inst\u271d\u00b2 : IsCofiltered J F : J \u2964 Profinite C : Cone F \u03b1 : Type u_1 inst\u271d\u00b9 : Finite \u03b1 inst\u271d : Nonempty \u03b1 hC : IsLimit C f : LocallyConstant (\u2191C.pt.toCompHaus.toTop) \u03b1 inhabited_h : Inhabited \u03b1 j : J gg : LocallyConstant (\u2191(F.obj j).toCompHaus.toTop) (\u03b1 \u2192 Fin 2) h : LocallyConstant.map (fun a b => if a = b then 0 else 1) f = LocallyConstant.comap (\u2191(C.\u03c0.app j)) gg \u03b9 : \u03b1 \u2192 \u03b1 \u2192 Fin 2 := fun a b => if a = b then 0 else 1 \u03c3 : (\u03b1 \u2192 Fin 2) \u2192 \u03b1 := fun f => if h : \u2203 a, \u03b9 a = f then Exists.choose h else default x : \u2191C.pt.toCompHaus.toTop h1 : \u03b9 (\u2191f x) = \u2191gg (\u2191(C.\u03c0.app j) x) h2 : \u2203 a, \u03b9 a = \u2191gg (\u2191(C.\u03c0.app j) x) a b : \u03b1 hh : \u03b9 a = \u03b9 b hh1 : \u00acb = a hhh : 0 = 1 \u22a2 a = b ** exact False.elim (bot_ne_top hhh) ** Qed", + "informal": "" + }, + { + "formal": "Int.neg_ediv_of_dvd ** b c : Int \u22a2 -(b * c) / b = -(b * c / b) ** if bz : b = 0 then simp [bz] else\nrw [Int.neg_mul_eq_mul_neg, Int.mul_ediv_cancel_left _ bz, Int.mul_ediv_cancel_left _ bz] ** b c : Int bz : b = 0 \u22a2 -(b * c) / b = -(b * c / b) ** simp [bz] ** b c : Int bz : \u00acb = 0 \u22a2 -(b * c) / b = -(b * c / b) ** rw [Int.neg_mul_eq_mul_neg, Int.mul_ediv_cancel_left _ bz, Int.mul_ediv_cancel_left _ bz] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Idempotents.whiskeringLeft_obj_preimage_app ** C : Type u_1 D : Type u_2 E : Type u_3 inst\u271d\u00b3 : Category.{u_4, u_1} C inst\u271d\u00b2 : Category.{u_5, u_2} D inst\u271d\u00b9 : Category.{?u.88402, u_3} E inst\u271d : IsIdempotentComplete D F G : Karoubi C \u2964 D \u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G P : Karoubi C \u22a2 (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4).app P = F.map (decompId_i P) \u226b \u03c4.app P.X \u226b G.map (decompId_p P) ** rw [natTrans_eq] ** C : Type u_1 D : Type u_2 E : Type u_3 inst\u271d\u00b3 : Category.{u_4, u_1} C inst\u271d\u00b2 : Category.{u_5, u_2} D inst\u271d\u00b9 : Category.{?u.88402, u_3} E inst\u271d : IsIdempotentComplete D F G : Karoubi C \u2964 D \u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G P : Karoubi C \u22a2 F.map (decompId_i P) \u226b (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4).app (Karoubi.mk P.X (\ud835\udfd9 P.X)) \u226b G.map (decompId_p P) = F.map (decompId_i P) \u226b \u03c4.app P.X \u226b G.map (decompId_p P) ** congr 2 ** case e_a.e_a C : Type u_1 D : Type u_2 E : Type u_3 inst\u271d\u00b3 : Category.{u_4, u_1} C inst\u271d\u00b2 : Category.{u_5, u_2} D inst\u271d\u00b9 : Category.{?u.88402, u_3} E inst\u271d : IsIdempotentComplete D F G : Karoubi C \u2964 D \u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G P : Karoubi C \u22a2 (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4).app (Karoubi.mk P.X (\ud835\udfd9 P.X)) = \u03c4.app P.X ** rw [\u2190 congr_app (((whiskeringLeft _ _ _).obj (toKaroubi _)).image_preimage \u03c4) P.X] ** case e_a.e_a C : Type u_1 D : Type u_2 E : Type u_3 inst\u271d\u00b3 : Category.{u_4, u_1} C inst\u271d\u00b2 : Category.{u_5, u_2} D inst\u271d\u00b9 : Category.{?u.88402, u_3} E inst\u271d : IsIdempotentComplete D F G : Karoubi C \u2964 D \u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G P : Karoubi C \u22a2 (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4).app (Karoubi.mk P.X (\ud835\udfd9 P.X)) = (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).map (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4)).app P.X ** dsimp ** case e_a.e_a C : Type u_1 D : Type u_2 E : Type u_3 inst\u271d\u00b3 : Category.{u_4, u_1} C inst\u271d\u00b2 : Category.{u_5, u_2} D inst\u271d\u00b9 : Category.{?u.88402, u_3} E inst\u271d : IsIdempotentComplete D F G : Karoubi C \u2964 D \u03c4 : toKaroubi C \u22d9 F \u27f6 toKaroubi C \u22d9 G P : Karoubi C \u22a2 (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4).app (Karoubi.mk P.X (\ud835\udfd9 P.X)) = (((whiskeringLeft C (Karoubi C) D).obj (toKaroubi C)).preimage \u03c4).app ((toKaroubi C).obj P.X) ** congr ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi ** V : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V P : Type u_2 inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d p : P h : Cospherical {a, b, c, d} hapb : \u2220 a p b = \u03c0 hcpd : \u2220 c p d = \u03c0 \u22a2 dist a p * dist b p = dist c p * dist d p ** obtain \u27e8-, k\u2081, _, hab\u27e9 := angle_eq_pi_iff.mp hapb ** case intro.intro.intro V : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V P : Type u_2 inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d p : P h : Cospherical {a, b, c, d} hapb : \u2220 a p b = \u03c0 hcpd : \u2220 c p d = \u03c0 k\u2081 : \u211d left\u271d : k\u2081 < 0 hab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p) \u22a2 dist a p * dist b p = dist c p * dist d p ** obtain \u27e8-, k\u2082, _, hcd\u27e9 := angle_eq_pi_iff.mp hcpd ** case intro.intro.intro.intro.intro.intro V : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V P : Type u_2 inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d p : P h : Cospherical {a, b, c, d} hapb : \u2220 a p b = \u03c0 hcpd : \u2220 c p d = \u03c0 k\u2081 : \u211d left\u271d\u00b9 : k\u2081 < 0 hab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p) k\u2082 : \u211d left\u271d : k\u2082 < 0 hcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p) \u22a2 dist a p * dist b p = dist c p * dist d p ** exact mul_dist_eq_mul_dist_of_cospherical h \u27e8k\u2081, by linarith, hab\u27e9 \u27e8k\u2082, by linarith, hcd\u27e9 ** V : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V P : Type u_2 inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d p : P h : Cospherical {a, b, c, d} hapb : \u2220 a p b = \u03c0 hcpd : \u2220 c p d = \u03c0 k\u2081 : \u211d left\u271d\u00b9 : k\u2081 < 0 hab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p) k\u2082 : \u211d left\u271d : k\u2082 < 0 hcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p) \u22a2 k\u2081 \u2260 1 ** linarith ** V : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V P : Type u_2 inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b c d p : P h : Cospherical {a, b, c, d} hapb : \u2220 a p b = \u03c0 hcpd : \u2220 c p d = \u03c0 k\u2081 : \u211d left\u271d\u00b9 : k\u2081 < 0 hab : b -\u1d65 p = k\u2081 \u2022 (a -\u1d65 p) k\u2082 : \u211d left\u271d : k\u2082 < 0 hcd : d -\u1d65 p = k\u2082 \u2022 (c -\u1d65 p) \u22a2 k\u2082 \u2260 1 ** linarith ** Qed", + "informal": "" + }, + { + "formal": "compare_eq_iff_eq ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 \u22a2 compare a b = Ordering.eq \u2194 a = b ** rw [LinearOrder.compare_eq_compareOfLessAndEq, compareOfLessAndEq] ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 \u22a2 (if a < b then Ordering.lt else if a = b then Ordering.eq else Ordering.gt) = Ordering.eq \u2194 a = b ** split_ifs <;> try simp only [] ** case pos \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d : a < b \u22a2 False \u2194 a = b case pos \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : a = b \u22a2 True \u2194 a = b case neg \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : \u00aca = b \u22a2 False \u2194 a = b ** case _ h => exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h ** case pos \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : a = b \u22a2 True \u2194 a = b case neg \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : \u00aca = b \u22a2 False \u2194 a = b ** case _ _ h => exact true_iff_iff.2 h ** case neg \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : \u00aca = b \u22a2 False \u2194 a = b ** case _ _ h => exact false_iff_iff.2 h ** case pos \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d\u00b9 : \u00aca < b h\u271d : a = b \u22a2 Ordering.eq = Ordering.eq \u2194 a = b ** simp only [] ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h : a < b \u22a2 False \u2194 a = b ** exact false_iff_iff.2 <| ne_iff_lt_or_gt.2 <| .inl h ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d : \u00aca < b h : a = b \u22a2 True \u2194 a = b ** exact true_iff_iff.2 h ** \u03b1 : Type u inst\u271d : LinearOrder \u03b1 a b : \u03b1 h\u271d : \u00aca < b h : \u00aca = b \u22a2 False \u2194 a = b ** exact false_iff_iff.2 h ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearMap.op_nnnorm_le_bound' ** \ud835\udd5c : Type u_1 \ud835\udd5c\u2082 : Type u_2 \ud835\udd5c\u2083 : Type u_3 E : Type u_4 E\u2097 : Type u_5 F : Type u_6 F\u2097 : Type u_7 G : Type u_8 G\u2097 : Type u_9 \ud835\udcd5 : Type u_10 inst\u271d\u00b9\u2077 : SeminormedAddCommGroup E inst\u271d\u00b9\u2076 : SeminormedAddCommGroup E\u2097 inst\u271d\u00b9\u2075 : SeminormedAddCommGroup F inst\u271d\u00b9\u2074 : SeminormedAddCommGroup F\u2097 inst\u271d\u00b9\u00b3 : SeminormedAddCommGroup G inst\u271d\u00b9\u00b2 : SeminormedAddCommGroup G\u2097 inst\u271d\u00b9\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c\u2082 inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c\u2083 inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : NormedSpace \ud835\udd5c E\u2097 inst\u271d\u2076 : NormedSpace \ud835\udd5c\u2082 F inst\u271d\u2075 : NormedSpace \ud835\udd5c F\u2097 inst\u271d\u2074 : NormedSpace \ud835\udd5c\u2083 G inst\u271d\u00b3 : NormedSpace \ud835\udd5c G\u2097 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b2 : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 inst\u271d\u00b9 : RingHomIsometric \u03c3\u2081\u2082 inst\u271d : RingHomIsometric \u03c3\u2082\u2083 f\u271d g : E \u2192SL[\u03c3\u2081\u2082] F h : F \u2192SL[\u03c3\u2082\u2083] G x\u271d : E f : E \u2192SL[\u03c3\u2081\u2082] F M : \u211d\u22650 hM : \u2200 (x : E), \u2016x\u2016\u208a \u2260 0 \u2192 \u2016\u2191f x\u2016\u208a \u2264 M * \u2016x\u2016\u208a x : E hx : \u2016x\u2016 \u2260 0 \u22a2 \u2016x\u2016\u208a \u2260 0 ** rwa [\u2190 NNReal.coe_ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "IsCoprime.of_mul_add_left_right ** R : Type u inst\u271d : CommSemiring R x y z : R h : IsCoprime x (x * z + y) \u22a2 IsCoprime x y ** rw [add_comm] at h ** R : Type u inst\u271d : CommSemiring R x y z : R h : IsCoprime x (y + x * z) \u22a2 IsCoprime x y ** exact h.of_add_mul_left_right ** Qed", + "informal": "" + }, + { + "formal": "Submodule.piQuotientLift_single ** \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i \u22a2 \u2191(piQuotientLift p q f hf) (Pi.single i x) = \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) x ** simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply] ** \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i \u22a2 (Finset.sum Finset.univ fun x_1 => \u2191(mapQ (p x_1) q (f x_1) (_ : p x_1 \u2264 comap (f x_1) q)) (Pi.single i x x_1)) = \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) x ** rw [Finset.sum_eq_single i] ** \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i \u22a2 \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) (Pi.single i x i) = \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) x ** rw [Pi.single_eq_same] ** case h\u2080 \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i \u22a2 \u2200 (b : \u03b9), b \u2208 Finset.univ \u2192 b \u2260 i \u2192 \u2191(mapQ (p b) q (f b) (_ : p b \u2264 comap (f b) q)) (Pi.single i x b) = 0 ** rintro j - hj ** case h\u2080 \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i j : \u03b9 hj : j \u2260 i \u22a2 \u2191(mapQ (p j) q (f j) (_ : p j \u2264 comap (f j) q)) (Pi.single i x j) = 0 ** rw [Pi.single_eq_of_ne hj, _root_.map_zero] ** case h\u2081 \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i \u22a2 \u00aci \u2208 Finset.univ \u2192 \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) (Pi.single i x i) = 0 ** intros ** case h\u2081 \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i a\u271d : \u00aci \u2208 Finset.univ \u22a2 \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) (Pi.single i x i) = 0 ** have := Finset.mem_univ i ** case h\u2081 \u03b9 : Type u_1 R : Type u_2 inst\u271d\u2078 : CommRing R Ms : \u03b9 \u2192 Type u_3 inst\u271d\u2077 : (i : \u03b9) \u2192 AddCommGroup (Ms i) inst\u271d\u2076 : (i : \u03b9) \u2192 Module R (Ms i) N : Type u_4 inst\u271d\u2075 : AddCommGroup N inst\u271d\u2074 : Module R N Ns : \u03b9 \u2192 Type u_5 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommGroup (Ns i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (Ns i) inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 p : (i : \u03b9) \u2192 Submodule R (Ms i) q : Submodule R N f : (i : \u03b9) \u2192 Ms i \u2192\u2097[R] N hf : \u2200 (i : \u03b9), p i \u2264 comap (f i) q i : \u03b9 x : Ms i \u29f8 p i a\u271d : \u00aci \u2208 Finset.univ this : i \u2208 Finset.univ \u22a2 \u2191(mapQ (p i) q (f i) (_ : p i \u2264 comap (f i) q)) (Pi.single i x i) = 0 ** contradiction ** Qed", + "informal": "" + }, + { + "formal": "orthonormalBasis_one_dim ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d \u22a2 (\u2191b = fun x => 1) \u2228 \u2191b = fun x => -1 ** have : Unique \u03b9 := b.toBasis.unique ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this : Unique \u03b9 \u22a2 (\u2191b = fun x => 1) \u2228 \u2191b = fun x => -1 ** have : b default = 1 \u2228 b default = -1 := by\n have : \u2016b default\u2016 = 1 := b.orthonormal.1 _\n rwa [Real.norm_eq_abs, abs_eq (zero_le_one' \u211d)] at this ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this\u271d : Unique \u03b9 this : \u2191b default = 1 \u2228 \u2191b default = -1 \u22a2 (\u2191b = fun x => 1) \u2228 \u2191b = fun x => -1 ** rw [eq_const_of_unique b] ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this\u271d : Unique \u03b9 this : \u2191b default = 1 \u2228 \u2191b default = -1 \u22a2 (const \u03b9 (\u2191b default) = fun x => 1) \u2228 const \u03b9 (\u2191b default) = fun x => -1 ** refine' this.imp _ _ <;> (intro; ext; simp [*]) ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this : Unique \u03b9 \u22a2 \u2191b default = 1 \u2228 \u2191b default = -1 ** have : \u2016b default\u2016 = 1 := b.orthonormal.1 _ ** \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this\u271d : Unique \u03b9 this : \u2016\u2191b default\u2016 = 1 \u22a2 \u2191b default = 1 \u2228 \u2191b default = -1 ** rwa [Real.norm_eq_abs, abs_eq (zero_le_one' \u211d)] at this ** case refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this\u271d : Unique \u03b9 this : \u2191b default = 1 \u2228 \u2191b default = -1 \u22a2 \u2191b default = -1 \u2192 const \u03b9 (\u2191b default) = fun x => -1 ** intro ** case refine'_2 \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this\u271d : Unique \u03b9 this : \u2191b default = 1 \u2228 \u2191b default = -1 a\u271d : \u2191b default = -1 \u22a2 const \u03b9 (\u2191b default) = fun x => -1 ** ext ** case refine'_2.h \u03b9 : Type u_1 \u03b9' : Type u_2 \ud835\udd5c : Type u_3 inst\u271d\u00b9\u2070 : IsROrC \ud835\udd5c E : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : InnerProductSpace \ud835\udd5c E E' : Type u_5 inst\u271d\u2077 : NormedAddCommGroup E' inst\u271d\u2076 : InnerProductSpace \ud835\udd5c E' F : Type u_6 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : InnerProductSpace \u211d F F' : Type u_7 inst\u271d\u00b3 : NormedAddCommGroup F' inst\u271d\u00b2 : InnerProductSpace \u211d F' inst\u271d\u00b9 : Fintype \u03b9 v : Set E A : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : FiniteDimensional \ud835\udd5c E b : OrthonormalBasis \u03b9 \u211d \u211d this\u271d : Unique \u03b9 this : \u2191b default = 1 \u2228 \u2191b default = -1 a\u271d : \u2191b default = -1 x\u271d : \u03b9 \u22a2 const \u03b9 (\u2191b default) x\u271d = -1 ** simp [*] ** Qed", + "informal": "" + }, + { + "formal": "TendstoLocallyUniformlyOn.differentiableOn ** E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U \u22a2 DifferentiableOn \u2102 f U ** rintro x hx ** E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U \u22a2 DifferentiableWithinAt \u2102 f U x ** obtain \u27e8K, \u27e8hKx, hK\u27e9, hKU\u27e9 := (compact_basis_nhds x).mem_iff.mp (hU.mem_nhds hx) ** case intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u22a2 DifferentiableWithinAt \u2102 f U x ** obtain \u27e8\u03b4, _, _, h1\u27e9 := exists_cthickening_tendstoUniformlyOn hf hF hK hU hKU ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K \u22a2 DifferentiableWithinAt \u2102 f U x ** have h2 : interior K \u2286 U := interior_subset.trans hKU ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U \u22a2 DifferentiableWithinAt \u2102 f U x ** have h3 : \u2200\u1da0 n in \u03c6, DifferentiableOn \u2102 (F n) (interior K) ** case h3 E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U \u22a2 \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U h3 : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) \u22a2 DifferentiableWithinAt \u2102 f U x ** filter_upwards [hF] with n h using h.mono h2 ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U h3 : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) \u22a2 DifferentiableWithinAt \u2102 f U x ** have h4 : TendstoLocallyUniformlyOn F f \u03c6 (interior K) := hf.mono h2 ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U h3 : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) h4 : TendstoLocallyUniformlyOn F f \u03c6 (interior K) \u22a2 DifferentiableWithinAt \u2102 f U x ** have h5 : TendstoLocallyUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 (interior K) :=\n h1.tendstoLocallyUniformlyOn.mono interior_subset ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U h3 : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) h4 : TendstoLocallyUniformlyOn F f \u03c6 (interior K) h5 : TendstoLocallyUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 (interior K) \u22a2 DifferentiableWithinAt \u2102 f U x ** have h6 : \u2200 x \u2208 interior K, HasDerivAt f (cderiv \u03b4 f x) x := fun x h =>\n hasDerivAt_of_tendsto_locally_uniformly_on' isOpen_interior h5 h3 (fun _ => h4.tendsto_at) h ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U h3 : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) h4 : TendstoLocallyUniformlyOn F f \u03c6 (interior K) h5 : TendstoLocallyUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 (interior K) h6 : \u2200 (x : \u2102), x \u2208 interior K \u2192 HasDerivAt f (cderiv \u03b4 f x) x \u22a2 DifferentiableWithinAt \u2102 f U x ** have h7 : DifferentiableOn \u2102 f (interior K) := fun x hx =>\n (h6 x hx).differentiableAt.differentiableWithinAt ** case intro.intro.intro.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E inst\u271d\u00b9 : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E inst\u271d : NeBot \u03c6 hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U x : \u2102 hx : x \u2208 U K : Set \u2102 hKU : K \u2286 U hKx : K \u2208 \ud835\udcdd x hK : IsCompact K \u03b4 : \u211d left\u271d\u00b9 : \u03b4 > 0 left\u271d : cthickening \u03b4 K \u2286 U h1 : TendstoUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K h2 : interior K \u2286 U h3 : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) (interior K) h4 : TendstoLocallyUniformlyOn F f \u03c6 (interior K) h5 : TendstoLocallyUniformlyOn (deriv \u2218 F) (cderiv \u03b4 f) \u03c6 (interior K) h6 : \u2200 (x : \u2102), x \u2208 interior K \u2192 HasDerivAt f (cderiv \u03b4 f x) x h7 : DifferentiableOn \u2102 f (interior K) \u22a2 DifferentiableWithinAt \u2102 f U x ** exact (h7.differentiableAt (interior_mem_nhds.mpr hKx)).differentiableWithinAt ** Qed", + "informal": "" + }, + { + "formal": "Complex.cosh_ofReal_im ** x\u271d y : \u2102 x : \u211d \u22a2 (cosh \u2191x).im = 0 ** rw [\u2190 ofReal_cosh_ofReal_re, ofReal_im] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.mem_ideal_smul_span_iff_exists_sum' ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 s : Set \u03b9 f : \u03b9 \u2192 M x : M \u22a2 x \u2208 I \u2022 span R (f '' s) \u2194 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f \u2191i) = x ** rw [\u2190 Submodule.mem_ideal_smul_span_iff_exists_sum, \u2190 Set.image_eq_range] ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.sumAddHom_apply ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : \u03a0\u2080 (i : \u03b9), \u03b2 i \u22a2 \u2191(sumAddHom \u03c6) f = sum f fun x => \u2191(\u03c6 x) ** rcases f with \u27e8f, s, hf\u27e9 ** case mk'.mk.mk \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 \u22a2 \u2191(sumAddHom \u03c6) { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } = sum { toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } fun x => \u2191(\u03c6 x) ** change (\u2211 i in _, _) = \u2211 i in Finset.filter _ _, _ ** case mk'.mk.mk \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 \u22a2 \u2211 i in Multiset.toFinset \u2191{ val := s, property := hf }, \u2191(\u03c6 i) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) = \u2211 i in Finset.filter (fun i => \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i \u2260 0) (Multiset.toFinset \u2191{ val := s, property := hf }), (fun x => \u2191(\u03c6 x)) i (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) ** rw [Finset.sum_filter, Finset.sum_congr rfl] ** case mk'.mk.mk \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 \u22a2 \u2200 (x : \u03b9), x \u2208 Multiset.toFinset \u2191{ val := s, property := hf } \u2192 \u2191(\u03c6 x) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x) = if \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x \u2260 0 then (fun x => \u2191(\u03c6 x)) x (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } x) else 0 ** intro i _ ** case mk'.mk.mk \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 i : \u03b9 a\u271d : i \u2208 Multiset.toFinset \u2191{ val := s, property := hf } \u22a2 \u2191(\u03c6 i) (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) = if \u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i \u2260 0 then (fun x => \u2191(\u03c6 x)) i (\u2191{ toFun := f, support' := Quot.mk Setoid.r { val := s, property := hf } } i) else 0 ** dsimp only [coe_mk', Subtype.coe_mk] at * ** case mk'.mk.mk \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 i : \u03b9 a\u271d : i \u2208 Multiset.toFinset s \u22a2 \u2191(\u03c6 i) (f i) = if f i \u2260 0 then \u2191(\u03c6 i) (f i) else 0 ** split_ifs with h ** case pos \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 i : \u03b9 a\u271d : i \u2208 Multiset.toFinset s h : f i \u2260 0 \u22a2 \u2191(\u03c6 i) (f i) = \u2191(\u03c6 i) (f i) case neg \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 i : \u03b9 a\u271d : i \u2208 Multiset.toFinset s h : \u00acf i \u2260 0 \u22a2 \u2191(\u03c6 i) (f i) = 0 ** rfl ** case neg \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddZeroClass (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d : AddCommMonoid \u03b3 \u03c6 : (i : \u03b9) \u2192 \u03b2 i \u2192+ \u03b3 f : (i : \u03b9) \u2192 \u03b2 i support'\u271d : Trunc { s // \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 } s : Multiset \u03b9 hf : \u2200 (i : \u03b9), i \u2208 s \u2228 f i = 0 i : \u03b9 a\u271d : i \u2208 Multiset.toFinset s h : \u00acf i \u2260 0 \u22a2 \u2191(\u03c6 i) (f i) = 0 ** rw [not_not.mp h, AddMonoidHom.map_zero] ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.detAux_def'' ** R : Type u_1 inst\u271d\u00b9\u00b9 : CommRing R M : Type u_2 inst\u271d\u00b9\u2070 : AddCommGroup M inst\u271d\u2079 : Module R M M' : Type u_3 inst\u271d\u2078 : AddCommGroup M' inst\u271d\u2077 : Module R M' \u03b9 : Type u_4 inst\u271d\u2076 : DecidableEq \u03b9 inst\u271d\u2075 : Fintype \u03b9 e : Basis \u03b9 R M A : Type u_5 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : Module A M \u03ba : Type u_6 inst\u271d\u00b2 : Fintype \u03ba \u03b9' : Type u_7 inst\u271d\u00b9 : Fintype \u03b9' inst\u271d : DecidableEq \u03b9' tb : Trunc (Basis \u03b9 A M) b' : Basis \u03b9' A M f : M \u2192\u2097[A] M \u22a2 \u2191(detAux tb) f = det (\u2191(toMatrix b' b') f) ** induction tb using Trunc.induction_on with\n| h b => rw [detAux_def', det_toMatrix_eq_det_toMatrix b b'] ** case h R : Type u_1 inst\u271d\u00b9\u00b9 : CommRing R M : Type u_2 inst\u271d\u00b9\u2070 : AddCommGroup M inst\u271d\u2079 : Module R M M' : Type u_3 inst\u271d\u2078 : AddCommGroup M' inst\u271d\u2077 : Module R M' \u03b9 : Type u_4 inst\u271d\u2076 : DecidableEq \u03b9 inst\u271d\u2075 : Fintype \u03b9 e : Basis \u03b9 R M A : Type u_5 inst\u271d\u2074 : CommRing A inst\u271d\u00b3 : Module A M \u03ba : Type u_6 inst\u271d\u00b2 : Fintype \u03ba \u03b9' : Type u_7 inst\u271d\u00b9 : Fintype \u03b9' inst\u271d : DecidableEq \u03b9' b' : Basis \u03b9' A M f : M \u2192\u2097[A] M b : Basis \u03b9 A M \u22a2 \u2191(detAux (Trunc.mk b)) f = det (\u2191(toMatrix b' b') f) ** rw [detAux_def', det_toMatrix_eq_det_toMatrix b b'] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.integralNormalization_coeff ** R : Type u S : Type v a b : R m n : \u2115 \u03b9 : Type y inst\u271d : Semiring R f : R[X] i : \u2115 \u22a2 coeff (integralNormalization f) i = if degree f = \u2191i then 1 else coeff f i * leadingCoeff f ^ (natDegree f - 1 - i) ** have : f.coeff i = 0 \u2192 f.degree \u2260 i := fun hc hd => coeff_ne_zero_of_eq_degree hd hc ** R : Type u S : Type v a b : R m n : \u2115 \u03b9 : Type y inst\u271d : Semiring R f : R[X] i : \u2115 this : coeff f i = 0 \u2192 degree f \u2260 \u2191i \u22a2 coeff (integralNormalization f) i = if degree f = \u2191i then 1 else coeff f i * leadingCoeff f ^ (natDegree f - 1 - i) ** simp (config := { contextual := true }) [integralNormalization, coeff_monomial, this,\n mem_support_iff] ** Qed", + "informal": "" + }, + { + "formal": "LinearPMap.mem_domain_iff_of_eq_graph ** R : Type u_1 inst\u271d\u2079 : Ring R E : Type u_2 inst\u271d\u2078 : AddCommGroup E inst\u271d\u2077 : Module R E F : Type u_3 inst\u271d\u2076 : AddCommGroup F inst\u271d\u2075 : Module R F G : Type u_4 inst\u271d\u2074 : AddCommGroup G inst\u271d\u00b3 : Module R G M : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : DistribMulAction M F inst\u271d : SMulCommClass R M F y : M f g : E \u2192\u2097.[R] F h : graph f = graph g x : E \u22a2 x \u2208 f.domain \u2194 x \u2208 g.domain ** simp_rw [mem_domain_iff, h] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.integral_comp_smul_deriv_Ioi ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** rw [integrableOn_Ici_iff_integrableOn_Ioi] at hg2 ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** have t2 := intervalIntegral_tendsto_integral_Ioi _ hg2 tendsto_id ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u t2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x)) \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** have : Ioi (f a) \u2286 f '' Ici a :=\n Ioi_subset_Ici_self.trans <|\n IsPreconnected.intermediate_value_Ici isPreconnected_Ici left_mem_Ici\n (le_principal_iff.mpr <| Ici_mem_atTop _) hf hft ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u t2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x)) this : Ioi (f a) \u2286 f '' Ici a \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** have t1 := (intervalIntegral_tendsto_integral_Ioi _ (hg1.mono_set this) tendsto_id).comp hft ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ioi a) eq : \u2200 (b : \u211d), a < b \u2192 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u t2 : Tendsto (fun i => \u222b (x : \u211d) in a..id i, f' x \u2022 (g \u2218 f) x) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x)) this : Ioi (f a) \u2286 f '' Ici a t1 : Tendsto ((fun i => \u222b (x : \u211d) in f a..id i, g x) \u2218 f) atTop (\ud835\udcdd (\u222b (x : \u211d) in Ioi (f a), g x)) \u22a2 \u222b (x : \u211d) in Ioi a, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in Ioi (f a), g u ** exact tendsto_nhds_unique (Tendsto.congr' (eventuallyEq_of_mem (Ioi_mem_atTop a) eq) t2) t1 ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** have i1 : Ioo (min a b) (max a b) \u2286 Ioi a := by\n rw [min_eq_left hb.le]\n exact Ioo_subset_Ioi_self ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** have i2 : [[a, b]] \u2286 Ici a := by rw [uIcc_of_le hb.le]; exact Icc_subset_Ici_self ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 \u222b (x : \u211d) in a..b, f' x \u2022 (g \u2218 f) x = \u222b (u : \u211d) in f a..f b, g u ** refine'\n intervalIntegral.integral_comp_smul_deriv''' (hf.mono i2)\n (fun x hx => hff' x <| mem_of_mem_of_subset hx i1) (hg_cont.mono <| image_subset _ _)\n (hg1.mono_set <| image_subset _ _) (hg2.mono_set i2) ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b \u22a2 Ioo (min a b) (max a b) \u2286 Ioi a ** rw [min_eq_left hb.le] ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b \u22a2 Ioo a (max a b) \u2286 Ioi a ** exact Ioo_subset_Ioi_self ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a \u22a2 [[a, b]] \u2286 Ici a ** rw [uIcc_of_le hb.le] ** E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a \u22a2 Icc a b \u2286 Ici a ** exact Icc_subset_Ici_self ** case refine'_1 E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 Ioo (min a b) (max a b) \u2286 Ioi a ** rw [min_eq_left hb.le] ** case refine'_1 E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 Ioo a (max a b) \u2286 Ioi a ** exact Ioo_subset_Ioi_self ** case refine'_2 E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 [[a, b]] \u2286 Ici a ** rw [uIcc_of_le hb.le] ** case refine'_2 E : Type u_1 f\u271d : \u211d \u2192 E inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f f' : \u211d \u2192 \u211d g : \u211d \u2192 E a : \u211d hf : ContinuousOn f (Ici a) hft : Tendsto f atTop atTop hff' : \u2200 (x : \u211d), x \u2208 Ioi a \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hg_cont : ContinuousOn g (f '' Ioi a) hg1 : IntegrableOn g (f '' Ici a) hg2 : IntegrableOn (fun x => f' x \u2022 (g \u2218 f) x) (Ici a) b : \u211d hb : a < b i1 : Ioo (min a b) (max a b) \u2286 Ioi a i2 : [[a, b]] \u2286 Ici a \u22a2 Icc a b \u2286 Ici a ** exact Icc_subset_Ici_self ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.abs_oangle_right_toReal_lt_pi_div_two_of_dist_eq ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P h : dist p\u2081 p\u2082 = dist p\u2081 p\u2083 \u22a2 |Real.Angle.toReal (\u2221 p\u2081 p\u2082 p\u2083)| < \u03c0 / 2 ** simp_rw [dist_eq_norm_vsub V] at h ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P h : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016 \u22a2 |Real.Angle.toReal (\u2221 p\u2081 p\u2082 p\u2083)| < \u03c0 / 2 ** rw [oangle, \u2190 vsub_sub_vsub_cancel_left p\u2083 p\u2082 p\u2081] ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P h : \u2016p\u2081 -\u1d65 p\u2082\u2016 = \u2016p\u2081 -\u1d65 p\u2083\u2016 \u22a2 |Real.Angle.toReal (Orientation.oangle o (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 p\u2082 - (p\u2081 -\u1d65 p\u2083)))| < \u03c0 / 2 ** exact o.abs_oangle_sub_right_toReal_lt_pi_div_two h ** Qed", + "informal": "" + }, + { + "formal": "nhds_basis_opens ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a\u271d : \u03b1 s s\u2081 s\u2082 t : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 a : \u03b1 \u22a2 HasBasis (\ud835\udcdd a) (fun s => a \u2208 s \u2227 IsOpen s) fun s => s ** rw [nhds_def] ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a\u271d : \u03b1 s s\u2081 s\u2082 t : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 a : \u03b1 \u22a2 HasBasis (\u2a05 s \u2208 {s | a \u2208 s \u2227 IsOpen s}, \ud835\udcdf s) (fun s => a \u2208 s \u2227 IsOpen s) fun s => s ** exact hasBasis_biInf_principal\n (fun s \u27e8has, hs\u27e9 t \u27e8hat, ht\u27e9 =>\n \u27e8s \u2229 t, \u27e8\u27e8has, hat\u27e9, IsOpen.inter hs ht\u27e9, \u27e8inter_subset_left _ _, inter_subset_right _ _\u27e9\u27e9)\n \u27e8univ, \u27e8mem_univ a, isOpen_univ\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_preimage ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x inst\u271d : CommMonoid \u03b2 f : \u03b1 \u2192 \u03b3 s : Finset \u03b3 hf : InjOn f (f \u207b\u00b9' \u2191s) g : \u03b3 \u2192 \u03b2 hg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 Set.range f \u2192 g x = 1 \u22a2 \u220f x in preimage s f hf, g (f x) = \u220f x in s, g x ** classical\n rw [prod_preimage', prod_filter_of_ne]\n exact fun x hx => Not.imp_symm (hg x hx) ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x inst\u271d : CommMonoid \u03b2 f : \u03b1 \u2192 \u03b3 s : Finset \u03b3 hf : InjOn f (f \u207b\u00b9' \u2191s) g : \u03b3 \u2192 \u03b2 hg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 Set.range f \u2192 g x = 1 \u22a2 \u220f x in preimage s f hf, g (f x) = \u220f x in s, g x ** rw [prod_preimage', prod_filter_of_ne] ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x inst\u271d : CommMonoid \u03b2 f : \u03b1 \u2192 \u03b3 s : Finset \u03b3 hf : InjOn f (f \u207b\u00b9' \u2191s) g : \u03b3 \u2192 \u03b2 hg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 Set.range f \u2192 g x = 1 \u22a2 \u2200 (x : \u03b3), x \u2208 s \u2192 g x \u2260 1 \u2192 x \u2208 Set.range f ** exact fun x hx => Not.imp_symm (hg x hx) ** Qed", + "informal": "" + }, + { + "formal": "AffineSubspace.wSameSide_of_left_mem ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : StrictOrderedCommRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' s : AffineSubspace R P x y : P hx : x \u2208 s \u22a2 WSameSide s x y ** refine' \u27e8x, hx, x, hx, _\u27e9 ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : StrictOrderedCommRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' s : AffineSubspace R P x y : P hx : x \u2208 s \u22a2 SameRay R (x -\u1d65 x) (y -\u1d65 x) ** rw [vsub_self] ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : StrictOrderedCommRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' s : AffineSubspace R P x y : P hx : x \u2208 s \u22a2 SameRay R 0 (y -\u1d65 x) ** apply SameRay.zero_left ** Qed", + "informal": "" + }, + { + "formal": "Multiset.countP_map ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 p\u271d : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p\u271d f : \u03b1 \u2192 \u03b2 s : Multiset \u03b1 p : \u03b2 \u2192 Prop inst\u271d : DecidablePred p \u22a2 countP p (map f s) = \u2191card (filter (fun a => p (f a)) s) ** refine' Multiset.induction_on s _ fun a t IH => _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 p\u271d : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p\u271d f : \u03b1 \u2192 \u03b2 s : Multiset \u03b1 p : \u03b2 \u2192 Prop inst\u271d : DecidablePred p \u22a2 countP p (map f 0) = \u2191card (filter (fun a => p (f a)) 0) ** rw [map_zero, countP_zero, filter_zero, card_zero] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 p\u271d : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p\u271d f : \u03b1 \u2192 \u03b2 s : Multiset \u03b1 p : \u03b2 \u2192 Prop inst\u271d : DecidablePred p a : \u03b1 t : Multiset \u03b1 IH : countP p (map f t) = \u2191card (filter (fun a => p (f a)) t) \u22a2 countP p (map f (a ::\u2098 t)) = \u2191card (filter (fun a => p (f a)) (a ::\u2098 t)) ** rw [map_cons, countP_cons, IH, filter_cons, card_add, apply_ite card, card_zero, card_singleton,\n add_comm] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Presieve.ofArrows_pUnit ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y Z : C f : Y \u27f6 X \u22a2 (ofArrows (fun x => Y) fun x => f) = singleton f ** funext Y ** case h C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y\u271d Z : C f : Y\u271d \u27f6 X Y : C \u22a2 (ofArrows (fun x => Y\u271d) fun x => f) = singleton f ** ext g ** case h.h C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y\u271d Z : C f : Y\u271d \u27f6 X Y : C g : Y \u27f6 X \u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2194 g \u2208 singleton f ** constructor ** case h.h.mp C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y\u271d Z : C f : Y\u271d \u27f6 X Y : C g : Y \u27f6 X \u22a2 (g \u2208 ofArrows (fun x => Y\u271d) fun x => f) \u2192 g \u2208 singleton f ** rintro \u27e8_\u27e9 ** case h.h.mp.mk C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y\u271d Z : C f : Y\u271d \u27f6 X Y : C i\u271d : PUnit.{u_1 + 1} \u22a2 f \u2208 singleton f ** apply singleton.mk ** case h.h.mpr C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y\u271d Z : C f : Y\u271d \u27f6 X Y : C g : Y \u27f6 X \u22a2 g \u2208 singleton f \u2192 g \u2208 ofArrows (fun x => Y\u271d) fun x => f ** rintro \u27e8_\u27e9 ** case h.h.mpr.mk C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D X Y\u271d Z : C f : Y\u271d \u27f6 X Y : C \u22a2 f \u2208 ofArrows (fun x => Y\u271d) fun x => f ** exact ofArrows.mk PUnit.unit ** Qed", + "informal": "" + }, + { + "formal": "Matrix.det_add_mul ** l : Type u_1 m : Type u_2 n : Type u_3 \u03b1 : Type u_4 inst\u271d\u2076 : Fintype l inst\u271d\u2075 : Fintype m inst\u271d\u2074 : Fintype n inst\u271d\u00b3 : DecidableEq l inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A : Matrix m m \u03b1 U : Matrix m n \u03b1 V : Matrix n m \u03b1 hA : IsUnit (det A) \u22a2 det (A + U * V) = det A * det (1 + V * A\u207b\u00b9 * U) ** nth_rewrite 1 [\u2190 Matrix.mul_one A] ** l : Type u_1 m : Type u_2 n : Type u_3 \u03b1 : Type u_4 inst\u271d\u2076 : Fintype l inst\u271d\u2075 : Fintype m inst\u271d\u2074 : Fintype n inst\u271d\u00b3 : DecidableEq l inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A : Matrix m m \u03b1 U : Matrix m n \u03b1 V : Matrix n m \u03b1 hA : IsUnit (det A) \u22a2 det (A * 1 + U * V) = det A * det (1 + V * A\u207b\u00b9 * U) ** rwa [\u2190 Matrix.mul_nonsing_inv_cancel_left A (U * V), \u2190Matrix.mul_add,\n det_mul, \u2190Matrix.mul_assoc, det_one_add_mul_comm, \u2190Matrix.mul_assoc] ** Qed", + "informal": "" + }, + { + "formal": "Set.biUnion_and ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 p : \u03b9 \u2192 Prop q : \u03b9 \u2192 \u03b9' \u2192 Prop s : (x : \u03b9) \u2192 (y : \u03b9') \u2192 p x \u2227 q x y \u2192 Set \u03b1 \u22a2 \u22c3 x, \u22c3 y, \u22c3 (h : p x \u2227 q x y), s x y h = \u22c3 x, \u22c3 (hx : p x), \u22c3 y, \u22c3 (hy : q x y), s x y (_ : p x \u2227 q x y) ** simp only [iUnion_and, @iUnion_comm _ \u03b9'] ** Qed", + "informal": "" + }, + { + "formal": "div_mul_cancel' ** \u03b1 : Type u_1 \u03b2 : Type u_2 G : Type u_3 inst\u271d : Group G a\u271d b\u271d c d a b : G \u22a2 a / b * b = a ** rw [div_eq_mul_inv, inv_mul_cancel_right a b] ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.toMatrixAlgEquiv_toLinAlgEquiv ** R : Type u_1 inst\u271d\u2079 : CommSemiring R l : Type u_2 m : Type u_3 n : Type u_4 inst\u271d\u2078 : Fintype n inst\u271d\u2077 : Fintype m inst\u271d\u2076 : DecidableEq n M\u2081 : Type u_5 M\u2082 : Type u_6 inst\u271d\u2075 : AddCommMonoid M\u2081 inst\u271d\u2074 : AddCommMonoid M\u2082 inst\u271d\u00b3 : Module R M\u2081 inst\u271d\u00b2 : Module R M\u2082 v\u2081 : Basis n R M\u2081 v\u2082 : Basis m R M\u2082 M\u2083 : Type u_7 inst\u271d\u00b9 : AddCommMonoid M\u2083 inst\u271d : Module R M\u2083 v\u2083 : Basis l R M\u2083 M : Matrix n n R \u22a2 \u2191(toMatrixAlgEquiv v\u2081) (\u2191(toLinAlgEquiv v\u2081) M) = M ** rw [\u2190 Matrix.toLinAlgEquiv_symm, AlgEquiv.symm_apply_apply] ** Qed", + "informal": "" + }, + { + "formal": "WithTop.mul_coe ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 b : \u03b1 hb : b \u2260 0 \u22a2 (if \u22a4 = 0 \u2228 \u2191b = 0 then 0 else \u22a4) = \u22a4 ** simp [hb] ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 b : \u03b1 hb : b \u2260 0 a : \u03b1 \u22a2 Option.some a * \u2191b = Option.bind (Option.some a) fun a => Option.some (a * b) ** rw [some_eq_coe, \u2190 coe_mul] ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 b : \u03b1 hb : b \u2260 0 a : \u03b1 \u22a2 \u2191(a * b) = Option.bind \u2191a fun a => Option.some (a * b) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.pairwise_aedisjoint_of_aedisjoint_forall_ne_one ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) \u22a2 Pairwise (AEDisjoint \u03bc on fun g => g \u2022 s) ** intro g\u2081 g\u2082 hg ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G hg : g\u2081 \u2260 g\u2082 \u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082 ** let g := g\u2082\u207b\u00b9 * g\u2081 ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G hg : g\u2081 \u2260 g\u2082 g : G := g\u2082\u207b\u00b9 * g\u2081 \u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082 ** replace hg : g \u2260 1 ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G g : G := g\u2082\u207b\u00b9 * g\u2081 hg : g \u2260 1 this : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s \u22a2 (AEDisjoint \u03bc on fun g => g \u2022 s) g\u2081 g\u2082 ** change \u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0 ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G g : G := g\u2082\u207b\u00b9 * g\u2081 hg : g \u2260 1 this : (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s \u22a2 \u2191\u2191\u03bc (g\u2081 \u2022 s \u2229 g\u2082 \u2022 s) = 0 ** exact this \u25b8 (h_qmp g\u2082\u207b\u00b9).preimage_null (h_ae_disjoint g hg) ** case hg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G hg : g\u2081 \u2260 g\u2082 g : G := g\u2082\u207b\u00b9 * g\u2081 \u22a2 g \u2260 1 ** rw [Ne.def, inv_mul_eq_one] ** case hg \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G hg : g\u2081 \u2260 g\u2082 g : G := g\u2082\u207b\u00b9 * g\u2081 \u22a2 \u00acg\u2082 = g\u2081 ** exact hg.symm ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1\u271d inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1\u271d s\u271d s' t : Set \u03b1\u271d G : Type u_8 \u03b1 : Type u_9 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : MulAction G \u03b1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 s : Set \u03b1 h_ae_disjoint : \u2200 (g : G), g \u2260 1 \u2192 AEDisjoint \u03bc (g \u2022 s) s h_qmp : \u2200 (g : G), QuasiMeasurePreserving ((fun x x_1 => x \u2022 x_1) g) g\u2081 g\u2082 : G g : G := g\u2082\u207b\u00b9 * g\u2081 hg : g \u2260 1 \u22a2 (fun x x_1 => x \u2022 x_1) g\u2082\u207b\u00b9 \u207b\u00b9' (g \u2022 s \u2229 s) = g\u2081 \u2022 s \u2229 g\u2082 \u2022 s ** rw [preimage_eq_iff_eq_image (MulAction.bijective g\u2082\u207b\u00b9), image_smul, smul_set_inter, smul_smul,\n smul_smul, inv_mul_self, one_smul] ** Qed", + "informal": "" + }, + { + "formal": "MulChar.IsQuadratic.comp ** R : Type u inst\u271d\u00b2 : CommRing R R' : Type v inst\u271d\u00b9 : CommRing R' R'' : Type w inst\u271d : CommRing R'' \u03c7 : MulChar R R' h\u03c7 : IsQuadratic \u03c7 f : R' \u2192+* R'' \u22a2 IsQuadratic (ringHomComp \u03c7 f) ** intro a ** R : Type u inst\u271d\u00b2 : CommRing R R' : Type v inst\u271d\u00b9 : CommRing R' R'' : Type w inst\u271d : CommRing R'' \u03c7 : MulChar R R' h\u03c7 : IsQuadratic \u03c7 f : R' \u2192+* R'' a : R \u22a2 \u2191(ringHomComp \u03c7 f) a = 0 \u2228 \u2191(ringHomComp \u03c7 f) a = 1 \u2228 \u2191(ringHomComp \u03c7 f) a = -1 ** rcases h\u03c7 a with (ha | ha | ha) <;> simp [ha] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SignedMeasure.bddBelow_measureOfNegatives ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 \u22a2 BddBelow (measureOfNegatives s) ** simp_rw [BddBelow, Set.Nonempty, mem_lowerBounds] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 \u22a2 \u2203 x, \u2200 (x_1 : \u211d), x_1 \u2208 measureOfNegatives s \u2192 x \u2264 x_1 ** by_contra' h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x \u22a2 False ** have h' : \u2200 n : \u2115, \u2203 y : \u211d, y \u2208 s.measureOfNegatives \u2227 y < -n := fun n => h (-n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x h' : \u2200 (n : \u2115), \u2203 y, y \u2208 measureOfNegatives s \u2227 y < -\u2191n \u22a2 False ** choose f hf using h' ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n \u22a2 False ** have hf' : \u2200 n : \u2115, \u2203 B, MeasurableSet B \u2227 s \u2264[B] 0 \u2227 s B < -n := by\n intro n\n rcases hf n with \u27e8\u27e8B, \u27e8hB\u2081, hBr\u27e9, hB\u2082\u27e9, hlt\u27e9\n exact \u27e8B, hB\u2081, hBr, hB\u2082.symm \u25b8 hlt\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n hf' : \u2200 (n : \u2115), \u2203 B, MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B \u2227 \u2191s B < -\u2191n \u22a2 False ** choose B hmeas hr h_lt using hf' ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n \u22a2 False ** set A := \u22c3 n, B n with hA ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hfalse : \u2200 (n : \u2115), \u2191s A \u2264 -\u2191n \u22a2 False ** rcases exists_nat_gt (-s A) with \u27e8n, hn\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n hfalse : \u2200 (n : \u2115), \u2191s A \u2264 -\u2191n n : \u2115 hn : -\u2191s A < \u2191n \u22a2 False ** exact lt_irrefl _ ((neg_lt.1 hn).trans_le (hfalse n)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n \u22a2 \u2200 (n : \u2115), \u2203 B, MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B \u2227 \u2191s B < -\u2191n ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n n : \u2115 \u22a2 \u2203 B, MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B \u2227 \u2191s B < -\u2191n ** rcases hf n with \u27e8\u27e8B, \u27e8hB\u2081, hBr\u27e9, hB\u2082\u27e9, hlt\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n n : \u2115 hlt : f n < -\u2191n B : Set \u03b1 hB\u2082 : \u2191s B = f n hB\u2081 : MeasurableSet B hBr : restrict s B \u2264 restrict 0 B \u22a2 \u2203 B, MeasurableSet B \u2227 restrict s B \u2264 restrict 0 B \u2227 \u2191s B < -\u2191n ** exact \u27e8B, hB\u2081, hBr, hB\u2082.symm \u25b8 hlt\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n \u22a2 \u2200 (n : \u2115), \u2191s A \u2264 -\u2191n ** intro n ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 \u22a2 \u2191s A \u2264 -\u2191n ** refine' le_trans _ (le_of_lt (h_lt _)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 \u22a2 \u2191s A \u2264 \u2191s (B n) ** rw [hA, \u2190 Set.diff_union_of_subset (Set.subset_iUnion _ n),\n of_union Set.disjoint_sdiff_left _ (hmeas n)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 \u22a2 \u2191s ((\u22c3 i, B i) \\ B n) + \u2191s (B n) \u2264 \u2191s (B n) ** refine' add_le_of_nonpos_left _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 \u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0 ** have : s \u2264[A] 0 := restrict_le_restrict_iUnion _ _ hmeas hr ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 this : restrict s A \u2264 restrict 0 A \u22a2 \u2191s ((\u22c3 i, B i) \\ B n) \u2264 0 ** refine' nonpos_of_restrict_le_zero _ (restrict_le_zero_subset _ _ (Set.diff_subset _ _) this) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 this : restrict s A \u2264 restrict 0 A \u22a2 MeasurableSet (\u22c3 i, B i) ** exact MeasurableSet.iUnion hmeas ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s : SignedMeasure \u03b1 i j : Set \u03b1 h : \u2200 (x : \u211d), \u2203 x_1, x_1 \u2208 measureOfNegatives s \u2227 x_1 < x f : \u2115 \u2192 \u211d hf : \u2200 (n : \u2115), f n \u2208 measureOfNegatives s \u2227 f n < -\u2191n B : \u2115 \u2192 Set \u03b1 hmeas : \u2200 (n : \u2115), MeasurableSet (B n) hr : \u2200 (n : \u2115), restrict s (B n) \u2264 restrict 0 (B n) h_lt : \u2200 (n : \u2115), \u2191s (B n) < -\u2191n A : Set \u03b1 := \u22c3 n, B n hA : A = \u22c3 n, B n n : \u2115 \u22a2 MeasurableSet ((\u22c3 i, B i) \\ B n) ** exact (MeasurableSet.iUnion hmeas).diff (hmeas n) ** Qed", + "informal": "" + }, + { + "formal": "Turing.PartrecToTM2.K'.elim_update_aux ** a b c d c' : List \u0393' \u22a2 update (elim a b c d) aux c' = elim a b c' d ** funext x ** case h a b c d c' : List \u0393' x : K' \u22a2 update (elim a b c d) aux c' x = elim a b c' d x ** cases x <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "AbsoluteValue.IsAdmissible.exists_approx ** R : Type u_1 inst\u271d\u00b9 : EuclideanDomain R abv : AbsoluteValue R \u2124 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 b : R hb : b \u2260 0 h : IsAdmissible abv A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R \u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : \u03b9), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5 ** let e := Fintype.equivFin \u03b9 ** R : Type u_1 inst\u271d\u00b9 : EuclideanDomain R abv : AbsoluteValue R \u2124 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 b : R hb : b \u2260 0 h : IsAdmissible abv A : Fin (Nat.succ (IsAdmissible.card h \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R e : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9 \u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : \u03b9), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5 ** obtain \u27e8i\u2080, i\u2081, ne, h\u27e9 := h.exists_approx_aux (Fintype.card \u03b9) h\u03b5 hb fun x y \u21a6 A x (e.symm y) ** case intro.intro.intro R : Type u_1 inst\u271d\u00b9 : EuclideanDomain R abv : AbsoluteValue R \u2124 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 b : R hb : b \u2260 0 h\u271d : IsAdmissible abv A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R e : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9 i\u2080 i\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) ne : i\u2080 \u2260 i\u2081 h : \u2200 (k : Fin (Fintype.card \u03b9)), \u2191(\u2191abv (A i\u2081 (\u2191e.symm k) % b - A i\u2080 (\u2191e.symm k) % b)) < \u2191abv b \u2022 \u03b5 \u22a2 \u2203 i\u2080 i\u2081, i\u2080 \u2260 i\u2081 \u2227 \u2200 (k : \u03b9), \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5 ** refine' \u27e8i\u2080, i\u2081, ne, fun k \u21a6 _\u27e9 ** case intro.intro.intro R : Type u_1 inst\u271d\u00b9 : EuclideanDomain R abv : AbsoluteValue R \u2124 \u03b9 : Type u_2 inst\u271d : Fintype \u03b9 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 b : R hb : b \u2260 0 h\u271d : IsAdmissible abv A : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) \u2192 \u03b9 \u2192 R e : \u03b9 \u2243 Fin (Fintype.card \u03b9) := Fintype.equivFin \u03b9 i\u2080 i\u2081 : Fin (Nat.succ (IsAdmissible.card h\u271d \u03b5 ^ Fintype.card \u03b9)) ne : i\u2080 \u2260 i\u2081 h : \u2200 (k : Fin (Fintype.card \u03b9)), \u2191(\u2191abv (A i\u2081 (\u2191e.symm k) % b - A i\u2080 (\u2191e.symm k) % b)) < \u2191abv b \u2022 \u03b5 k : \u03b9 \u22a2 \u2191(\u2191abv (A i\u2081 k % b - A i\u2080 k % b)) < \u2191abv b \u2022 \u03b5 ** convert h (e k) <;> simp only [e.symm_apply_apply] ** Qed", + "informal": "" + }, + { + "formal": "Fintype.prod_unique ** \u03b9 : Type u_1 \u03b2\u271d : Type u \u03b1\u271d : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1\u271d a : \u03b1\u271d f\u271d g : \u03b1\u271d \u2192 \u03b2\u271d \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b2 : CommMonoid \u03b2 inst\u271d\u00b9 : Unique \u03b1 inst\u271d : Fintype \u03b1 f : \u03b1 \u2192 \u03b2 \u22a2 \u220f x : \u03b1, f x = f default ** rw [univ_unique, prod_singleton] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.extend_iUnion_le_tsum_nat ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) \u22a2 \u2200 (s : \u2115 \u2192 Set \u03b1), extend m (\u22c3 i, s i) \u2264 \u2211' (i : \u2115), extend m (s i) ** refine' extend_iUnion_le_tsum_nat' MeasurableSet.iUnion _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) \u22a2 \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) \u2264 \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) ** intro f h ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) f : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), MeasurableSet (f i) \u22a2 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) \u2264 \u2211' (i : \u2115), m (f i) (_ : ?m.314738 (f i)) ** simp (config := { singlePass := true }) [iUnion_disjointed.symm] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) f : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), MeasurableSet (f i) \u22a2 m (\u22c3 n, disjointed (fun n => f n) n) (_ : MeasurableSet (\u22c3 n, disjointed (fun n => f n) n)) \u2264 \u2211' (i : \u2115), m (f i) (_ : ?m.314738 (f i)) ** rw [mU (MeasurableSet.disjointed h) (disjoint_disjointed _)] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) f : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), MeasurableSet (f i) \u22a2 \u2211' (i : \u2115), m (disjointed (fun i => f i) i) (_ : MeasurableSet (disjointed (fun i => f i) i)) \u2264 \u2211' (i : \u2115), m (f i) (_ : ?m.314738 (f i)) ** refine' ENNReal.tsum_le_tsum fun i => _ ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) f : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), MeasurableSet (f i) i : \u2115 \u22a2 m (disjointed (fun i => f i) i) (_ : MeasurableSet (disjointed (fun i => f i) i)) \u2264 m (f i) (_ : ?m.314738 (f i)) ** rw [\u2190 extend_eq m, \u2190 extend_eq m] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 m : (s : Set \u03b1) \u2192 MeasurableSet s \u2192 \u211d\u22650\u221e m0 : m \u2205 (_ : MeasurableSet \u2205) = 0 mU : \u2200 \u2983f : \u2115 \u2192 Set \u03b1\u2984 (hm : \u2200 (i : \u2115), MeasurableSet (f i)), Pairwise (Disjoint on f) \u2192 m (\u22c3 i, f i) (_ : MeasurableSet (\u22c3 b, f b)) = \u2211' (i : \u2115), m (f i) (_ : MeasurableSet (f i)) f : \u2115 \u2192 Set \u03b1 h : \u2200 (i : \u2115), MeasurableSet (f i) i : \u2115 \u22a2 extend m (disjointed (fun i => f i) i) \u2264 extend m (f i) ** exact extend_mono m0 mU (MeasurableSet.disjointed h _) (disjointed_le f _) ** Qed", + "informal": "" + }, + { + "formal": "NNReal.summable_nat_add_iff ** \u03b1 : Type u_1 f : \u2115 \u2192 \u211d\u22650 k : \u2115 \u22a2 (Summable fun i => f (i + k)) \u2194 Summable f ** rw [\u2190 summable_coe, \u2190 summable_coe] ** \u03b1 : Type u_1 f : \u2115 \u2192 \u211d\u22650 k : \u2115 \u22a2 (Summable fun a => \u2191(f (a + k))) \u2194 Summable fun a => \u2191(f a) ** exact @summable_nat_add_iff \u211d _ _ _ (fun i => (f i : \u211d)) k ** Qed", + "informal": "" + }, + { + "formal": "MaximalSpectrum.iInf_localization_eq_bot ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K \u22a2 \u2a05 v, Localization.subalgebra.ofField K (Ideal.primeCompl v.asIdeal) (_ : Ideal.primeCompl v.asIdeal \u2264 nonZeroDivisors R) = \u22a5 ** ext x ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K \u22a2 x \u2208 \u2a05 v, Localization.subalgebra.ofField K (Ideal.primeCompl v.asIdeal) (_ : Ideal.primeCompl v.asIdeal \u2264 nonZeroDivisors R) \u2194 x \u2208 \u22a5 ** rw [Algebra.mem_bot, Algebra.mem_iInf] ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K \u22a2 (\u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R)) \u2194 x \u2208 range \u2191(algebraMap R K) ** constructor ** case h.mp R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K \u22a2 (\u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R)) \u2192 x \u2208 range \u2191(algebraMap R K) ** contrapose ** case h.mp R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K \u22a2 \u00acx \u2208 range \u2191(algebraMap R K) \u2192 \u00ac\u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) ** intro hrange hlocal ** case h.mp R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) \u22a2 False ** let denom : Ideal R := (Submodule.span R {1} : Submodule R K).colon (Submodule.span R {x}) ** case h.mp R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) \u22a2 False ** have hdenom : (1 : R) \u2209 denom := by\n intro hdenom\n rcases Submodule.mem_span_singleton.mp\n (Submodule.mem_colon.mp hdenom x <| Submodule.mem_span_singleton_self x) with \u27e8y, hy\u27e9\n exact hrange \u27e8y, by rw [\u2190 mul_one <| algebraMap R K y, \u2190 Algebra.smul_def, hy, one_smul]\u27e9 ** case h.mp R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) hdenom : \u00ac1 \u2208 denom \u22a2 False ** rcases denom.exists_le_maximal fun h => (h \u25b8 hdenom) Submodule.mem_top with \u27e8max, hmax, hle\u27e9 ** case h.mp.intro.intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) hdenom : \u00ac1 \u2208 denom max : Ideal R hmax : Ideal.IsMaximal max hle : denom \u2264 max \u22a2 False ** rcases hlocal \u27e8max, hmax\u27e9 with \u27e8n, d, hd, rfl\u27e9 ** case h.mp.intro.intro.intro.intro.intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K max : Ideal R hmax : Ideal.IsMaximal max n d : R hd : d \u2208 Ideal.primeCompl { asIdeal := max, IsMaximal := hmax }.asIdeal hrange : \u00ac\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), \u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9}) hdenom : \u00ac1 \u2208 denom hle : denom \u2264 max \u22a2 False ** apply hd (hle <| Submodule.mem_colon.mpr fun _ hy => _) ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K max : Ideal R hmax : Ideal.IsMaximal max n d : R hd : d \u2208 Ideal.primeCompl { asIdeal := max, IsMaximal := hmax }.asIdeal hrange : \u00ac\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), \u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9}) hdenom : \u00ac1 \u2208 denom hle : denom \u2264 max \u22a2 \u2200 (x : K), x \u2208 Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9} \u2192 d \u2022 x \u2208 Submodule.span R {1} ** intro _ hy ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K max : Ideal R hmax : Ideal.IsMaximal max n d : R hd : d \u2208 Ideal.primeCompl { asIdeal := max, IsMaximal := hmax }.asIdeal hrange : \u00ac\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), \u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9}) hdenom : \u00ac1 \u2208 denom hle : denom \u2264 max x\u271d : K hy : x\u271d \u2208 Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9} \u22a2 d \u2022 x\u271d \u2208 Submodule.span R {1} ** rcases Submodule.mem_span_singleton.mp hy with \u27e8y, rfl\u27e9 ** case intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K max : Ideal R hmax : Ideal.IsMaximal max n d : R hd : d \u2208 Ideal.primeCompl { asIdeal := max, IsMaximal := hmax }.asIdeal hrange : \u00ac\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), \u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9}) hdenom : \u00ac1 \u2208 denom hle : denom \u2264 max y : R hy : y \u2022 (\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9) \u2208 Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9} \u22a2 d \u2022 y \u2022 (\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9) \u2208 Submodule.span R {1} ** exact Submodule.mem_span_singleton.mpr \u27e8y * n, by\n rw [Algebra.smul_def, mul_one, map_mul, smul_comm, Algebra.smul_def, Algebra.smul_def,\n mul_comm <| algebraMap R K d,\n inv_mul_cancel_right\u2080 <|\n (map_ne_zero_iff _ <| NoZeroSMulDivisors.algebraMap_injective R K).mpr fun h =>\n (h \u25b8 hd) max.zero_mem]\u27e9 ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) \u22a2 \u00ac1 \u2208 denom ** intro hdenom ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) hdenom : 1 \u2208 denom \u22a2 False ** rcases Submodule.mem_span_singleton.mp\n (Submodule.mem_colon.mp hdenom x <| Submodule.mem_span_singleton_self x) with \u27e8y, hy\u27e9 ** case intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) hdenom : 1 \u2208 denom y : R hy : y \u2022 1 = 1 \u2022 x \u22a2 False ** exact hrange \u27e8y, by rw [\u2190 mul_one <| algebraMap R K y, \u2190 Algebra.smul_def, hy, one_smul]\u27e9 ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K hrange : \u00acx \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {x}) hdenom : 1 \u2208 denom y : R hy : y \u2022 1 = 1 \u2022 x \u22a2 \u2191(algebraMap R K) y = x ** rw [\u2190 mul_one <| algebraMap R K y, \u2190 Algebra.smul_def, hy, one_smul] ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K max : Ideal R hmax : Ideal.IsMaximal max n d : R hd : d \u2208 Ideal.primeCompl { asIdeal := max, IsMaximal := hmax }.asIdeal hrange : \u00ac\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 range \u2191(algebraMap R K) hlocal : \u2200 (i : MaximalSpectrum R), \u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9 \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) denom : Ideal R := Submodule.colon (Submodule.span R {1}) (Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9}) hdenom : \u00ac1 \u2208 denom hle : denom \u2264 max y : R hy : y \u2022 (\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9) \u2208 Submodule.span R {\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9} \u22a2 (y * n) \u2022 1 = d \u2022 y \u2022 (\u2191(algebraMap R K) n * (\u2191(algebraMap R K) d)\u207b\u00b9) ** rw [Algebra.smul_def, mul_one, map_mul, smul_comm, Algebra.smul_def, Algebra.smul_def,\n mul_comm <| algebraMap R K d,\n inv_mul_cancel_right\u2080 <|\n (map_ne_zero_iff _ <| NoZeroSMulDivisors.algebraMap_injective R K).mpr fun h =>\n (h \u25b8 hd) max.zero_mem] ** case h.mpr R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K x : K \u22a2 x \u2208 range \u2191(algebraMap R K) \u2192 \u2200 (i : MaximalSpectrum R), x \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl i.asIdeal) (_ : Ideal.primeCompl i.asIdeal \u2264 nonZeroDivisors R) ** rintro \u27e8y, rfl\u27e9 \u27e8v, hv\u27e9 ** case h.mpr.intro.mk R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K y : R v : Ideal R hv : Ideal.IsMaximal v \u22a2 \u2191(algebraMap R K) y \u2208 Localization.subalgebra.ofField K (Ideal.primeCompl { asIdeal := v, IsMaximal := hv }.asIdeal) (_ : Ideal.primeCompl { asIdeal := v, IsMaximal := hv }.asIdeal \u2264 nonZeroDivisors R) ** exact \u27e8y, 1, v.ne_top_iff_one.mp hv.ne_top, by rw [map_one, inv_one, mul_one]\u27e9 ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsDomain R K : Type v inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Algebra R K inst\u271d : IsFractionRing R K y : R v : Ideal R hv : Ideal.IsMaximal v \u22a2 \u2191(algebraMap R K) y = \u2191(algebraMap R K) y * (\u2191(algebraMap R K) 1)\u207b\u00b9 ** rw [map_one, inv_one, mul_one] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.tan_oangle_add_right_smul_rotation_pi_div_two ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V hd2 : Fact (finrank \u211d V = 2) o : Orientation \u211d V (Fin 2) x : V h : x \u2260 0 r : \u211d \u22a2 Real.Angle.tan (oangle o x (x + r \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x)) = r ** rw [o.oangle_add_right_smul_rotation_pi_div_two h, Real.Angle.tan_coe, Real.tan_arctan] ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.card_uIcc ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : DecidableEq \u03b9 inst\u271d\u00b3 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Lattice (\u03b1 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i) f g : \u03a0\u2080 (i : \u03b9), \u03b1 i \u22a2 card (uIcc f g) = \u220f i in support f \u222a support g, card (uIcc (\u2191f i) (\u2191g i)) ** rw [\u2190support_inf_union_support_sup] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u2074 : DecidableEq \u03b9 inst\u271d\u00b3 : (i : \u03b9) \u2192 DecidableEq (\u03b1 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Lattice (\u03b1 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b1 i) inst\u271d : (i : \u03b9) \u2192 LocallyFiniteOrder (\u03b1 i) f g : \u03a0\u2080 (i : \u03b9), \u03b1 i \u22a2 card (uIcc f g) = \u220f i in support (f \u2293 g) \u222a support (f \u2294 g), card (uIcc (\u2191f i) (\u2191g i)) ** exact card_Icc _ _ ** Qed", + "informal": "" + }, + { + "formal": "AbsoluteValue.abs_abv_sub_le_abv_sub ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommRing S abv : AbsoluteValue R S a b : R \u22a2 \u2191abv b - \u2191abv a \u2264 \u2191abv (a - b) ** rw [abv.map_sub] ** R : Type u_1 S : Type u_2 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommRing S abv : AbsoluteValue R S a b : R \u22a2 \u2191abv b - \u2191abv a \u2264 \u2191abv (b - a) ** apply abv.le_sub ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.GrothendieckTopology.diagramNatTrans_comp ** C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b9 : Category.{max v u, w} D inst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R X : C \u22a2 diagramNatTrans J (\u03b7 \u226b \u03b3) X = diagramNatTrans J \u03b7 X \u226b diagramNatTrans J \u03b3 X ** ext : 2 ** case w.h C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b9 : Category.{max v u, w} D inst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R X : C x\u271d : (Cover J X)\u1d52\u1d56 \u22a2 (diagramNatTrans J (\u03b7 \u226b \u03b3) X).app x\u271d = (diagramNatTrans J \u03b7 X \u226b diagramNatTrans J \u03b3 X).app x\u271d ** refine' Multiequalizer.hom_ext _ _ _ (fun i => _) ** case w.h C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b9 : Category.{max v u, w} D inst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R X : C x\u271d : (Cover J X)\u1d52\u1d56 i : (Cover.index x\u271d.unop R).L \u22a2 (diagramNatTrans J (\u03b7 \u226b \u03b3) X).app x\u271d \u226b Multiequalizer.\u03b9 (Cover.index x\u271d.unop R) i = (diagramNatTrans J \u03b7 X \u226b diagramNatTrans J \u03b3 X).app x\u271d \u226b Multiequalizer.\u03b9 (Cover.index x\u271d.unop R) i ** dsimp ** case w.h C : Type u inst\u271d\u00b2 : Category.{v, u} C J : GrothendieckTopology C D : Type w inst\u271d\u00b9 : Category.{max v u, w} D inst\u271d : \u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P) P\u271d P Q R : C\u1d52\u1d56 \u2964 D \u03b7 : P \u27f6 Q \u03b3 : Q \u27f6 R X : C x\u271d : (Cover J X)\u1d52\u1d56 i : (Cover.index x\u271d.unop R).L \u22a2 Multiequalizer.lift (Cover.index x\u271d.unop R) (multiequalizer (Cover.index x\u271d.unop P)) (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop P) i \u226b \u03b7.app (op i.Y) \u226b \u03b3.app (op i.Y)) (_ : \u2200 (i : (Cover.index x\u271d.unop R).R), (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop P) i \u226b (\u03b7 \u226b \u03b3).app (op i.Y)) (MulticospanIndex.fstTo (Cover.index x\u271d.unop R) i) \u226b MulticospanIndex.fst (Cover.index x\u271d.unop R) i = (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop P) i \u226b (\u03b7 \u226b \u03b3).app (op i.Y)) (MulticospanIndex.sndTo (Cover.index x\u271d.unop R) i) \u226b MulticospanIndex.snd (Cover.index x\u271d.unop R) i) \u226b Multiequalizer.\u03b9 (Cover.index x\u271d.unop R) i = (Multiequalizer.lift (Cover.index x\u271d.unop Q) (multiequalizer (Cover.index x\u271d.unop P)) (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop P) i \u226b \u03b7.app (op i.Y)) (_ : \u2200 (i : (Cover.index x\u271d.unop Q).R), (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop P) i \u226b \u03b7.app (op i.Y)) (MulticospanIndex.fstTo (Cover.index x\u271d.unop Q) i) \u226b MulticospanIndex.fst (Cover.index x\u271d.unop Q) i = (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop P) i \u226b \u03b7.app (op i.Y)) (MulticospanIndex.sndTo (Cover.index x\u271d.unop Q) i) \u226b MulticospanIndex.snd (Cover.index x\u271d.unop Q) i) \u226b Multiequalizer.lift (Cover.index x\u271d.unop R) (multiequalizer (Cover.index x\u271d.unop Q)) (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop Q) i \u226b \u03b3.app (op i.Y)) (_ : \u2200 (i : (Cover.index x\u271d.unop R).R), (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop Q) i \u226b \u03b3.app (op i.Y)) (MulticospanIndex.fstTo (Cover.index x\u271d.unop R) i) \u226b MulticospanIndex.fst (Cover.index x\u271d.unop R) i = (fun i => Multiequalizer.\u03b9 (Cover.index x\u271d.unop Q) i \u226b \u03b3.app (op i.Y)) (MulticospanIndex.sndTo (Cover.index x\u271d.unop R) i) \u226b MulticospanIndex.snd (Cover.index x\u271d.unop R) i)) \u226b Multiequalizer.\u03b9 (Cover.index x\u271d.unop R) i ** simp ** Qed", + "informal": "" + }, + { + "formal": "YoungDiagram.mem_col_iff ** \u03bc : YoungDiagram j : \u2115 c : \u2115 \u00d7 \u2115 \u22a2 c \u2208 col \u03bc j \u2194 c \u2208 \u03bc \u2227 c.2 = j ** simp [col] ** Qed", + "informal": "" + }, + { + "formal": "Computation.orElse_empty ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w c : Computation \u03b1 \u22a2 (HOrElse.hOrElse c fun x => empty \u03b1) = c ** apply eq_of_bisim (fun c\u2081 c\u2082 => (c\u2082 <|> empty \u03b1) = c\u2081) _ rfl ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w c : Computation \u03b1 \u22a2 IsBisimulation fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081 ** intro s' s h ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w c s' s : Computation \u03b1 h : (HOrElse.hOrElse s fun x => empty \u03b1) = s' \u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081) (destruct s') (destruct s) ** rw [\u2190 h] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w c s' s : Computation \u03b1 h : (HOrElse.hOrElse s fun x => empty \u03b1) = s' \u22a2 BisimO (fun c\u2081 c\u2082 => (HOrElse.hOrElse c\u2082 fun x => empty \u03b1) = c\u2081) (destruct (HOrElse.hOrElse s fun x => empty \u03b1)) (destruct s) ** apply recOn s <;> intro s <;> rw [think_empty] <;> simp ** case h2 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w c s' s\u271d : Computation \u03b1 h : (HOrElse.hOrElse s\u271d fun x => empty \u03b1) = s' s : Computation \u03b1 \u22a2 (HOrElse.hOrElse s fun x => think (empty \u03b1)) = HOrElse.hOrElse s fun x => empty \u03b1 ** rw [\u2190 think_empty] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.ofBoxProdLeft_boxProdRight ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 G : SimpleGraph \u03b1 H : SimpleGraph \u03b2 a : \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableRel G.Adj b\u2081 b\u2082 x z y : \u03b1 h : Adj G x y w : Walk G y z \u22a2 ofBoxProdRight (Walk.boxProdRight G a (cons' x y z h w)) = cons' x y z h w ** rw [Walk.boxProdRight, map_cons, ofBoxProdRight, Or.by_cases, dif_pos, \u2190\n Walk.boxProdRight] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 G : SimpleGraph \u03b1 H : SimpleGraph \u03b2 a : \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableRel G.Adj b\u2081 b\u2082 x z y : \u03b1 h : Adj G x y w : Walk G y z \u22a2 cons (_ : Adj G (a, x).2 (\u2191(Embedding.toHom (boxProdRight G G a)) y).2) (ofBoxProdRight (Walk.boxProdRight G a w)) = cons' x y z h w case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 G : SimpleGraph \u03b1 H : SimpleGraph \u03b2 a : \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableRel G.Adj b\u2081 b\u2082 x z y : \u03b1 h : Adj G x y w : Walk G y z \u22a2 Adj G (a, x).2 (\u2191(Embedding.toHom (boxProdRight G G a)) y).2 \u2227 (a, x).1 = (\u2191(Embedding.toHom (boxProdRight G G a)) y).1 case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 G : SimpleGraph \u03b1 H : SimpleGraph \u03b2 a : \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableRel G.Adj b\u2081 b\u2082 x z y : \u03b1 h : Adj G x y w : Walk G y z \u22a2 Adj G (a, x).2 (\u2191(Embedding.toHom (boxProdRight G G a)) y).2 \u2227 (a, x).1 = (\u2191(Embedding.toHom (boxProdRight G G a)) y).1 ** simp [ofBoxProdLeft_boxProdRight] ** case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 G : SimpleGraph \u03b1 H : SimpleGraph \u03b2 a : \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableRel G.Adj b\u2081 b\u2082 x z y : \u03b1 h : Adj G x y w : Walk G y z \u22a2 Adj G (a, x).2 (\u2191(Embedding.toHom (boxProdRight G G a)) y).2 \u2227 (a, x).1 = (\u2191(Embedding.toHom (boxProdRight G G a)) y).1 case hc \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 G : SimpleGraph \u03b1 H : SimpleGraph \u03b2 a : \u03b1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableRel G.Adj b\u2081 b\u2082 x z y : \u03b1 h : Adj G x y w : Walk G y z \u22a2 Adj G (a, x).2 (\u2191(Embedding.toHom (boxProdRight G G a)) y).2 \u2227 (a, x).1 = (\u2191(Embedding.toHom (boxProdRight G G a)) y).1 ** exact \u27e8h, rfl\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Abelian.Pseudoelement.apply_eq_zero_of_comp_eq_zero ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : Abelian C P Q R : C f : Q \u27f6 R a : P \u27f6 Q h : a \u226b f = 0 \u22a2 pseudoApply f (Quot.mk (PseudoEqual Q) (Over.mk a)) = 0 ** simp [over_coe_def, pseudoApply_mk', Over.coe_hom, h] ** Qed", + "informal": "" + }, + { + "formal": "coe_lowerCentralSeries_ideal_quot_eq ** R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 \u22a2 \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) ** induction' k with k ih ** case zero R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L \u22a2 \u2191(lowerCentralSeries R L (L \u29f8 I) Nat.zero) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) Nat.zero) ** simp only [Nat.zero_eq, LieModule.lowerCentralSeries_zero, LieSubmodule.top_coeSubmodule,\n LieIdeal.top_coe_lieSubalgebra, LieSubalgebra.top_coe_submodule] ** case succ R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) \u22a2 \u2191(lowerCentralSeries R L (L \u29f8 I) (Nat.succ k)) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) (Nat.succ k)) ** simp only [LieModule.lowerCentralSeries_succ, LieSubmodule.lieIdeal_oper_eq_linear_span] ** case succ R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) \u22a2 Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} = Submodule.span R {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** congr ** case succ.e_s R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) \u22a2 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} = {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** ext x ** case succ.e_s.h R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) x : L \u29f8 I \u22a2 x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2194 x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** constructor ** case succ.e_s.h.mp R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) x : L \u29f8 I \u22a2 x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2192 x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** rintro \u27e8\u27e8y, -\u27e9, \u27e8z, hz\u27e9, rfl : \u2045y, z\u2046 = x\u27e9 ** case succ.e_s.h.mp.intro.mk.intro.mk R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) y : L z : L \u29f8 I hz : z \u2208 lowerCentralSeries R L (L \u29f8 I) k \u22a2 \u2045y, z\u2046 \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** erw [\u2190 LieSubmodule.mem_coeSubmodule, ih, LieSubmodule.mem_coeSubmodule] at hz ** case succ.e_s.h.mp.intro.mk.intro.mk R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) y : L z : L \u29f8 I hz : z \u2208 lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k \u22a2 \u2045y, z\u2046 \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** exact \u27e8\u27e8LieSubmodule.Quotient.mk y, LieSubmodule.mem_top _\u27e9, \u27e8z, hz\u27e9, rfl\u27e9 ** case succ.e_s.h.mpr R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) x : L \u29f8 I \u22a2 x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} \u2192 x \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** rintro \u27e8\u27e8\u27e8y\u27e9, -\u27e9, \u27e8z, hz\u27e9, rfl : \u2045y, z\u2046 = x\u27e9 ** case succ.e_s.h.mpr.intro.mk.mk.intro.mk R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) val\u271d : L \u29f8 I y : L z : L \u29f8 I hz : z \u2208 lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k \u22a2 \u2045y, z\u2046 \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** erw [\u2190 LieSubmodule.mem_coeSubmodule, \u2190 ih, LieSubmodule.mem_coeSubmodule] at hz ** case succ.e_s.h.mpr.intro.mk.mk.intro.mk R : Type u L : Type v L' : Type w inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : LieRing L inst\u271d\u00b2 : LieAlgebra R L inst\u271d\u00b9 : LieRing L' inst\u271d : LieAlgebra R L' I : LieIdeal R L k : \u2115 ih : \u2191(lowerCentralSeries R L (L \u29f8 I) k) = \u2191(lowerCentralSeries R (L \u29f8 I) (L \u29f8 I) k) val\u271d : L \u29f8 I y : L z : L \u29f8 I hz : z \u2208 lowerCentralSeries R L (L \u29f8 I) k \u22a2 \u2045y, z\u2046 \u2208 {m | \u2203 x n, \u2045\u2191x, \u2191n\u2046 = m} ** exact \u27e8\u27e8y, LieSubmodule.mem_top _\u27e9, \u27e8z, hz\u27e9, rfl\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "WithBot.map_le_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 a b : \u03b1 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 f : \u03b1 \u2192 \u03b2 mono_iff : \u2200 {a b : \u03b1}, f a \u2264 f b \u2194 a \u2264 b x\u271d : WithBot \u03b1 \u22a2 map f \u22a5 \u2264 map f x\u271d \u2194 \u22a5 \u2264 x\u271d ** simp only [map_bot, bot_le] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 a\u271d b : \u03b1 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 f : \u03b1 \u2192 \u03b2 mono_iff : \u2200 {a b : \u03b1}, f a \u2264 f b \u2194 a \u2264 b a : \u03b1 \u22a2 map f \u2191a \u2264 map f \u22a5 \u2194 \u2191a \u2264 \u22a5 ** simp only [map_coe, map_bot, coe_ne_bot, not_coe_le_bot _] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 a\u271d b\u271d : \u03b1 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 f : \u03b1 \u2192 \u03b2 mono_iff : \u2200 {a b : \u03b1}, f a \u2264 f b \u2194 a \u2264 b a b : \u03b1 \u22a2 map f \u2191a \u2264 map f \u2191b \u2194 \u2191a \u2264 \u2191b ** simpa only [map_coe, coe_le_coe] using mono_iff ** Qed", + "informal": "" + }, + { + "formal": "List.revzip_map_fst ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 l : List \u03b1 \u22a2 map Prod.fst (revzip l) = l ** rw [\u2190 unzip_left, unzip_revzip] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.MorphismProperty.universally_le ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u_1 inst\u271d : Category.{?u.117055, u_1} D P : MorphismProperty C \u22a2 universally P \u2264 P ** intro X Y f hf ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u_1 inst\u271d : Category.{?u.117055, u_1} D P : MorphismProperty C X Y : C f : X \u27f6 Y hf : universally P f \u22a2 P f ** exact hf (\ud835\udfd9 _) (\ud835\udfd9 _) _ (IsPullback.of_vert_isIso \u27e8by rw [Category.comp_id, Category.id_comp]\u27e9) ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u_1 inst\u271d : Category.{?u.117055, u_1} D P : MorphismProperty C X Y : C f : X \u27f6 Y hf : universally P f \u22a2 f \u226b \ud835\udfd9 Y = \ud835\udfd9 X \u226b f ** rw [Category.comp_id, Category.id_comp] ** Qed", + "informal": "" + }, + { + "formal": "leftCoset_rightCoset ** \u03b1 : Type u_1 inst\u271d : Semigroup \u03b1 s : Set \u03b1 a b : \u03b1 \u22a2 a *l s *r b = a *l (s *r b) ** simp [leftCoset, rightCoset, (image_comp _ _ _).symm, Function.comp, mul_assoc] ** Qed", + "informal": "" + }, + { + "formal": "CliffordAlgebra.changeForm.associated_neg_proof ** R : Type u1 inst\u271d\u00b3 : CommRing R M : Type u2 inst\u271d\u00b2 : AddCommGroup M inst\u271d\u00b9 : Module R M Q Q' Q'' : QuadraticForm R M B B' : BilinForm R M h : BilinForm.toQuadraticForm B = Q' - Q h' : BilinForm.toQuadraticForm B' = Q'' - Q' inst\u271d : Invertible 2 \u22a2 BilinForm.toQuadraticForm (\u2191QuadraticForm.associated (-Q)) = 0 - Q ** simp [QuadraticForm.toQuadraticForm_associated] ** Qed", + "informal": "" + }, + { + "formal": "Set.ncard_eq_ncard_iff_ncard_diff_eq_ncard_diff ** \u03b1 : Type u_1 s t : Set \u03b1 hs : autoParam (Set.Finite s) _auto\u271d ht : autoParam (Set.Finite t) _auto\u271d \u22a2 ncard s = ncard t \u2194 ncard (s \\ t) = ncard (t \\ s) ** rw [\u2190 ncard_inter_add_ncard_diff_eq_ncard s t hs, \u2190 ncard_inter_add_ncard_diff_eq_ncard t s ht,\n inter_comm, add_right_inj] ** Qed", + "informal": "" + }, + { + "formal": "Algebra.TensorProduct.intCast_def' ** R : Type uR S : Type uS A : Type uA B : Type uB C : Type uC D : Type uD E : Type uE F : Type uF inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : Ring A inst\u271d\u00b2 : Algebra R A inst\u271d\u00b9 : Ring B inst\u271d : Algebra R B z : \u2124 \u22a2 \u2191z = 1 \u2297\u209c[R] \u2191z ** rw [intCast_def, \u2190zsmul_one, smul_tmul, zsmul_one] ** Qed", + "informal": "" + }, + { + "formal": "ENorm.map_sub_rev ** \ud835\udd5c : Type u_1 V : Type u_2 inst\u271d\u00b2 : NormedField \ud835\udd5c inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module \ud835\udd5c V e : ENorm \ud835\udd5c V x y : V \u22a2 \u2191e (x - y) = \u2191e (y - x) ** rw [\u2190 neg_sub, e.map_neg] ** Qed", + "informal": "" + }, + { + "formal": "linearIndependent_le_span_aux' ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) \u22a2 Fintype.card \u03b9 \u2264 Fintype.card \u2191w ** fapply card_le_of_injective' R ** case f K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) \u22a2 (\u03b9 \u2192\u2080 R) \u2192\u2097[R] \u2191w \u2192\u2080 R ** apply Finsupp.total ** case f.v K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) \u22a2 \u03b9 \u2192 \u2191w \u2192\u2080 R ** exact fun i => Span.repr R w \u27e8v i, s (mem_range_self i)\u27e9 ** case i K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) \u22a2 Injective \u2191(Finsupp.total \u03b9 (\u2191w \u2192\u2080 R) R fun i => Span.repr R w { val := v i, property := (_ : v i \u2208 \u2191(span R w)) }) ** intro f g h ** case i K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) f g : \u03b9 \u2192\u2080 R h : \u2191(Finsupp.total \u03b9 (\u2191w \u2192\u2080 R) R fun i => Span.repr R w { val := v i, property := (_ : v i \u2208 \u2191(span R w)) }) f = \u2191(Finsupp.total \u03b9 (\u2191w \u2192\u2080 R) R fun i => Span.repr R w { val := v i, property := (_ : v i \u2208 \u2191(span R w)) }) g \u22a2 f = g ** apply_fun Finsupp.total w M R (\u2191) at h ** case i K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) f g : \u03b9 \u2192\u2080 R h : \u2191(Finsupp.total (\u2191w) M R Subtype.val) (\u2191(Finsupp.total \u03b9 (\u2191w \u2192\u2080 R) R fun i => Span.repr R w { val := v i, property := (_ : v i \u2208 \u2191(span R w)) }) f) = \u2191(Finsupp.total (\u2191w) M R Subtype.val) (\u2191(Finsupp.total \u03b9 (\u2191w \u2192\u2080 R) R fun i => Span.repr R w { val := v i, property := (_ : v i \u2208 \u2191(span R w)) }) g) \u22a2 f = g ** simp only [Finsupp.total_total, Submodule.coe_mk, Span.finsupp_total_repr] at h ** case i K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) f g : \u03b9 \u2192\u2080 R h : \u2191(Finsupp.total \u03b9 M R fun b => v b) f = \u2191(Finsupp.total \u03b9 M R fun b => v b) g \u22a2 f = g ** rw [\u2190 sub_eq_zero, \u2190 LinearMap.map_sub] at h ** case i K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2075 : Ring R inst\u271d\u2074 : StrongRankCondition R M : Type v inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 v : \u03b9 \u2192 M i : LinearIndependent R v w : Set M inst\u271d : Fintype \u2191w s : range v \u2264 \u2191(span R w) f g : \u03b9 \u2192\u2080 R h\u271d : \u2191(Finsupp.total \u03b9 M R fun b => v b) f = \u2191(Finsupp.total \u03b9 M R fun b => v b) g h : \u2191(Finsupp.total \u03b9 M R fun b => v b) (f - g) = 0 \u22a2 f = g ** exact sub_eq_zero.mp (linearIndependent_iff.mp i _ h) ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.BinaryCofan.isVanKampen_iff ** J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y \u22a2 IsVanKampenColimit c \u2194 \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) ** constructor ** case mp J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y \u22a2 IsVanKampenColimit c \u2192 \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) ** introv H h\u03b1X h\u03b1Y ** case mp J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u22a2 Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) ** rw [H c' (mapPair \u03b1X \u03b1Y) f (by ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> assumption) (mapPair_equifibered _)] ** case mp J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u22a2 (\u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) ((mapPair \u03b1X \u03b1Y).app j) f (c.\u03b9.app j)) \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) ** constructor ** J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u22a2 mapPair \u03b1X \u03b1Y \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f ** ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> assumption ** case mp.mp J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u22a2 (\u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) ((mapPair \u03b1X \u03b1Y).app j) f (c.\u03b9.app j)) \u2192 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) ** intro H ** case mp.mp J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H\u271d : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f H : \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) ((mapPair \u03b1X \u03b1Y).app j) f (c.\u03b9.app j) \u22a2 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) ** exact \u27e8H _, H _\u27e9 ** case mp.mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u22a2 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) \u2192 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) ((mapPair \u03b1X \u03b1Y).app j) f (c.\u03b9.app j) ** rintro H \u27e8\u27e8\u27e9\u27e9 ** case mp.mpr.mk.left J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H\u271d : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f H : IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) \u22a2 IsPullback (c'.\u03b9.app { as := WalkingPair.left }) ((mapPair \u03b1X \u03b1Y).app { as := WalkingPair.left }) f (c.\u03b9.app { as := WalkingPair.left }) case mp.mpr.mk.right J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H\u271d : IsVanKampenColimit c X' Y' : C c' : BinaryCofan X' Y' \u03b1X : X' \u27f6 X \u03b1Y : Y' \u27f6 Y f : c'.pt \u27f6 c.pt h\u03b1X : \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f h\u03b1Y : \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f H : IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c) \u22a2 IsPullback (c'.\u03b9.app { as := WalkingPair.right }) ((mapPair \u03b1X \u03b1Y).app { as := WalkingPair.right }) f (c.\u03b9.app { as := WalkingPair.right }) ** exacts [H.1, H.2] ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y \u22a2 (\u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c))) \u2192 IsVanKampenColimit c ** introv H F' h\u03b1 h ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 \u22a2 Nonempty (IsColimit c') \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** let X' := F'.obj \u27e8WalkingPair.left\u27e9 ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 X' : C := F'.obj { as := WalkingPair.left } \u22a2 Nonempty (IsColimit c') \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** let Y' := F'.obj \u27e8WalkingPair.right\u27e9 ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 X' : C := F'.obj { as := WalkingPair.left } Y' : C := F'.obj { as := WalkingPair.right } this : F' = pair X' Y' \u22a2 Nonempty (IsColimit c') \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** clear_value X' Y' ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 Y' X' : C this : F' = pair X' Y' \u22a2 Nonempty (IsColimit c') \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** subst this ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C c' : Cocone (pair X' Y') \u03b1 : pair X' Y' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 \u22a2 Nonempty (IsColimit c') \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** change BinaryCofan X' Y' at c' ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f \u22a2 Nonempty (IsColimit c') \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** rw [H c' _ _ _ (NatTrans.congr_app h\u03b1 \u27e8WalkingPair.left\u27e9)\n (NatTrans.congr_app h\u03b1 \u27e8WalkingPair.right\u27e9)] ** case mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f \u22a2 IsPullback (BinaryCofan.inl c') (\u03b1.app { as := WalkingPair.left }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') (\u03b1.app { as := WalkingPair.right }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inr c) \u2194 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** constructor ** J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 X' : C := F'.obj { as := WalkingPair.left } Y' : C := F'.obj { as := WalkingPair.right } \u22a2 F' = pair X' Y' ** apply Functor.hext ** case h_obj J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 X' : C := F'.obj { as := WalkingPair.left } Y' : C := F'.obj { as := WalkingPair.right } \u22a2 \u2200 (X : Discrete WalkingPair), F'.obj X = (pair X' Y').obj X ** rintro \u27e8\u27e8\u27e9\u27e9 <;> rfl ** case h_map J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) F' : Discrete WalkingPair \u2964 C c' : Cocone F' \u03b1 : F' \u27f6 pair X Y f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f h : NatTrans.Equifibered \u03b1 X' : C := F'.obj { as := WalkingPair.left } Y' : C := F'.obj { as := WalkingPair.right } \u22a2 \u2200 (X Y : Discrete WalkingPair) (f : X \u27f6 Y), HEq (F'.map f) ((pair X' Y').map f) ** rintro \u27e8\u27e8\u27e9\u27e9 \u27e8j\u27e9 \u27e8\u27e8rfl : _ = j\u27e9\u27e9 <;> simp ** case mpr.mp J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f \u22a2 IsPullback (BinaryCofan.inl c') (\u03b1.app { as := WalkingPair.left }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') (\u03b1.app { as := WalkingPair.right }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inr c) \u2192 \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) ** rintro H \u27e8\u27e8\u27e9\u27e9 ** case mpr.mp.mk.left J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H\u271d : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f H : IsPullback (BinaryCofan.inl c') (\u03b1.app { as := WalkingPair.left }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') (\u03b1.app { as := WalkingPair.right }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inr c) \u22a2 IsPullback (c'.\u03b9.app { as := WalkingPair.left }) (\u03b1.app { as := WalkingPair.left }) f (c.\u03b9.app { as := WalkingPair.left }) case mpr.mp.mk.right J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H\u271d : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f H : IsPullback (BinaryCofan.inl c') (\u03b1.app { as := WalkingPair.left }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') (\u03b1.app { as := WalkingPair.right }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inr c) \u22a2 IsPullback (c'.\u03b9.app { as := WalkingPair.right }) (\u03b1.app { as := WalkingPair.right }) f (c.\u03b9.app { as := WalkingPair.right }) ** exacts [H.1, H.2] ** case mpr.mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f \u22a2 (\u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j)) \u2192 IsPullback (BinaryCofan.inl c') (\u03b1.app { as := WalkingPair.left }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') (\u03b1.app { as := WalkingPair.right }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inr c) ** intro H ** case mpr.mpr J : Type v' inst\u271d\u00b9 : Category.{u', v'} J C : Type u inst\u271d : Category.{v, u} C X Y : C c : BinaryCofan X Y H\u271d : \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt), \u03b1X \u226b BinaryCofan.inl c = BinaryCofan.inl c' \u226b f \u2192 \u03b1Y \u226b BinaryCofan.inr c = BinaryCofan.inr c' \u226b f \u2192 (Nonempty (IsColimit c') \u2194 IsPullback (BinaryCofan.inl c') \u03b1X f (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') \u03b1Y f (BinaryCofan.inr c)) Y' X' : C \u03b1 : pair X' Y' \u27f6 pair X Y h : NatTrans.Equifibered \u03b1 c' : BinaryCofan X' Y' f : c'.pt \u27f6 c.pt h\u03b1 : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f H : \u2200 (j : Discrete WalkingPair), IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j) \u22a2 IsPullback (BinaryCofan.inl c') (\u03b1.app { as := WalkingPair.left }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inl c) \u2227 IsPullback (BinaryCofan.inr c') (\u03b1.app { as := WalkingPair.right }) (((Functor.const (Discrete WalkingPair)).map f).app { as := WalkingPair.left }) (BinaryCofan.inr c) ** exact \u27e8H _, H _\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Matrix.SpecialLinearGroup.SL2_inv_expl ** n : Type u inst\u271d\u00b3 : DecidableEq n inst\u271d\u00b2 : Fintype n R : Type v inst\u271d\u00b9 : CommRing R S : Type u_1 inst\u271d : CommRing S A : SL(2, R) \u22a2 A\u207b\u00b9 = { val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]], property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) } ** ext ** case a n : Type u inst\u271d\u00b3 : DecidableEq n inst\u271d\u00b2 : Fintype n R : Type v inst\u271d\u00b9 : CommRing R S : Type u_1 inst\u271d : CommRing S A : SL(2, R) i\u271d j\u271d : Fin 2 \u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d = \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]], property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) } i\u271d j\u271d ** have := Matrix.adjugate_fin_two A.1 ** case a n : Type u inst\u271d\u00b3 : DecidableEq n inst\u271d\u00b2 : Fintype n R : Type v inst\u271d\u00b9 : CommRing R S : Type u_1 inst\u271d : CommRing S A : SL(2, R) i\u271d j\u271d : Fin 2 this : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] \u22a2 \u2191A\u207b\u00b9 i\u271d j\u271d = \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]], property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) } i\u271d j\u271d ** rw [coe_inv, this] ** case a n : Type u inst\u271d\u00b3 : DecidableEq n inst\u271d\u00b2 : Fintype n R : Type v inst\u271d\u00b9 : CommRing R S : Type u_1 inst\u271d : CommRing S A : SL(2, R) i\u271d j\u271d : Fin 2 this : adjugate \u2191A = \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] \u22a2 \u2191of ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] i\u271d j\u271d = \u2191{ val := ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]], property := (_ : det ![![\u2191A 1 1, -\u2191A 0 1], ![-\u2191A 1 0, \u2191A 0 0]] = 1) } i\u271d j\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.restrict_const_lintegral ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 c : \u211d\u22650\u221e s : Set \u03b1 hs : MeasurableSet s \u22a2 lintegral (restrict (const \u03b1 c) s) \u03bc = c * \u2191\u2191\u03bc s ** rw [restrict_lintegral_eq_lintegral_restrict _ hs, const_lintegral_restrict] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.MeasurePreserving.lintegral_comp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd\u271d : Measure \u03b1 mb : MeasurableSpace \u03b2 \u03bd : Measure \u03b2 g : \u03b1 \u2192 \u03b2 hg : MeasurePreserving g f : \u03b2 \u2192 \u211d\u22650\u221e hf : Measurable f \u22a2 \u222b\u207b (a : \u03b1), f (g a) \u2202\u03bc = \u222b\u207b (b : \u03b2), f b \u2202\u03bd ** rw [\u2190 hg.map_eq, lintegral_map hf hg.measurable] ** Qed", + "informal": "" + }, + { + "formal": "Prime.pow_dvd_of_dvd_mul_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : CancelCommMonoidWithZero \u03b1 p a b : \u03b1 hp : Prime p n : \u2115 h : \u00acp \u2223 b h' : p ^ n \u2223 a * b \u22a2 p ^ n \u2223 a ** rw [mul_comm] at h' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : CancelCommMonoidWithZero \u03b1 p a b : \u03b1 hp : Prime p n : \u2115 h : \u00acp \u2223 b h' : p ^ n \u2223 b * a \u22a2 p ^ n \u2223 a ** exact hp.pow_dvd_of_dvd_mul_left n h h' ** Qed", + "informal": "" + }, + { + "formal": "Algebra.TensorProduct.productMap_range ** R : Type uR S : Type uS A : Type uA B : Type uB C : Type uC D : Type uD E : Type uE F : Type uF inst\u271d\u2076 : CommSemiring R inst\u271d\u2075 : Semiring A inst\u271d\u2074 : Semiring B inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra R A inst\u271d\u00b9 : Algebra R B inst\u271d : Algebra R S f : A \u2192\u2090[R] S g : B \u2192\u2090[R] S \u22a2 AlgHom.range (productMap f g) = AlgHom.range f \u2294 AlgHom.range g ** rw [productMap_eq_comp_map, AlgHom.range_comp, map_range, map_sup, \u2190 AlgHom.range_comp,\n \u2190 AlgHom.range_comp,\n \u2190 AlgHom.comp_assoc, \u2190 AlgHom.comp_assoc, lmul'_comp_includeLeft, lmul'_comp_includeRight,\n AlgHom.id_comp, AlgHom.id_comp] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.Represents.mul ** \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A A' : Matrix \u03b9 \u03b9 R f f' : Module.End R M h : Represents b A f h' : Represents b A' f' \u22a2 Represents b (A * A') (f * f') ** delta Matrix.Represents PiToModule.fromMatrix ** \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A A' : Matrix \u03b9 \u03b9 R f f' : Module.End R M h : Represents b A f h' : Represents b A' f' \u22a2 \u2191(LinearMap.comp (\u2191(LinearMap.llcomp R (\u03b9 \u2192 R) (\u03b9 \u2192 R) M) (\u2191(Fintype.total R R) b)) (AlgEquiv.toLinearMap (AlgEquiv.symm algEquivMatrix'))) (A * A') = \u2191(PiToModule.fromEnd R b) (f * f') ** rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_mul] ** \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A A' : Matrix \u03b9 \u03b9 R f f' : Module.End R M h : Represents b A f h' : Represents b A' f' \u22a2 \u2191(\u2191(LinearMap.llcomp R (\u03b9 \u2192 R) (\u03b9 \u2192 R) M) (\u2191(Fintype.total R R) b)) (\u2191(AlgEquiv.symm algEquivMatrix') A * \u2191(AlgEquiv.symm algEquivMatrix') A') = \u2191(PiToModule.fromEnd R b) (f * f') ** ext ** case h.h \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A A' : Matrix \u03b9 \u03b9 R f f' : Module.End R M h : Represents b A f h' : Represents b A' f' i\u271d : \u03b9 \u22a2 \u2191(LinearMap.comp (\u2191(\u2191(LinearMap.llcomp R (\u03b9 \u2192 R) (\u03b9 \u2192 R) M) (\u2191(Fintype.total R R) b)) (\u2191(AlgEquiv.symm algEquivMatrix') A * \u2191(AlgEquiv.symm algEquivMatrix') A')) (LinearMap.single i\u271d)) 1 = \u2191(LinearMap.comp (\u2191(PiToModule.fromEnd R b) (f * f')) (LinearMap.single i\u271d)) 1 ** dsimp [PiToModule.fromEnd] ** case h.h \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A A' : Matrix \u03b9 \u03b9 R f f' : Module.End R M h : Represents b A f h' : Represents b A' f' i\u271d : \u03b9 \u22a2 \u2191(\u2191(Fintype.total R R) b) (\u2191(\u2191(AlgEquiv.symm algEquivMatrix') A) (\u2191(\u2191(AlgEquiv.symm algEquivMatrix') A') (Pi.single i\u271d 1))) = \u2191f (\u2191f' (\u2191(\u2191(Fintype.total R R) b) (Pi.single i\u271d 1))) ** rw [\u2190 h'.congr_fun, \u2190 h.congr_fun] ** case h.h \u03b9 : Type u_1 inst\u271d\u2074 : Fintype \u03b9 M : Type u_2 inst\u271d\u00b3 : AddCommGroup M R : Type u_3 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : Module R M I : Ideal R b : \u03b9 \u2192 M hb : Submodule.span R (Set.range b) = \u22a4 inst\u271d : DecidableEq \u03b9 A A' : Matrix \u03b9 \u03b9 R f f' : Module.End R M h : Represents b A f h' : Represents b A' f' i\u271d : \u03b9 \u22a2 \u2191(\u2191(Fintype.total R R) b) (\u2191(\u2191(AlgEquiv.symm algEquivMatrix') A) (\u2191(\u2191(AlgEquiv.symm algEquivMatrix') A') (Pi.single i\u271d 1))) = \u2191(\u2191(Fintype.total R R) b) (mulVec A (mulVec A' (Pi.single i\u271d 1))) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "padicNorm.sum_le' ** p : \u2115 hp : Fact (Nat.Prime p) \u03b1 : Type u_1 F : \u03b1 \u2192 \u211a t : \u211a s : Finset \u03b1 hF : \u2200 (i : \u03b1), i \u2208 s \u2192 padicNorm p (F i) \u2264 t ht : 0 \u2264 t \u22a2 padicNorm p (\u2211 i in s, F i) \u2264 t ** obtain rfl | hs := Finset.eq_empty_or_nonempty s ** case inl p : \u2115 hp : Fact (Nat.Prime p) \u03b1 : Type u_1 F : \u03b1 \u2192 \u211a t : \u211a ht : 0 \u2264 t hF : \u2200 (i : \u03b1), i \u2208 \u2205 \u2192 padicNorm p (F i) \u2264 t \u22a2 padicNorm p (\u2211 i in \u2205, F i) \u2264 t ** simp [ht] ** case inr p : \u2115 hp : Fact (Nat.Prime p) \u03b1 : Type u_1 F : \u03b1 \u2192 \u211a t : \u211a s : Finset \u03b1 hF : \u2200 (i : \u03b1), i \u2208 s \u2192 padicNorm p (F i) \u2264 t ht : 0 \u2264 t hs : Finset.Nonempty s \u22a2 padicNorm p (\u2211 i in s, F i) \u2264 t ** exact sum_le hs hF ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.quasiMeasurePreserving_div_left ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc g : G \u22a2 QuasiMeasurePreserving fun h => g / h ** simp_rw [div_eq_mul_inv] ** G : Type u_1 inst\u271d\u2076 : MeasurableSpace G inst\u271d\u2075 : Group G inst\u271d\u2074 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b3 : SigmaFinite \u03bd inst\u271d\u00b2 : SigmaFinite \u03bc s : Set G inst\u271d\u00b9 : MeasurableInv G inst\u271d : IsMulLeftInvariant \u03bc g : G \u22a2 QuasiMeasurePreserving fun h => g * h\u207b\u00b9 ** exact\n (measurePreserving_mul_left \u03bc g).quasiMeasurePreserving.comp (quasiMeasurePreserving_inv \u03bc) ** Qed", + "informal": "" + }, + { + "formal": "minpoly.root ** A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x\u271d x : B hx : IsIntegral A x y : A h : IsRoot (minpoly A x) y \u22a2 \u2191(algebraMap A B) y = x ** have key : minpoly A x = X - C y := eq_of_monic_of_associated (monic hx) (monic_X_sub_C y)\n (associated_of_dvd_dvd ((irreducible_X_sub_C y).dvd_symm (irreducible hx) (dvd_iff_isRoot.2 h))\n (dvd_iff_isRoot.2 h)) ** A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x\u271d x : B hx : IsIntegral A x y : A h : IsRoot (minpoly A x) y key : minpoly A x = X - \u2191C y \u22a2 \u2191(algebraMap A B) y = x ** have := aeval A x ** A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x\u271d x : B hx : IsIntegral A x y : A h : IsRoot (minpoly A x) y key : minpoly A x = X - \u2191C y this : \u2191(Polynomial.aeval x) (minpoly A x) = 0 \u22a2 \u2191(algebraMap A B) y = x ** rwa [key, AlgHom.map_sub, aeval_X, aeval_C, sub_eq_zero, eq_comm] at this ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.graph_eq_ker_coprod ** R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : AddCommMonoid M inst\u271d\u2076 : AddCommMonoid M\u2082 inst\u271d\u2075 : AddCommGroup M\u2083 inst\u271d\u2074 : AddCommGroup M\u2084 inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : Module R M\u2083 inst\u271d : Module R M\u2084 f : M \u2192\u2097[R] M\u2082 g : M\u2083 \u2192\u2097[R] M\u2084 \u22a2 graph g = ker (coprod (-g) id) ** ext x ** case h R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : AddCommMonoid M inst\u271d\u2076 : AddCommMonoid M\u2082 inst\u271d\u2075 : AddCommGroup M\u2083 inst\u271d\u2074 : AddCommGroup M\u2084 inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : Module R M\u2083 inst\u271d : Module R M\u2084 f : M \u2192\u2097[R] M\u2082 g : M\u2083 \u2192\u2097[R] M\u2084 x : M\u2083 \u00d7 M\u2084 \u22a2 x \u2208 graph g \u2194 x \u2208 ker (coprod (-g) id) ** change _ = _ \u2194 -g x.1 + x.2 = _ ** case h R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2078 : Semiring R inst\u271d\u2077 : AddCommMonoid M inst\u271d\u2076 : AddCommMonoid M\u2082 inst\u271d\u2075 : AddCommGroup M\u2083 inst\u271d\u2074 : AddCommGroup M\u2084 inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : Module R M\u2083 inst\u271d : Module R M\u2084 f : M \u2192\u2097[R] M\u2082 g : M\u2083 \u2192\u2097[R] M\u2084 x : M\u2083 \u00d7 M\u2084 \u22a2 x.2 = \u2191g x.1 \u2194 -\u2191g x.1 + x.2 = 0 ** rw [add_comm, add_neg_eq_zero] ** Qed", + "informal": "" + }, + { + "formal": "SemiconjBy.sub_left ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w R : Type x inst\u271d : NonUnitalNonAssocRing R a b x y x' y' : R ha : SemiconjBy a x y hb : SemiconjBy b x y \u22a2 SemiconjBy (a - b) x y ** simpa only [sub_eq_add_neg] using ha.add_left hb.neg_left ** Qed", + "informal": "" + }, + { + "formal": "IsPrimitiveRoot.card_nthRoots ** M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n \u22a2 \u2191Multiset.card (nthRoots n 1) = n ** cases' Nat.eq_zero_or_pos n with hzero hpos ** case inr M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hpos : n > 0 \u22a2 \u2191Multiset.card (nthRoots n 1) = n ** rw [eq_iff_le_not_lt] ** case inr M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hpos : n > 0 \u22a2 \u2191Multiset.card (nthRoots n 1) \u2264 n \u2227 \u00ac\u2191Multiset.card (nthRoots n 1) < n ** use card_nthRoots n 1 ** case inl M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hzero : n = 0 \u22a2 \u2191Multiset.card (nthRoots n 1) = n ** simp only [hzero, Multiset.card_zero, nthRoots_zero] ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hpos : n > 0 \u22a2 \u00ac\u2191Multiset.card (nthRoots n 1) < n ** rw [not_lt] ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hpos : n > 0 \u22a2 n \u2264 \u2191Multiset.card (nthRoots n 1) ** have hcard :\n Fintype.card { x // x \u2208 nthRoots n (1 : R) } \u2264 Multiset.card (nthRoots n (1 : R)).attach :=\n Multiset.card_le_of_le (Multiset.dedup_le _) ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hpos : n > 0 hcard : Fintype.card { x // x \u2208 nthRoots n 1 } \u2264 \u2191Multiset.card (Multiset.attach (nthRoots n 1)) \u22a2 n \u2264 \u2191Multiset.card (nthRoots n 1) ** rw [Multiset.card_attach] at hcard ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 n hpos : n > 0 hcard : Fintype.card { x // x \u2208 nthRoots n 1 } \u2264 \u2191Multiset.card (nthRoots n 1) \u22a2 n \u2264 \u2191Multiset.card (nthRoots n 1) ** rw [\u2190 PNat.toPNat'_coe hpos] at hcard h \u22a2 ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 h : IsPrimitiveRoot \u03b6 \u2191(Nat.toPNat' n) hpos : n > 0 hcard : Fintype.card { x // x \u2208 nthRoots (\u2191(Nat.toPNat' n)) 1 } \u2264 \u2191Multiset.card (nthRoots (\u2191(Nat.toPNat' n)) 1) \u22a2 \u2191(Nat.toPNat' n) \u2264 \u2191Multiset.card (nthRoots (\u2191(Nat.toPNat' n)) 1) ** set m := Nat.toPNat' n ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : n > 0 m : \u2115+ := Nat.toPNat' n h : IsPrimitiveRoot \u03b6 \u2191m hcard : Fintype.card { x // x \u2208 nthRoots (\u2191m) 1 } \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1) \u22a2 \u2191m \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1) ** rw [\u2190 Fintype.card_congr (rootsOfUnityEquivNthRoots R m), card_rootsOfUnity h] at hcard ** case right M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2074 : CommMonoid M inst\u271d\u00b3 : CommMonoid N inst\u271d\u00b2 : DivisionCommMonoid G k l : \u2115 inst\u271d\u00b9 : CommRing R \u03b6\u271d : R\u02e3 h\u271d : IsPrimitiveRoot \u03b6\u271d k inst\u271d : IsDomain R \u03b6 : R n : \u2115 hpos : n > 0 m : \u2115+ := Nat.toPNat' n h : IsPrimitiveRoot \u03b6 \u2191m hcard : \u2191m \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1) \u22a2 \u2191m \u2264 \u2191Multiset.card (nthRoots (\u2191m) 1) ** exact hcard ** Qed", + "informal": "" + }, + { + "formal": "Nat.cast_choose_eq_ascPochhammer_div ** K : Type u_1 inst\u271d\u00b9 : DivisionRing K inst\u271d : CharZero K a b : \u2115 \u22a2 \u2191(choose a b) = Polynomial.eval (\u2191(a - (b - 1))) (ascPochhammer K b) / \u2191b ! ** rw [eq_div_iff_mul_eq (cast_ne_zero.2 b.factorial_ne_zero : (b ! : K) \u2260 0), \u2190 cast_mul,\n mul_comm, \u2190 descFactorial_eq_factorial_mul_choose, \u2190 cast_descFactorial] ** Qed", + "informal": "" + }, + { + "formal": "le_nhdsAdjoint_iff' ** \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 \u22a2 t \u2264 nhdsAdjoint a f \u2194 \ud835\udcdd a \u2264 pure a \u2294 f \u2227 \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b ** rw [le_iff_nhds] ** \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 \u22a2 (\u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x) \u2194 \ud835\udcdd a \u2264 pure a \u2294 f \u2227 \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b ** constructor ** case mp \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 \u22a2 (\u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x) \u2192 \ud835\udcdd a \u2264 pure a \u2294 f \u2227 \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b ** intro h ** case mp \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x \u22a2 \ud835\udcdd a \u2264 pure a \u2294 f \u2227 \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b ** constructor ** case mp.left \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x \u22a2 \ud835\udcdd a \u2264 pure a \u2294 f ** specialize h a ** case mp.left \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \ud835\udcdd a \u2264 \ud835\udcdd a \u22a2 \ud835\udcdd a \u2264 pure a \u2294 f ** rwa [nhdsAdjoint_nhds] at h ** case mp.right \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x \u22a2 \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b ** intro b hb ** case mp.right \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x b : \u03b1 hb : b \u2260 a \u22a2 \ud835\udcdd b = pure b ** apply le_antisymm _ (pure_le_nhds b) ** \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x b : \u03b1 hb : b \u2260 a \u22a2 \ud835\udcdd b \u2264 pure b ** specialize h b ** \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 b : \u03b1 hb : b \u2260 a h : \ud835\udcdd b \u2264 \ud835\udcdd b \u22a2 \ud835\udcdd b \u2264 pure b ** rwa [nhdsAdjoint_nhds_of_ne a f hb] at h ** case mpr \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 \u22a2 (\ud835\udcdd a \u2264 pure a \u2294 f \u2227 \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b) \u2192 \u2200 (x : \u03b1), \ud835\udcdd x \u2264 \ud835\udcdd x ** rintro \u27e8h, h'\u27e9 b ** case mpr.intro \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \ud835\udcdd a \u2264 pure a \u2294 f h' : \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b b : \u03b1 \u22a2 \ud835\udcdd b \u2264 \ud835\udcdd b ** by_cases hb : b = a ** case pos \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \ud835\udcdd a \u2264 pure a \u2294 f h' : \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b b : \u03b1 hb : b = a \u22a2 \ud835\udcdd b \u2264 \ud835\udcdd b ** rwa [hb, nhdsAdjoint_nhds] ** case neg \u03b1\u271d : Type u \u03b2 : Type v \u03b1 : Type u_1 a : \u03b1 f : Filter \u03b1 t : TopologicalSpace \u03b1 h : \ud835\udcdd a \u2264 pure a \u2294 f h' : \u2200 (b : \u03b1), b \u2260 a \u2192 \ud835\udcdd b = pure b b : \u03b1 hb : \u00acb = a \u22a2 \ud835\udcdd b \u2264 \ud835\udcdd b ** simp [nhdsAdjoint_nhds_of_ne a f hb, h' b hb] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.tendsto_approxOn_Lp_snorm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** by_cases hp_zero : p = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** have hp : 0 < p.toReal := toReal_pos hp_zero hp_ne_top ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** suffices\n Tendsto (fun n => \u222b\u207b x, (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) atTop\n (\ud835\udcdd 0) by\n simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top]\n convert continuous_rpow_const.continuousAt.tendsto.comp this\n simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have hF_meas :\n \u2200 n, Measurable fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal := by\n simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n (measurable_edist_right.comp hf).pow_const p.toReal ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h_bound :\n \u2200 n, (fun x => (\u2016approxOn f hf s y\u2080 h\u2080 n x - f x\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) \u2264\u1d50[\u03bc] fun x =>\n (\u2016f x - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal :=\n fun n =>\n eventually_of_forall fun x =>\n rpow_le_rpow (coe_mono (nnnorm_approxOn_le hf h\u2080 x n)) toReal_nonneg ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h_fin : (\u222b\u207b a : \u03b2, (\u2016f a - y\u2080\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal \u2202\u03bc) \u2260 \u22a4 :=\n (lintegral_rpow_nnnorm_lt_top_of_snorm_lt_top hp_zero hp_ne_top hi).ne ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** have h_lim :\n \u2200\u1d50 a : \u03b2 \u2202\u03bc,\n Tendsto (fun n => (\u2016approxOn f hf s y\u2080 h\u2080 n a - f a\u2016\u208a : \u211d\u22650\u221e) ^ p.toReal) atTop (\ud835\udcdd 0) := by\n filter_upwards [h\u03bc] with a ha\n have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds\n convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm)\n simp [zero_rpow_of_pos hp] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 h_lim : \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) ** simpa using tendsto_lintegral_of_dominated_convergence _ hF_meas h_bound h_fin h_lim ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : p = 0 \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** simpa only [hp_zero, snorm_exponent_zero] using tendsto_const_nhds ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p this : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => snorm (\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) p \u03bc) atTop (\ud835\udcdd 0) ** simp only [snorm_eq_lintegral_rpow_nnnorm hp_zero hp_ne_top] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p this : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 Tendsto (fun n => (\u222b\u207b (x : \u03b2), \u2191\u2016(\u2191(approxOn f hf s y\u2080 h\u2080 n) - f) x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) ^ (1 / ENNReal.toReal p)) atTop (\ud835\udcdd 0) ** convert continuous_rpow_const.continuousAt.tendsto.comp this ** case h.e'_5.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p this : Tendsto (fun n => \u222b\u207b (x : \u03b2), \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc) atTop (\ud835\udcdd 0) \u22a2 0 = 0 ^ (1 / ENNReal.toReal p) ** simp [zero_rpow_of_pos (_root_.inv_pos.mpr hp)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p \u22a2 \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p ** simpa only [\u2190 edist_eq_coe_nnnorm_sub] using fun n =>\n (approxOn f hf s y\u2080 h\u2080 n).measurable_bind (fun y x => edist y (f x) ^ p.toReal) fun y =>\n (measurable_edist_right.comp hf).pow_const p.toReal ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 \u22a2 \u2200\u1d50 (a : \u03b2) \u2202\u03bc, Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) ** filter_upwards [h\u03bc] with a ha ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 a : \u03b2 ha : f a \u2208 closure s \u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) ** have : Tendsto (fun n => (approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) :=\n (tendsto_approxOn hf h\u2080 ha).sub tendsto_const_nhds ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 a : \u03b2 ha : f a \u2208 closure s this : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) \u22a2 Tendsto (fun n => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a\u2016\u208a ^ ENNReal.toReal p) atTop (\ud835\udcdd 0) ** convert continuous_rpow_const.continuousAt.tendsto.comp (tendsto_coe.mpr this.nnnorm) ** case h.e'_5.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : MeasurableSpace E inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e inst\u271d\u00b9 : OpensMeasurableSpace E f : \u03b2 \u2192 E hf : Measurable f s : Set E y\u2080 : E h\u2080 : y\u2080 \u2208 s inst\u271d : SeparableSpace \u2191s hp_ne_top : p \u2260 \u22a4 \u03bc : Measure \u03b2 h\u03bc : \u2200\u1d50 (x : \u03b2) \u2202\u03bc, f x \u2208 closure s hi : snorm (fun x => f x - y\u2080) p \u03bc < \u22a4 hp_zero : \u00acp = 0 hp : 0 < ENNReal.toReal p hF_meas : \u2200 (n : \u2115), Measurable fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p h_bound : \u2200 (n : \u2115), (fun x => \u2191\u2016\u2191(approxOn f hf s y\u2080 h\u2080 n) x - f x\u2016\u208a ^ ENNReal.toReal p) \u2264\u1d50[\u03bc] fun x => \u2191\u2016f x - y\u2080\u2016\u208a ^ ENNReal.toReal p h_fin : \u222b\u207b (a : \u03b2), \u2191\u2016f a - y\u2080\u2016\u208a ^ ENNReal.toReal p \u2202\u03bc \u2260 \u22a4 a : \u03b2 ha : f a \u2208 closure s this : Tendsto (fun n => \u2191(approxOn f hf s y\u2080 h\u2080 n) a - f a) atTop (\ud835\udcdd (f a - f a)) \u22a2 0 = \u2191\u2016f a - f a\u2016\u208a ^ ENNReal.toReal p ** simp [zero_rpow_of_pos hp] ** Qed", + "informal": "" + }, + { + "formal": "IsometryEquiv.completeSpace_iff ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : PseudoEMetricSpace \u03b1 inst\u271d\u00b9 : PseudoEMetricSpace \u03b2 inst\u271d : PseudoEMetricSpace \u03b3 e : \u03b1 \u2243\u1d62 \u03b2 \u22a2 CompleteSpace \u03b1 \u2194 CompleteSpace \u03b2 ** simp only [completeSpace_iff_isComplete_univ, \u2190 e.range_eq_univ, \u2190 image_univ,\n isComplete_image_iff e.isometry.uniformInducing] ** Qed", + "informal": "" + }, + { + "formal": "Asymptotics.superpolynomialDecay_iff_isLittleO ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop \u22a2 SuperpolynomialDecay l k f \u2194 \u2200 (z : \u2124), f =o[l] fun a => k a ^ z ** refine' \u27e8fun h z => _, fun h => (superpolynomialDecay_iff_isBigO f hk).2 fun z => (h z).isBigO\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 \u22a2 f =o[l] fun a => k a ^ z ** have hk0 : \u2200\u1da0 x in l, k x \u2260 0 := hk.eventually_ne_atTop 0 ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 hk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0 \u22a2 f =o[l] fun a => k a ^ z ** have : (fun _ : \u03b1 => (1 : \u03b2)) =o[l] k :=\n isLittleO_of_tendsto' (hk0.mono fun x hkx hkx' => absurd hkx' hkx)\n (by simpa using hk.inv_tendsto_atTop) ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 hk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0 this : (fun x => 1) =o[l] k \u22a2 f =o[l] fun a => k a ^ z ** have : f =o[l] fun x : \u03b1 => k x * k x ^ (z - 1) := by\n simpa using this.mul_isBigO ((superpolynomialDecay_iff_isBigO f hk).1 h <| z - 1) ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 hk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0 this\u271d : (fun x => 1) =o[l] k this : f =o[l] fun x => k x * k x ^ (z - 1) \u22a2 f =o[l] fun a => k a ^ z ** refine' this.trans_isBigO (IsBigO.of_bound 1 (hk0.mono fun x hkx => le_of_eq _)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 hk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0 this\u271d : (fun x => 1) =o[l] k this : f =o[l] fun x => k x * k x ^ (z - 1) x : \u03b1 hkx : k x \u2260 0 \u22a2 \u2016k x * k x ^ (z - 1)\u2016 = 1 * \u2016k x ^ z\u2016 ** rw [one_mul, zpow_sub_one\u2080 hkx, mul_comm (k x), mul_assoc, inv_mul_cancel hkx, mul_one] ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 hk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0 \u22a2 Tendsto (fun x => 1 / k x) l (\ud835\udcdd 0) ** simpa using hk.inv_tendsto_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 l : Filter \u03b1 k f g g' : \u03b1 \u2192 \u03b2 inst\u271d\u00b9 : NormedLinearOrderedField \u03b2 inst\u271d : OrderTopology \u03b2 hk : Tendsto k l atTop h : SuperpolynomialDecay l k f z : \u2124 hk0 : \u2200\u1da0 (x : \u03b1) in l, k x \u2260 0 this : (fun x => 1) =o[l] k \u22a2 f =o[l] fun x => k x * k x ^ (z - 1) ** simpa using this.mul_isBigO ((superpolynomialDecay_iff_isBigO f hk).1 h <| z - 1) ** Qed", + "informal": "" + }, + { + "formal": "List.count_concat ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 a : \u03b1 l : List \u03b1 \u22a2 count a (concat l a) = succ (count a l) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Filter.HasBasis.nhds ** \u03b9 : Sort u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 X : Type u_4 Y : Type u_5 l : Filter \u03b1 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 h : HasBasis l p s \u22a2 HasBasis (\ud835\udcdd l) p fun i => Iic (\ud835\udcdf (s i)) ** rw [nhds_eq] ** \u03b9 : Sort u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 X : Type u_4 Y : Type u_5 l : Filter \u03b1 p : \u03b9 \u2192 Prop s : \u03b9 \u2192 Set \u03b1 h : HasBasis l p s \u22a2 HasBasis (Filter.lift' l (Iic \u2218 \ud835\udcdf)) p fun i => Iic (\ud835\udcdf (s i)) ** exact h.lift' monotone_principal.Iic ** Qed", + "informal": "" + }, + { + "formal": "Nat.gcd_mul_right_right ** m n : Nat \u22a2 gcd n (n * m) = n ** rw [gcd_comm, gcd_mul_right_left] ** Qed", + "informal": "" + }, + { + "formal": "Ico_subset_closure_interior ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : DenselyOrdered \u03b1 a\u271d b\u271d : \u03b1 s : Set \u03b1 a b : \u03b1 \u22a2 Ico a b \u2286 closure (interior (Ico a b)) ** simpa only [dual_Ioc] using Ioc_subset_closure_interior (OrderDual.toDual b) (OrderDual.toDual a) ** Qed", + "informal": "" + }, + { + "formal": "\u03b5NFA.evalFrom_append_singleton ** \u03b1 : Type u \u03c3 \u03c3' : Type v M : \u03b5NFA \u03b1 \u03c3 S\u271d : Set \u03c3 x\u271d : List \u03b1 s : \u03c3 a\u271d : \u03b1 S : Set \u03c3 x : List \u03b1 a : \u03b1 \u22a2 evalFrom M S (x ++ [a]) = stepSet M (evalFrom M S x) a ** rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil] ** Qed", + "informal": "" + }, + { + "formal": "NonarchAddGroupSeminorm.add_bddBelow_range_add ** \u03b9 : Type u_1 R : Type u_2 R' : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b9 : AddCommGroup E inst\u271d : AddCommGroup F p\u271d q\u271d : NonarchAddGroupSeminorm E x\u271d y : E p q : NonarchAddGroupSeminorm E x : E \u22a2 0 \u2208 lowerBounds (range fun y => \u2191p y + \u2191q (x - y)) ** rintro _ \u27e8x, rfl\u27e9 ** case intro \u03b9 : Type u_1 R : Type u_2 R' : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b9 : AddCommGroup E inst\u271d : AddCommGroup F p\u271d q\u271d : NonarchAddGroupSeminorm E x\u271d\u00b9 y : E p q : NonarchAddGroupSeminorm E x\u271d x : E \u22a2 0 \u2264 (fun y => \u2191p y + \u2191q (x\u271d - y)) x ** dsimp ** case intro \u03b9 : Type u_1 R : Type u_2 R' : Type u_3 E : Type u_4 F : Type u_5 G : Type u_6 inst\u271d\u00b9 : AddCommGroup E inst\u271d : AddCommGroup F p\u271d q\u271d : NonarchAddGroupSeminorm E x\u271d\u00b9 y : E p q : NonarchAddGroupSeminorm E x\u271d x : E \u22a2 0 \u2264 \u2191p x + \u2191q (x\u271d - x) ** positivity ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.prod.lift_fst_comp_snd_comp ** C : Type u inst\u271d\u00b2 : Category.{v, u} C X\u271d Y\u271d W X Y Z : C inst\u271d\u00b9 : HasBinaryProduct W Y inst\u271d : HasBinaryProduct X Z g : W \u27f6 X g' : Y \u27f6 Z \u22a2 lift (fst \u226b g) (snd \u226b g') = map g g' ** rw [\u2190 prod.lift_map] ** C : Type u inst\u271d\u00b2 : Category.{v, u} C X\u271d Y\u271d W X Y Z : C inst\u271d\u00b9 : HasBinaryProduct W Y inst\u271d : HasBinaryProduct X Z g : W \u27f6 X g' : Y \u27f6 Z \u22a2 lift fst snd \u226b map g g' = map g g' ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.exists_simpleFunc_snorm_sub_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** borelize E ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** let f' := hf.1.mk f ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** rsuffices \u27e8g, hg, g_mem\u27e9 : \u2203 g : \u03b2 \u2192\u209b E, snorm (f' - \u21d1g) p \u03bc < \u03b5 \u2227 Mem\u2112p g p \u03bc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) \u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** have hf' : Mem\u2112p f' p \u03bc := hf.ae_eq hf.1.ae_eq_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hf' : Mem\u2112p f' p \u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** have f'meas : Measurable f' := hf.1.measurable_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hf' : Mem\u2112p f' p f'meas : Measurable f' \u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** have : SeparableSpace (range f' \u222a {0} : Set E) :=\n StronglyMeasurable.separableSpace_range_union_singleton hf.1.stronglyMeasurable_mk ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hf' : Mem\u2112p f' p f'meas : Measurable f' this : SeparableSpace \u2191(Set.range f' \u222a {0}) \u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** rcases ((tendsto_approxOn_range_Lp_snorm hp_ne_top f'meas hf'.2).eventually <|\n gt_mem_nhds h\u03b5.bot_lt).exists with \u27e8n, hn\u27e9 ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hf' : Mem\u2112p f' p f'meas : Measurable f' this : SeparableSpace \u2191(Set.range f' \u222a {0}) n : \u2115 hn : snorm (\u2191(approxOn f' f'meas (Set.range f' \u222a {0}) 0 (_ : 0 \u2208 Set.range f' \u222a {0}) n) - f') p \u03bc < \u03b5 \u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** rw [\u2190 snorm_neg, neg_sub] at hn ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) hf' : Mem\u2112p f' p f'meas : Measurable f' this : SeparableSpace \u2191(Set.range f' \u222a {0}) n : \u2115 hn : snorm (f' - \u2191(approxOn f' f'meas (Set.range f' \u222a {0}) 0 (_ : 0 \u2208 Set.range f' \u222a {0}) n)) p \u03bc < \u03b5 \u22a2 \u2203 g, snorm (f' - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** exact \u27e8_, hn, mem\u2112p_approxOn_range f'meas hf' _\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) g : \u03b2 \u2192\u209b E hg : snorm (f' - \u2191g) p \u03bc < \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 \u2203 g, snorm (f - \u2191g) p \u03bc < \u03b5 \u2227 Mem\u2112p (\u2191g) p ** refine' \u27e8g, _, g_mem\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) g : \u03b2 \u2192\u209b E hg : snorm (f' - \u2191g) p \u03bc < \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 snorm (f - \u2191g) p \u03bc < \u03b5 ** suffices snorm (f - \u21d1g) p \u03bc = snorm (f' - \u21d1g) p \u03bc by rwa [this] ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) g : \u03b2 \u2192\u209b E hg : snorm (f' - \u2191g) p \u03bc < \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 snorm (f - \u2191g) p \u03bc = snorm (f' - \u2191g) p \u03bc ** apply snorm_congr_ae ** case intro.intro.hfg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) g : \u03b2 \u2192\u209b E hg : snorm (f' - \u2191g) p \u03bc < \u03b5 g_mem : Mem\u2112p (\u2191g) p \u22a2 f - \u2191g =\u1d50[\u03bc] f' - \u2191g ** filter_upwards [hf.1.ae_eq_mk] with x hx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) g : \u03b2 \u2192\u209b E hg : snorm (f' - \u2191g) p \u03bc < \u03b5 g_mem : Mem\u2112p (\u2191g) p x : \u03b2 hx : f x = AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) x \u22a2 (f - \u2191g) x = (f' - \u2191g) x ** simpa only [Pi.sub_apply, sub_left_inj] using hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 E\u271d : Type u_4 F : Type u_5 \ud835\udd5c : Type u_6 inst\u271d\u2074 : MeasurableSpace \u03b2 inst\u271d\u00b3 : MeasurableSpace E\u271d inst\u271d\u00b2 : NormedAddCommGroup E\u271d inst\u271d\u00b9 : NormedAddCommGroup F q : \u211d p : \u211d\u22650\u221e E : Type u_7 inst\u271d : NormedAddCommGroup E f : \u03b2 \u2192 E \u03bc : Measure \u03b2 hf : Mem\u2112p f p hp_ne_top : p \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e h\u03b5 : \u03b5 \u2260 0 this\u271d\u00b9 : MeasurableSpace E := borel E this\u271d : BorelSpace E f' : \u03b2 \u2192 E := AEStronglyMeasurable.mk f (_ : AEStronglyMeasurable f \u03bc) g : \u03b2 \u2192\u209b E hg : snorm (f' - \u2191g) p \u03bc < \u03b5 g_mem : Mem\u2112p (\u2191g) p this : snorm (f - \u2191g) p \u03bc = snorm (f' - \u2191g) p \u03bc \u22a2 snorm (f - \u2191g) p \u03bc < \u03b5 ** rwa [this] ** Qed", + "informal": "" + }, + { + "formal": "le_suffixLevenshtein_cons_minimum ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 \u22a2 List.minimum \u2191(suffixLevenshtein C xs ys) \u2264 List.minimum \u2191(suffixLevenshtein C xs (y :: ys)) ** apply List.le_minimum_of_forall_le ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 \u22a2 \u2200 (a : \u03b4), a \u2208 \u2191(suffixLevenshtein C xs (y :: ys)) \u2192 List.minimum \u2191(suffixLevenshtein C xs ys) \u2264 \u2191a ** simp only [suffixLevenshtein_eq_tails_map] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 \u22a2 \u2200 (a : \u03b4), a \u2208 List.map (fun xs' => levenshtein C xs' (y :: ys)) (List.tails xs) \u2192 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191a ** simp only [List.mem_map, List.mem_tails, forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 \u22a2 \u2200 (a : List \u03b1), a <:+ xs \u2192 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191(levenshtein C a (y :: ys)) ** intro a suff ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs \u22a2 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191(levenshtein C a (y :: ys)) ** refine (?_ : _ \u2264 _).trans (suffixLevenshtein_minimum_le_levenshtein_cons _ _ _) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs \u22a2 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 List.minimum \u2191(suffixLevenshtein C a ys) ** simp only [suffixLevenshtein_eq_tails_map] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs \u22a2 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails a)) ** apply List.le_minimum_of_forall_le ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs \u22a2 \u2200 (a_1 : \u03b4), a_1 \u2208 List.map (fun xs' => levenshtein C xs' ys) (List.tails a) \u2192 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191a_1 ** intro b m ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs b : \u03b4 m : b \u2208 List.map (fun xs' => levenshtein C xs' ys) (List.tails a) \u22a2 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191b ** replace m : \u2203 a_1, a_1 <:+ a \u2227 levenshtein C a_1 ys = b ** case h.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs b : \u03b4 m : \u2203 a_1, a_1 <:+ a \u2227 levenshtein C a_1 ys = b \u22a2 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191b ** obtain \u27e8a', suff', rfl\u27e9 := m ** case h.h.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs a' : List \u03b1 suff' : a' <:+ a \u22a2 List.minimum (List.map (fun xs' => levenshtein C xs' ys) (List.tails xs)) \u2264 \u2191(levenshtein C a' ys) ** apply List.minimum_le_of_mem' ** case h.h.intro.intro.ha \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs a' : List \u03b1 suff' : a' <:+ a \u22a2 levenshtein C a' ys \u2208 List.map (fun xs' => levenshtein C xs' ys) (List.tails xs) ** simp only [List.mem_map, List.mem_tails] ** case h.h.intro.intro.ha \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs a' : List \u03b1 suff' : a' <:+ a \u22a2 \u2203 a, a <:+ xs \u2227 levenshtein C a ys = levenshtein C a' ys ** exact \u27e8a', suff'.trans suff, rfl\u27e9 ** case m \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs b : \u03b4 m : b \u2208 List.map (fun xs' => levenshtein C xs' ys) (List.tails a) \u22a2 \u2203 a_1, a_1 <:+ a \u2227 levenshtein C a_1 ys = b ** simpa using m ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b4 : Type u_3 C : Levenshtein.Cost \u03b1 \u03b2 \u03b4 inst\u271d : CanonicallyLinearOrderedAddCommMonoid \u03b4 xs : List \u03b1 y : \u03b2 ys : List \u03b2 a : List \u03b1 suff : a <:+ xs a' : List \u03b1 suff' : a' <:+ a this : \u2203 a, a <:+ xs \u2227 levenshtein C a ys = levenshtein C a' ys \u22a2 \u2203 a, a <:+ xs \u2227 levenshtein C a ys = levenshtein C a' ys ** simpa ** Qed", + "informal": "" + }, + { + "formal": "Complex.cos_ofReal_im ** x\u271d y : \u2102 x : \u211d \u22a2 (cos \u2191x).im = 0 ** rw [\u2190 ofReal_cos_ofReal_re, ofReal_im] ** Qed", + "informal": "" + }, + { + "formal": "Real.continuousAt_log_iff ** x y : \u211d \u22a2 ContinuousAt log x \u2194 x \u2260 0 ** refine' \u27e8_, continuousAt_log\u27e9 ** x y : \u211d \u22a2 ContinuousAt log x \u2192 x \u2260 0 ** rintro h rfl ** y : \u211d h : ContinuousAt log 0 \u22a2 False ** exact not_tendsto_nhds_of_tendsto_atBot tendsto_log_nhdsWithin_zero _\n (h.tendsto.mono_left inf_le_left) ** Qed", + "informal": "" + }, + { + "formal": "MonoidAlgebra.support_mul_single ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b9 : Semiring k inst\u271d : RightCancelSemigroup G f : MonoidAlgebra k G r : k hr : \u2200 (y : k), y * r = 0 \u2194 y = 0 x : G \u22a2 (f * single x r).support = map (mulRightEmbedding x) f.support ** classical\n ext\n simp only [support_mul_single_eq_image f hr (isRightRegular_of_rightCancelSemigroup x),\n mem_image, mem_map, mulRightEmbedding_apply] ** k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b9 : Semiring k inst\u271d : RightCancelSemigroup G f : MonoidAlgebra k G r : k hr : \u2200 (y : k), y * r = 0 \u2194 y = 0 x : G \u22a2 (f * single x r).support = map (mulRightEmbedding x) f.support ** ext ** case a k : Type u\u2081 G : Type u\u2082 inst\u271d\u00b9 : Semiring k inst\u271d : RightCancelSemigroup G f : MonoidAlgebra k G r : k hr : \u2200 (y : k), y * r = 0 \u2194 y = 0 x a\u271d : G \u22a2 a\u271d \u2208 (f * single x r).support \u2194 a\u271d \u2208 map (mulRightEmbedding x) f.support ** simp only [support_mul_single_eq_image f hr (isRightRegular_of_rightCancelSemigroup x),\n mem_image, mem_map, mulRightEmbedding_apply] ** Qed", + "informal": "" + }, + { + "formal": "edist_inv ** M : Type u G : Type v X : Type w inst\u271d\u2076 : PseudoEMetricSpace X inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G X inst\u271d\u00b3 : IsometricSMul G X inst\u271d\u00b2 : PseudoEMetricSpace G inst\u271d\u00b9 : IsometricSMul G G inst\u271d : IsometricSMul G\u1d50\u1d52\u1d56 G x y : G \u22a2 edist x\u207b\u00b9 y = edist x y\u207b\u00b9 ** rw [\u2190 edist_inv_inv, inv_inv] ** Qed", + "informal": "" + }, + { + "formal": "Filter.tendstoIxxClass_principal ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Preorder \u03b1 s t : Set \u03b1 Ixx : \u03b1 \u2192 \u03b1 \u2192 Set \u03b1 \u22a2 Tendsto (fun p => Ixx p.1 p.2) (\ud835\udcdf s \u00d7\u02e2 \ud835\udcdf s) (smallSets (\ud835\udcdf t)) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 Ixx x y \u2286 t ** simp only [smallSets_principal, prod_principal_principal, tendsto_principal_principal,\n forall_prod_set, mem_powerset_iff, mem_principal] ** Qed", + "informal": "" + }, + { + "formal": "smul_ite ** M : Type u_1 N : Type u_2 G : Type u_3 A : Type u_4 B : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \u03b4 : Type u_9 inst\u271d\u00b9 : SMul M \u03b1 p : Prop inst\u271d : Decidable p a : M b\u2081 b\u2082 : \u03b1 \u22a2 (a \u2022 if p then b\u2081 else b\u2082) = if p then a \u2022 b\u2081 else a \u2022 b\u2082 ** split_ifs <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "IsHilbertSum.mkInternal ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u2075 : IsROrC \ud835\udd5c E : Type u_3 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E cplt : CompleteSpace E G : \u03b9 \u2192 Type u_4 inst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i) inst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i) V : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E F : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : \u2200 (i : \u03b9), CompleteSpace { x // x \u2208 F i } hFortho : OrthogonalFamily \ud835\udd5c (fun i => { x // x \u2208 F i }) fun i => subtype\u2097\u1d62 (F i) hFtotal : \u22a4 \u2264 topologicalClosure (\u2a06 i, F i) \u22a2 \u22a4 \u2264 topologicalClosure (\u2a06 i, LinearMap.range (subtype\u2097\u1d62 (F i)).toLinearMap) ** simpa [subtype\u2097\u1d62_toLinearMap, range_subtype] using hFtotal ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.natDegree_mul_mirror ** R : Type u_1 inst\u271d\u00b9 : Semiring R p q : R[X] inst\u271d : NoZeroDivisors R \u22a2 natDegree (p * mirror p) = 2 * natDegree p ** by_cases hp : p = 0 ** case neg R : Type u_1 inst\u271d\u00b9 : Semiring R p q : R[X] inst\u271d : NoZeroDivisors R hp : \u00acp = 0 \u22a2 natDegree (p * mirror p) = 2 * natDegree p ** rw [natDegree_mul hp (mt mirror_eq_zero.mp hp), mirror_natDegree, two_mul] ** case pos R : Type u_1 inst\u271d\u00b9 : Semiring R p q : R[X] inst\u271d : NoZeroDivisors R hp : p = 0 \u22a2 natDegree (p * mirror p) = 2 * natDegree p ** rw [hp, zero_mul, natDegree_zero, mul_zero] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.integral_finset_sum ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) \u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc ** by_cases hG : CompleteSpace G ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) hG : CompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc ** simp only [integral, hG, L1.integral] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) hG : CompleteSpace G \u22a2 (if h : True then if hf : Integrable fun a => \u2211 i in s, f i a then \u2191L1.integralCLM (Integrable.toL1 (fun a => \u2211 i in s, f i a) hf) else 0 else 0) = \u2211 x in s, if h : True then if hf : Integrable fun a => f x a then \u2191L1.integralCLM (Integrable.toL1 (fun a => f x a) hf) else 0 else 0 ** exact setToFun_finset_sum (dominatedFinMeasAdditive_weightedSMul _) s hf ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace F G : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03b9 : Type u_6 s : Finset \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 G hf : \u2200 (i : \u03b9), i \u2208 s \u2192 Integrable (f i) hG : \u00acCompleteSpace G \u22a2 \u222b (a : \u03b1), \u2211 i in s, f i a \u2202\u03bc = \u2211 i in s, \u222b (a : \u03b1), f i a \u2202\u03bc ** simp [integral, hG] ** Qed", + "informal": "" + }, + { + "formal": "Affine.Simplex.sum_pointsWithCircumcenter ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 h : univ = insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) \u22a2 \u2211 i : PointsWithCircumcenterIndex n, f i = \u2211 i : Fin (n + 1), f (point_index i) + f circumcenter_index ** change _ = (\u2211 i, f (pointIndexEmbedding n i)) + _ ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 h : univ = insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) \u22a2 \u2211 i : PointsWithCircumcenterIndex n, f i = \u2211 i : Fin (n + 1), f (\u2191(pointIndexEmbedding n) i) + f circumcenter_index ** rw [add_comm, h, \u2190 sum_map, sum_insert] ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 h : univ = insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) \u22a2 \u00accircumcenter_index \u2208 Finset.map (pointIndexEmbedding n) univ ** simp_rw [Finset.mem_map, not_exists] ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 h : univ = insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) \u22a2 \u2200 (x : Fin (n + 1)), \u00ac(x \u2208 univ \u2227 \u2191(pointIndexEmbedding n) x = circumcenter_index) ** rintro x \u27e8_, h\u27e9 ** case intro V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 h\u271d : univ = insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) x : Fin (n + 1) left\u271d : x \u2208 univ h : \u2191(pointIndexEmbedding n) x = circumcenter_index \u22a2 False ** injection h ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 \u22a2 univ = insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) ** ext x ** case a V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 x : PointsWithCircumcenterIndex n \u22a2 x \u2208 univ \u2194 x \u2208 insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) ** refine' \u27e8fun h => _, fun _ => mem_univ _\u27e9 ** case a V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 x : PointsWithCircumcenterIndex n h : x \u2208 univ \u22a2 x \u2208 insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) ** cases' x with i ** case a.point_index V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 i : Fin (n + 1) h : point_index i \u2208 univ \u22a2 point_index i \u2208 insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) ** exact mem_insert_of_mem (mem_map_of_mem _ (mem_univ i)) ** case a.circumcenter_index V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P \u03b1 : Type u_3 inst\u271d : AddCommMonoid \u03b1 n : \u2115 f : PointsWithCircumcenterIndex n \u2192 \u03b1 h : circumcenter_index \u2208 univ \u22a2 circumcenter_index \u2208 insert circumcenter_index (Finset.map (pointIndexEmbedding n) univ) ** exact mem_insert_self _ _ ** Qed", + "informal": "" + }, + { + "formal": "ZNum.zneg_zneg ** \u03b1 : Type u_1 n : ZNum \u22a2 - -n = n ** cases n <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "AddChar.pow_mulShift ** R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' n : \u2115 \u22a2 \u03c8 ^ n = mulShift \u03c8 \u2191n ** ext x ** case h R : Type u inst\u271d\u00b9 : CommRing R R' : Type v inst\u271d : CommRing R' \u03c8 : AddChar R R' n : \u2115 x : R \u22a2 \u2191(\u03c8 ^ n) x = \u2191(mulShift \u03c8 \u2191n) x ** rw [show (\u03c8 ^ n) x = \u03c8 x ^ n from rfl, \u2190 mulShift_spec'] ** Qed", + "informal": "" + }, + { + "formal": "List.mem_zip ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 a : \u03b1 b : \u03b2 head\u271d\u00b9 : \u03b1 l\u2081 : List \u03b1 head\u271d : \u03b2 l\u2082 : List \u03b2 h : (a, b) \u2208 zip (head\u271d\u00b9 :: l\u2081) (head\u271d :: l\u2082) \u22a2 a \u2208 head\u271d\u00b9 :: l\u2081 \u2227 b \u2208 head\u271d :: l\u2082 ** cases' h with _ _ _ h ** case head \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 a : \u03b1 b : \u03b2 l\u2081 : List \u03b1 l\u2082 : List \u03b2 \u22a2 a \u2208 a :: l\u2081 \u2227 b \u2208 b :: l\u2082 ** simp ** case tail \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 a : \u03b1 b : \u03b2 head\u271d\u00b9 : \u03b1 l\u2081 : List \u03b1 head\u271d : \u03b2 l\u2082 : List \u03b2 h : Mem (a, b) (zipWith Prod.mk l\u2081 l\u2082) \u22a2 a \u2208 head\u271d\u00b9 :: l\u2081 \u2227 b \u2208 head\u271d :: l\u2082 ** have := mem_zip h ** case tail \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 \u03b4 : Type u_3 \u03b5 : Type u_4 a : \u03b1 b : \u03b2 head\u271d\u00b9 : \u03b1 l\u2081 : List \u03b1 head\u271d : \u03b2 l\u2082 : List \u03b2 h : Mem (a, b) (zipWith Prod.mk l\u2081 l\u2082) this : a \u2208 l\u2081 \u2227 b \u2208 l\u2082 \u22a2 a \u2208 head\u271d\u00b9 :: l\u2081 \u2227 b \u2208 head\u271d :: l\u2082 ** exact \u27e8Mem.tail _ this.1, Mem.tail _ this.2\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Function.mulSupport_prod ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d : CommMonoid M s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 \u2192 M \u22a2 (mulSupport fun x => \u220f i in s, f i x) \u2286 \u22c3 i \u2208 s, mulSupport (f i) ** rw [mulSupport_subset_iff'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d : CommMonoid M s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 \u2192 M \u22a2 \u2200 (x : \u03b2), \u00acx \u2208 \u22c3 i \u2208 s, mulSupport (f i) \u2192 \u220f i in s, f i x = 1 ** simp only [mem_iUnion, not_exists, nmem_mulSupport] ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d : CommMonoid M s : Finset \u03b1 f : \u03b1 \u2192 \u03b2 \u2192 M \u22a2 \u2200 (x : \u03b2), (\u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 f x_1 x = 1) \u2192 \u220f i in s, f i x = 1 ** exact fun x => Finset.prod_eq_one ** Qed", + "informal": "" + }, + { + "formal": "Finpartition.mem_avoid ** \u03b1 : Type u_1 inst\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 inst\u271d : DecidableEq \u03b1 a b c : \u03b1 P : Finpartition a \u22a2 c \u2208 (avoid P b).parts \u2194 \u2203 d, d \u2208 P.parts \u2227 \u00acd \u2264 b \u2227 d \\ b = c ** simp only [avoid, ofErase, mem_erase, Ne.def, mem_image, exists_prop, \u2190 exists_and_left,\n @and_left_comm (c \u2260 \u22a5)] ** \u03b1 : Type u_1 inst\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 inst\u271d : DecidableEq \u03b1 a b c : \u03b1 P : Finpartition a \u22a2 (\u2203 x, x \u2208 P.parts \u2227 \u00acc = \u22a5 \u2227 x \\ b = c) \u2194 \u2203 d, d \u2208 P.parts \u2227 \u00acd \u2264 b \u2227 d \\ b = c ** refine' exists_congr fun d \u21a6 and_congr_right' <| and_congr_left _ ** \u03b1 : Type u_1 inst\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 inst\u271d : DecidableEq \u03b1 a b c : \u03b1 P : Finpartition a d : \u03b1 \u22a2 d \\ b = c \u2192 (\u00acc = \u22a5 \u2194 \u00acd \u2264 b) ** rintro rfl ** \u03b1 : Type u_1 inst\u271d\u00b9 : GeneralizedBooleanAlgebra \u03b1 inst\u271d : DecidableEq \u03b1 a b : \u03b1 P : Finpartition a d : \u03b1 \u22a2 \u00acd \\ b = \u22a5 \u2194 \u00acd \u2264 b ** rw [sdiff_eq_bot_iff] ** Qed", + "informal": "" + }, + { + "formal": "Subtype.coind_injective ** \u03b1\u271d : Sort u_1 \u03b2\u271d : Sort u_2 \u03b3 : Sort u_3 p\u271d q : \u03b1\u271d \u2192 Prop \u03b1 : Sort u_4 \u03b2 : Sort u_5 f : \u03b1 \u2192 \u03b2 p : \u03b2 \u2192 Prop h : \u2200 (a : \u03b1), p (f a) hf : Injective f x y : \u03b1 hxy : coind f h x = coind f h y \u22a2 f x = f y ** apply congr_arg Subtype.val hxy ** Qed", + "informal": "" + }, + { + "formal": "Ideal.subset_union_prime ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R this : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i \u22a2 \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2194 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** have aux := fun h => (bex_def.2 <| this h) ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R this : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i aux : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2192 \u2203 x x_1, I \u2264 f x \u22a2 \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2194 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** simp_rw [exists_prop] at aux ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R this aux : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2192 \u2203 x, x \u2208 s \u2227 I \u2264 f x \u22a2 \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2194 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** refine \u27e8aux, fun \u27e8i, his, hi\u27e9 \u21a6 Set.Subset.trans hi ?_\u27e9 ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R this aux : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u2192 \u2203 x, x \u2208 s \u2227 I \u2264 f x x\u271d : \u2203 i, i \u2208 s \u2227 I \u2264 f i i : \u03b9 his : i \u2208 s hi : I \u2264 f i \u22a2 \u2191(f i) \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) ** apply Set.subset_biUnion_of_mem (show i \u2208 (\u2191s : Set \u03b9) from his) ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** by_cases has : a \u2208 s ** case pos R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : a \u2208 s \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** obtain \u27e8t, hat, rfl\u27e9 : \u2203 t, a \u2209 t \u2227 insert a t = s :=\n \u27e8s.erase a, Finset.not_mem_erase a s, Finset.insert_erase has\u27e9 ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a t), \u2191(f i) has : a \u2208 insert a t \u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i ** by_cases hbt : b \u2208 t ** case pos R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a t), \u2191(f i) has : a \u2208 insert a t hbt : b \u2208 t \u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i ** obtain \u27e8u, hbu, rfl\u27e9 : \u2203 u, b \u2209 u \u2227 insert b u = t :=\n \u27e8t.erase b, Finset.not_mem_erase b t, Finset.insert_erase hbt\u27e9 ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R u : Finset \u03b9 hbu : \u00acb \u2208 u hat : \u00aca \u2208 insert b u hp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a (insert b u)), \u2191(f i) has : a \u2208 insert a (insert b u) hbt : b \u2208 insert b u \u22a2 \u2203 i, i \u2208 insert a (insert b u) \u2227 I \u2264 f i ** have hp' : \u2200 i \u2208 u, IsPrime (f i) := by\n intro i hiu\n refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;>\n rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R u : Finset \u03b9 hbu : \u00acb \u2208 u hat : \u00aca \u2208 insert b u hp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a (insert b u)), \u2191(f i) has : a \u2208 insert a (insert b u) hbt : b \u2208 insert b u hp' : \u2200 (i : \u03b9), i \u2208 u \u2192 IsPrime (f i) \u22a2 \u2203 i, i \u2208 insert a (insert b u) \u2227 I \u2264 f i ** rw [Finset.coe_insert, Finset.coe_insert, Set.biUnion_insert, Set.biUnion_insert, \u2190\n Set.union_assoc, subset_union_prime' hp'] at h ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R u : Finset \u03b9 hbu : \u00acb \u2208 u hat : \u00aca \u2208 insert b u hp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i, i \u2208 u \u2227 I \u2264 f i has : a \u2208 insert a (insert b u) hbt : b \u2208 insert b u hp' : \u2200 (i : \u03b9), i \u2208 u \u2192 IsPrime (f i) \u22a2 \u2203 i, i \u2208 insert a (insert b u) \u2227 I \u2264 f i ** rwa [Finset.exists_mem_insert, Finset.exists_mem_insert] ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R u : Finset \u03b9 hbu : \u00acb \u2208 u hat : \u00aca \u2208 insert b u hp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a (insert b u)), \u2191(f i) has : a \u2208 insert a (insert b u) hbt : b \u2208 insert b u \u22a2 \u2200 (i : \u03b9), i \u2208 u \u2192 IsPrime (f i) ** intro i hiu ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R u : Finset \u03b9 hbu : \u00acb \u2208 u hat : \u00aca \u2208 insert b u hp : \u2200 (i : \u03b9), i \u2208 insert a (insert b u) \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a (insert b u)), \u2191(f i) has : a \u2208 insert a (insert b u) hbt : b \u2208 insert b u i : \u03b9 hiu : i \u2208 u \u22a2 IsPrime (f i) ** refine' hp i (Finset.mem_insert_of_mem (Finset.mem_insert_of_mem hiu)) _ _ <;>\n rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case neg R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a t), \u2191(f i) has : a \u2208 insert a t hbt : \u00acb \u2208 t \u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i ** have hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n intro j hj\n refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case neg R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a t), \u2191(f i) has : a \u2208 insert a t hbt : \u00acb \u2208 t hp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) \u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i ** rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f a : Set R),\n subset_union_prime' hp', \u2190 or_assoc, or_self_iff] at h ** case neg R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : I \u2264 f a \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i has : a \u2208 insert a t hbt : \u00acb \u2208 t hp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) \u22a2 \u2203 i, i \u2208 insert a t \u2227 I \u2264 f i ** rwa [Finset.exists_mem_insert] ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a t), \u2191(f i) has : a \u2208 insert a t hbt : \u00acb \u2208 t \u22a2 \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) ** intro j hj ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hat : \u00aca \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert a t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert a t), \u2191(f i) has : a \u2208 insert a t hbt : \u00acb \u2208 t j : \u03b9 hj : j \u2208 t \u22a2 IsPrime (f j) ** refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case neg R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : \u00aca \u2208 s \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** by_cases hbs : b \u2208 s ** case neg R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : \u00aca \u2208 s hbs : \u00acb \u2208 s \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** cases' s.eq_empty_or_nonempty with hse hsne ** case pos R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : \u00aca \u2208 s hbs : b \u2208 s \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** obtain \u27e8t, hbt, rfl\u27e9 : \u2203 t, b \u2209 t \u2227 insert b t = s :=\n \u27e8s.erase b, Finset.not_mem_erase b s, Finset.insert_erase hbs\u27e9 ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hbt : \u00acb \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert b t), \u2191(f i) has : \u00aca \u2208 insert b t hbs : b \u2208 insert b t \u22a2 \u2203 i, i \u2208 insert b t \u2227 I \u2264 f i ** have hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n intro j hj\n refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hbt : \u00acb \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert b t), \u2191(f i) has : \u00aca \u2208 insert b t hbs : b \u2208 insert b t hp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) \u22a2 \u2203 i, i \u2208 insert b t \u2227 I \u2264 f i ** rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f b : Set R),\n subset_union_prime' hp', \u2190 or_assoc, or_self_iff] at h ** case pos.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hbt : \u00acb \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : I \u2264 f b \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i has : \u00aca \u2208 insert b t hbs : b \u2208 insert b t hp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) \u22a2 \u2203 i, i \u2208 insert b t \u2227 I \u2264 f i ** rwa [Finset.exists_mem_insert] ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hbt : \u00acb \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert b t), \u2191(f i) has : \u00aca \u2208 insert b t hbs : b \u2208 insert b t \u22a2 \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) ** intro j hj ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R t : Finset \u03b9 hbt : \u00acb \u2208 t hp : \u2200 (i : \u03b9), i \u2208 insert b t \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191(insert b t), \u2191(f i) has : \u00aca \u2208 insert b t hbs : b \u2208 insert b t j : \u03b9 hj : j \u2208 t \u22a2 IsPrime (f j) ** refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case neg.inl R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : \u00aca \u2208 s hbs : \u00acb \u2208 s hse : s = \u2205 \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** subst hse ** case neg.inl R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R hp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I \u2286 \u22c3 i \u2208 \u2191\u2205, \u2191(f i) has : \u00aca \u2208 \u2205 hbs : \u00acb \u2208 \u2205 \u22a2 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i ** rw [Finset.coe_empty, Set.biUnion_empty, Set.subset_empty_iff] at h ** case neg.inl R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R hp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I = \u2205 has : \u00aca \u2208 \u2205 hbs : \u00acb \u2208 \u2205 \u22a2 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i ** have : (I : Set R) \u2260 \u2205 := Set.Nonempty.ne_empty (Set.nonempty_of_mem I.zero_mem) ** case neg.inl R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R hp : \u2200 (i : \u03b9), i \u2208 \u2205 \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) h : \u2191I = \u2205 has : \u00aca \u2208 \u2205 hbs : \u00acb \u2208 \u2205 this : \u2191I \u2260 \u2205 \u22a2 \u2203 i, i \u2208 \u2205 \u2227 I \u2264 f i ** exact absurd h this ** case neg.inr R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : \u00aca \u2208 s hbs : \u00acb \u2208 s hsne : Finset.Nonempty s \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** cases' hsne.bex with i his ** case neg.inr.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R s : Finset \u03b9 f : \u03b9 \u2192 Ideal R a b : \u03b9 hp : \u2200 (i : \u03b9), i \u2208 s \u2192 i \u2260 a \u2192 i \u2260 b \u2192 IsPrime (f i) I : Ideal R h : \u2191I \u2286 \u22c3 i \u2208 \u2191s, \u2191(f i) has : \u00aca \u2208 s hbs : \u00acb \u2208 s hsne : Finset.Nonempty s i : \u03b9 his : i \u2208 s \u22a2 \u2203 i, i \u2208 s \u2227 I \u2264 f i ** obtain \u27e8t, _, rfl\u27e9 : \u2203 t, i \u2209 t \u2227 insert i t = s :=\n \u27e8s.erase i, Finset.not_mem_erase i s, Finset.insert_erase his\u27e9 ** case neg.inr.intro.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R i : \u03b9 t : Finset \u03b9 left\u271d : \u00aci \u2208 t hp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1) h : \u2191I \u2286 \u22c3 i_1 \u2208 \u2191(insert i t), \u2191(f i_1) has : \u00aca \u2208 insert i t hbs : \u00acb \u2208 insert i t hsne : Finset.Nonempty (insert i t) his : i \u2208 insert i t \u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1 ** have hp' : \u2200 j \u2208 t, IsPrime (f j) := by\n intro j hj\n refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** case neg.inr.intro.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R i : \u03b9 t : Finset \u03b9 left\u271d : \u00aci \u2208 t hp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1) h : \u2191I \u2286 \u22c3 i_1 \u2208 \u2191(insert i t), \u2191(f i_1) has : \u00aca \u2208 insert i t hbs : \u00acb \u2208 insert i t hsne : Finset.Nonempty (insert i t) his : i \u2208 insert i t hp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) \u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1 ** rw [Finset.coe_insert, Set.biUnion_insert, \u2190 Set.union_self (f i : Set R),\n subset_union_prime' hp', \u2190 or_assoc, or_self_iff] at h ** case neg.inr.intro.intro.intro R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R i : \u03b9 t : Finset \u03b9 left\u271d : \u00aci \u2208 t hp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1) h : I \u2264 f i \u2228 \u2203 i, i \u2208 t \u2227 I \u2264 f i has : \u00aca \u2208 insert i t hbs : \u00acb \u2208 insert i t hsne : Finset.Nonempty (insert i t) his : i \u2208 insert i t hp' : \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) \u22a2 \u2203 i_1, i_1 \u2208 insert i t \u2227 I \u2264 f i_1 ** rwa [Finset.exists_mem_insert] ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R i : \u03b9 t : Finset \u03b9 left\u271d : \u00aci \u2208 t hp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1) h : \u2191I \u2286 \u22c3 i_1 \u2208 \u2191(insert i t), \u2191(f i_1) has : \u00aca \u2208 insert i t hbs : \u00acb \u2208 insert i t hsne : Finset.Nonempty (insert i t) his : i \u2208 insert i t \u22a2 \u2200 (j : \u03b9), j \u2208 t \u2192 IsPrime (f j) ** intro j hj ** R\u271d : Type u \u03b9 : Type u_1 inst\u271d\u00b9 : CommSemiring R\u271d I\u271d J K L : Ideal R\u271d R : Type u inst\u271d : CommRing R f : \u03b9 \u2192 Ideal R a b : \u03b9 I : Ideal R i : \u03b9 t : Finset \u03b9 left\u271d : \u00aci \u2208 t hp : \u2200 (i_1 : \u03b9), i_1 \u2208 insert i t \u2192 i_1 \u2260 a \u2192 i_1 \u2260 b \u2192 IsPrime (f i_1) h : \u2191I \u2286 \u22c3 i_1 \u2208 \u2191(insert i t), \u2191(f i_1) has : \u00aca \u2208 insert i t hbs : \u00acb \u2208 insert i t hsne : Finset.Nonempty (insert i t) his : i \u2208 insert i t j : \u03b9 hj : j \u2208 t \u22a2 IsPrime (f j) ** refine' hp j (Finset.mem_insert_of_mem hj) _ _ <;> rintro rfl <;>\n solve_by_elim only [Finset.mem_insert_of_mem, *] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.PushoutCocone.condition_zero ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D W X Y Z : C f : X \u27f6 Y g : X \u27f6 Z t : PushoutCocone f g \u22a2 t.\u03b9.app WalkingSpan.zero = f \u226b inl t ** have w := t.\u03b9.naturality WalkingSpan.Hom.fst ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D W X Y Z : C f : X \u27f6 Y g : X \u27f6 Z t : PushoutCocone f g w : (span f g).map fst \u226b t.\u03b9.app WalkingSpan.left = t.\u03b9.app WalkingSpan.zero \u226b ((Functor.const WalkingSpan).obj t.pt).map fst \u22a2 t.\u03b9.app WalkingSpan.zero = f \u226b inl t ** dsimp at w ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D W X Y Z : C f : X \u27f6 Y g : X \u27f6 Z t : PushoutCocone f g w : f \u226b inl t = t.\u03b9.app WalkingSpan.zero \u226b \ud835\udfd9 t.pt \u22a2 t.\u03b9.app WalkingSpan.zero = f \u226b inl t ** simpa using w.symm ** Qed", + "informal": "" + }, + { + "formal": "Matrix.linfty_op_norm_def ** R : Type u_1 l : Type u_2 m : Type u_3 n : Type u_4 \u03b1 : Type u_5 \u03b2 : Type u_6 inst\u271d\u00b3 : Fintype l inst\u271d\u00b2 : Fintype m inst\u271d\u00b9 : Fintype n inst\u271d : SeminormedAddCommGroup \u03b1 A : Matrix m n \u03b1 \u22a2 \u2016A\u2016 = \u2191(Finset.sup Finset.univ fun i => \u2211 j : n, \u2016A i j\u2016\u208a) ** change \u2016fun i => (WithLp.equiv 1 _).symm (A i)\u2016 = _ ** R : Type u_1 l : Type u_2 m : Type u_3 n : Type u_4 \u03b1 : Type u_5 \u03b2 : Type u_6 inst\u271d\u00b3 : Fintype l inst\u271d\u00b2 : Fintype m inst\u271d\u00b9 : Fintype n inst\u271d : SeminormedAddCommGroup \u03b1 A : Matrix m n \u03b1 \u22a2 \u2016fun i => \u2191(WithLp.equiv 1 (n \u2192 \u03b1)).symm (A i)\u2016 = \u2191(Finset.sup Finset.univ fun i => \u2211 j : n, \u2016A i j\u2016\u208a) ** simp [Pi.norm_def, PiLp.nnnorm_eq_sum ENNReal.one_ne_top] ** Qed", + "informal": "" + }, + { + "formal": "BoxIntegral.Box.isSome_iff ** \u03b9 : Type u_1 I J : Box \u03b9 x y : \u03b9 \u2192 \u211d \u22a2 Option.isSome \u22a5 = true \u2194 Set.Nonempty \u2191\u22a5 ** erw [Option.isSome] ** \u03b9 : Type u_1 I J : Box \u03b9 x y : \u03b9 \u2192 \u211d \u22a2 (match \u22a5 with | some val => true | none => false) = true \u2194 Set.Nonempty \u2191\u22a5 ** simp ** \u03b9 : Type u_1 I\u271d J : Box \u03b9 x y : \u03b9 \u2192 \u211d I : Box \u03b9 \u22a2 Option.isSome \u2191I = true \u2194 Set.Nonempty \u2191\u2191I ** erw [Option.isSome] ** \u03b9 : Type u_1 I\u271d J : Box \u03b9 x y : \u03b9 \u2192 \u211d I : Box \u03b9 \u22a2 (match \u2191I with | some val => true | none => false) = true \u2194 Set.Nonempty \u2191\u2191I ** simp [I.nonempty_coe] ** Qed", + "informal": "" + }, + { + "formal": "Nat.factorization_le_factorization_mul_left ** a b : \u2115 hb : b \u2260 0 \u22a2 factorization a \u2264 factorization (a * b) ** rcases eq_or_ne a 0 with (rfl | ha) ** case inr a b : \u2115 hb : b \u2260 0 ha : a \u2260 0 \u22a2 factorization a \u2264 factorization (a * b) ** rw [factorization_le_iff_dvd ha <| mul_ne_zero ha hb] ** case inr a b : \u2115 hb : b \u2260 0 ha : a \u2260 0 \u22a2 a \u2223 a * b ** exact Dvd.intro b rfl ** case inl b : \u2115 hb : b \u2260 0 \u22a2 factorization 0 \u2264 factorization (0 * b) ** simp ** Qed", + "informal": "" + }, + { + "formal": "isCoatomistic_dual_iff_isAtomistic ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : CompleteLattice \u03b1 h : IsCoatomistic \u03b1\u1d52\u1d48 b : \u03b1 \u22a2 \u2203 s, b = sSup s \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 IsAtom a ** apply h.eq_sInf_coatoms ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : CompleteLattice \u03b1 h : IsAtomistic \u03b1 b : \u03b1\u1d52\u1d48 \u22a2 \u2203 s, b = sInf s \u2227 \u2200 (a : \u03b1\u1d52\u1d48), a \u2208 s \u2192 IsCoatom a ** apply h.eq_sSup_atoms ** Qed", + "informal": "" + }, + { + "formal": "Set.ncard_eq_toFinset_card' ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 inst\u271d : Fintype \u2191s \u22a2 ncard s = Finset.card (toFinset s) ** simp [\u2190Nat.card_coe_set_eq, Nat.card_eq_fintype_card] ** Qed", + "informal": "" + }, + { + "formal": "WithTop.wellFounded_gt ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a : WithTop \u03b1 this : Acc (fun x x_1 => x < x_1) (\u2191WithTop.toDual a) \u22a2 Acc (fun x x_1 => x > x_1) a ** revert this ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a : WithTop \u03b1 \u22a2 Acc (fun x x_1 => x < x_1) (\u2191WithTop.toDual a) \u2192 Acc (fun x x_1 => x > x_1) a ** generalize ha : WithBot.toDual a = b ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a : WithTop \u03b1 b : WithTop \u03b1\u1d52\u1d48 ha : \u2191WithBot.toDual a = b \u22a2 Acc (fun x x_1 => x < x_1) b \u2192 Acc (fun x x_1 => x > x_1) a ** intro ac ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a : WithTop \u03b1 b : WithTop \u03b1\u1d52\u1d48 ha : \u2191WithBot.toDual a = b ac : Acc (fun x x_1 => x < x_1) b \u22a2 Acc (fun x x_1 => x > x_1) a ** induction' ac with _ H IH generalizing a ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a\u271d : WithTop \u03b1 b : WithTop \u03b1\u1d52\u1d48 ha\u271d : \u2191WithBot.toDual a\u271d = b x\u271d : WithTop \u03b1\u1d52\u1d48 H : \u2200 (y : WithTop \u03b1\u1d52\u1d48), y < x\u271d \u2192 Acc (fun x x_1 => x < x_1) y IH : \u2200 (y : WithTop \u03b1\u1d52\u1d48), y < x\u271d \u2192 \u2200 (a : WithTop \u03b1), \u2191WithBot.toDual a = y \u2192 Acc (fun x x_1 => x > x_1) a a : WithTop \u03b1 ha : \u2191WithBot.toDual a = x\u271d \u22a2 Acc (fun x x_1 => x > x_1) a ** subst ha ** case intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a\u271d : WithTop \u03b1 b : WithTop \u03b1\u1d52\u1d48 ha : \u2191WithBot.toDual a\u271d = b a : WithTop \u03b1 H : \u2200 (y : WithTop \u03b1\u1d52\u1d48), y < \u2191WithBot.toDual a \u2192 Acc (fun x x_1 => x < x_1) y IH : \u2200 (y : WithTop \u03b1\u1d52\u1d48), y < \u2191WithBot.toDual a \u2192 \u2200 (a : WithTop \u03b1), \u2191WithBot.toDual a = y \u2192 Acc (fun x x_1 => x > x_1) a \u22a2 Acc (fun x x_1 => x > x_1) a ** exact \u27e8_, fun a' h => IH (WithTop.toDual a') (toDual_lt_toDual.mpr h) _ rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 inst\u271d : LT \u03b1 h : WellFounded fun x x_1 => x > x_1 a : WithTop \u03b1 \u22a2 WellFounded fun x x_1 => x < x_1 ** convert h using 1 ** Qed", + "informal": "" + }, + { + "formal": "Orientation.linearIsometryEquiv_comp_rightAngleRotation ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u03c6 : E \u2243\u2097\u1d62[\u211d] E h\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv x : E \u22a2 \u2191\u03c6 (\u2191(rightAngleRotation o) x) = \u2191(rightAngleRotation o) (\u2191\u03c6 x) ** convert (o.rightAngleRotation_map \u03c6 (\u03c6 x)).symm ** case h.e'_2.h.e'_6.h.e'_6 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u03c6 : E \u2243\u2097\u1d62[\u211d] E h\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv x : E \u22a2 x = \u2191(LinearIsometryEquiv.symm \u03c6) (\u2191\u03c6 x) ** simp ** case h.e'_3.h.e'_5.h.e'_5 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u03c6 : E \u2243\u2097\u1d62[\u211d] E h\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv x : E \u22a2 o = \u2191(map (Fin 2) \u03c6.toLinearEquiv) o ** symm ** case h.e'_3.h.e'_5.h.e'_5 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u03c6 : E \u2243\u2097\u1d62[\u211d] E h\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv x : E \u22a2 \u2191(map (Fin 2) \u03c6.toLinearEquiv) o = o ** rwa [\u2190 o.map_eq_iff_det_pos \u03c6.toLinearEquiv] at h\u03c6 ** case h.e'_3.h.e'_5.h.e'_5 E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u03c6 : E \u2243\u2097\u1d62[\u211d] E h\u03c6 : 0 < \u2191LinearMap.det \u2191\u03c6.toLinearEquiv x : E \u22a2 Fintype.card (Fin 2) = finrank \u211d E ** rw [@Fact.out (finrank \u211d E = 2), Fintype.card_fin] ** Qed", + "informal": "" + }, + { + "formal": "isPathConnected_range ** X : Type u_1 Y : Type u_2 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : TopologicalSpace Y x y z : X \u03b9 : Type u_3 F : Set X inst\u271d : PathConnectedSpace X f : X \u2192 Y hf : Continuous f \u22a2 IsPathConnected (range f) ** rw [\u2190 image_univ] ** X : Type u_1 Y : Type u_2 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : TopologicalSpace Y x y z : X \u03b9 : Type u_3 F : Set X inst\u271d : PathConnectedSpace X f : X \u2192 Y hf : Continuous f \u22a2 IsPathConnected (f '' univ) ** exact isPathConnected_univ.image hf ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_expand_mul' ** R : Type u inst\u271d\u00b9 : CommSemiring R S : Type v inst\u271d : CommSemiring S p\u271d q p : \u2115 hp : 0 < p f : R[X] n : \u2115 \u22a2 coeff (\u2191(expand R p) f) (p * n) = coeff f n ** rw [mul_comm, coeff_expand_mul hp] ** Qed", + "informal": "" + }, + { + "formal": "Set.not_nontrivial_singleton ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1 x : \u03b1 H : Set.Nontrivial {x} \u22a2 False ** rw [nontrivial_iff_exists_ne (mem_singleton x)] at H ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1 x : \u03b1 H : \u2203 y, y \u2208 {x} \u2227 y \u2260 x \u22a2 False ** let \u27e8y, hy, hya\u27e9 := H ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x a b : \u03b1 s s\u2081 s\u2082 t t\u2081 t\u2082 u : Set \u03b1 x : \u03b1 H : \u2203 y, y \u2208 {x} \u2227 y \u2260 x y : \u03b1 hy : y \u2208 {x} hya : y \u2260 x \u22a2 False ** exact hya (mem_singleton_iff.1 hy) ** Qed", + "informal": "" + }, + { + "formal": "UniformCauchySeqOn.prod_map ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' \u22a2 UniformCauchySeqOn (fun i => Prod.map (F i.1) (F' i.2)) (p \u00d7\u02e2 p') (s \u00d7\u02e2 s') ** intro u hu ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') hu : u \u2208 \ud835\udce4 (\u03b2 \u00d7 \u03b2') \u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9') in (p \u00d7\u02e2 p') \u00d7\u02e2 p \u00d7\u02e2 p', \u2200 (x : \u03b1 \u00d7 \u03b1'), x \u2208 s \u00d7\u02e2 s' \u2192 ((fun i => Prod.map (F i.1) (F' i.2)) m.1 x, (fun i => Prod.map (F i.1) (F' i.2)) m.2 x) \u2208 u ** rw [uniformity_prod_eq_prod, mem_map, mem_prod_iff] at hu ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') hu : \u2203 t\u2081, t\u2081 \u2208 \ud835\udce4 \u03b2 \u2227 \u2203 t\u2082, t\u2082 \u2208 \ud835\udce4 \u03b2' \u2227 t\u2081 \u00d7\u02e2 t\u2082 \u2286 (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) \u207b\u00b9' u \u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9') in (p \u00d7\u02e2 p') \u00d7\u02e2 p \u00d7\u02e2 p', \u2200 (x : \u03b1 \u00d7 \u03b1'), x \u2208 s \u00d7\u02e2 s' \u2192 ((fun i => Prod.map (F i.1) (F' i.2)) m.1 x, (fun i => Prod.map (F i.1) (F' i.2)) m.2 x) \u2208 u ** obtain \u27e8v, hv, w, hw, hvw\u27e9 := hu ** case intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') v : Set (\u03b2 \u00d7 \u03b2) hv : v \u2208 \ud835\udce4 \u03b2 w : Set (\u03b2' \u00d7 \u03b2') hw : w \u2208 \ud835\udce4 \u03b2' hvw : v \u00d7\u02e2 w \u2286 (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) \u207b\u00b9' u \u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9') in (p \u00d7\u02e2 p') \u00d7\u02e2 p \u00d7\u02e2 p', \u2200 (x : \u03b1 \u00d7 \u03b1'), x \u2208 s \u00d7\u02e2 s' \u2192 ((fun i => Prod.map (F i.1) (F' i.2)) m.1 x, (fun i => Prod.map (F i.1) (F' i.2)) m.2 x) \u2208 u ** simp_rw [mem_prod, Prod_map, and_imp, Prod.forall] ** case intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') v : Set (\u03b2 \u00d7 \u03b2) hv : v \u2208 \ud835\udce4 \u03b2 w : Set (\u03b2' \u00d7 \u03b2') hw : w \u2208 \ud835\udce4 \u03b2' hvw : v \u00d7\u02e2 w \u2286 (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) \u207b\u00b9' u \u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9') in (p \u00d7\u02e2 p') \u00d7\u02e2 p \u00d7\u02e2 p', \u2200 (a : \u03b1) (b : \u03b1'), a \u2208 s \u2192 b \u2208 s' \u2192 ((F m.1.1 a, F' m.1.2 b), F m.2.1 a, F' m.2.2 b) \u2208 u ** rw [\u2190 Set.image_subset_iff] at hvw ** case intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') v : Set (\u03b2 \u00d7 \u03b2) hv : v \u2208 \ud835\udce4 \u03b2 w : Set (\u03b2' \u00d7 \u03b2') hw : w \u2208 \ud835\udce4 \u03b2' hvw : (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) '' v \u00d7\u02e2 w \u2286 u \u22a2 \u2200\u1da0 (m : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9') in (p \u00d7\u02e2 p') \u00d7\u02e2 p \u00d7\u02e2 p', \u2200 (a : \u03b1) (b : \u03b1'), a \u2208 s \u2192 b \u2208 s' \u2192 ((F m.1.1 a, F' m.1.2 b), F m.2.1 a, F' m.2.2 b) \u2208 u ** apply (tendsto_swap4_prod.eventually ((h v hv).prod_mk (h' w hw))).mono ** case intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') v : Set (\u03b2 \u00d7 \u03b2) hv : v \u2208 \ud835\udce4 \u03b2 w : Set (\u03b2' \u00d7 \u03b2') hw : w \u2208 \ud835\udce4 \u03b2' hvw : (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) '' v \u00d7\u02e2 w \u2286 u \u22a2 \u2200 (x : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9'), ((\u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 (F ((x.1.1, x.2.1), x.1.2, x.2.2).1.1 x_1, F ((x.1.1, x.2.1), x.1.2, x.2.2).1.2 x_1) \u2208 v) \u2227 \u2200 (x_1 : \u03b1'), x_1 \u2208 s' \u2192 (F' ((x.1.1, x.2.1), x.1.2, x.2.2).2.1 x_1, F' ((x.1.1, x.2.1), x.1.2, x.2.2).2.2 x_1) \u2208 w) \u2192 \u2200 (a : \u03b1) (b : \u03b1'), a \u2208 s \u2192 b \u2208 s' \u2192 ((F x.1.1 a, F' x.1.2 b), F x.2.1 a, F' x.2.2 b) \u2208 u ** intro x hx a b ha hb ** case intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s'\u271d : Set \u03b1 x\u271d : \u03b1 p : Filter \u03b9 p'\u271d : Filter \u03b1 g : \u03b9 \u2192 \u03b1 \u03b9' : Type u_1 \u03b1' : Type u_2 \u03b2' : Type u_3 inst\u271d : UniformSpace \u03b2' F' : \u03b9' \u2192 \u03b1' \u2192 \u03b2' p' : Filter \u03b9' s' : Set \u03b1' h : UniformCauchySeqOn F p s h' : UniformCauchySeqOn F' p' s' u : Set ((\u03b2 \u00d7 \u03b2') \u00d7 \u03b2 \u00d7 \u03b2') v : Set (\u03b2 \u00d7 \u03b2) hv : v \u2208 \ud835\udce4 \u03b2 w : Set (\u03b2' \u00d7 \u03b2') hw : w \u2208 \ud835\udce4 \u03b2' hvw : (fun p => ((p.1.1, p.2.1), p.1.2, p.2.2)) '' v \u00d7\u02e2 w \u2286 u x : (\u03b9 \u00d7 \u03b9') \u00d7 \u03b9 \u00d7 \u03b9' hx : (\u2200 (x_1 : \u03b1), x_1 \u2208 s \u2192 (F ((x.1.1, x.2.1), x.1.2, x.2.2).1.1 x_1, F ((x.1.1, x.2.1), x.1.2, x.2.2).1.2 x_1) \u2208 v) \u2227 \u2200 (x_1 : \u03b1'), x_1 \u2208 s' \u2192 (F' ((x.1.1, x.2.1), x.1.2, x.2.2).2.1 x_1, F' ((x.1.1, x.2.1), x.1.2, x.2.2).2.2 x_1) \u2208 w a : \u03b1 b : \u03b1' ha : a \u2208 s hb : b \u2208 s' \u22a2 ((F x.1.1 a, F' x.1.2 b), F x.2.1 a, F' x.2.2 b) \u2208 u ** refine' hvw \u27e8_, mk_mem_prod (hx.1 a ha) (hx.2 b hb), rfl\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Int.ModEq.cancel_right_div_gcd ** m n a b c d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** letI d := gcd m c ** m n a b c d\u271d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] d : \u2115 := gcd m c \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** have hmd := gcd_dvd_left m c ** m n a b c d\u271d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** have hcd := gcd_dvd_right m c ** m n a b c d\u271d : \u2124 hm : 0 < m h : a * c \u2261 b * c [ZMOD m] d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 a \u2261 b [ZMOD m / \u2191(gcd m c)] ** rw [modEq_iff_dvd] at h \u22a2 ** m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 m / \u2191(gcd m c) \u2223 b - a ** refine Int.dvd_of_dvd_mul_right_of_gcd_one (?_ : m / d \u2223 c / d * (b - a)) ?_ ** case refine_1 m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 m / \u2191d \u2223 c / \u2191d * (b - a) ** rw [mul_comm, \u2190 Int.mul_ediv_assoc (b - a) hcd, sub_mul] ** case refine_1 m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 m / \u2191d \u2223 (b * c - a * c) / \u2191(gcd m c) ** exact Int.ediv_dvd_ediv hmd h ** case refine_2 m n a b c d\u271d : \u2124 hm : 0 < m h : m \u2223 b * c - a * c d : \u2115 := gcd m c hmd : \u2191(gcd m c) \u2223 m hcd : \u2191(gcd m c) \u2223 c \u22a2 gcd (m / \u2191(gcd m c)) (c / \u2191d) = 1 ** rw [gcd_div hmd hcd, natAbs_ofNat, Nat.div_self (gcd_pos_of_ne_zero_left c hm.ne')] ** Qed", + "informal": "" + }, + { + "formal": "inv_pow_lt_inv_pow_of_lt ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d : LinearOrderedSemifield \u03b1 a b c d e : \u03b1 m\u271d n\u271d : \u2124 a1 : 1 < a m n : \u2115 mn : m < n \u22a2 (a ^ n)\u207b\u00b9 < (a ^ m)\u207b\u00b9 ** convert one_div_pow_lt_one_div_pow_of_lt a1 mn using 1 <;> simp ** Qed", + "informal": "" + }, + { + "formal": "RatFunc.liftMonoidWithZeroHom_apply_div ** K : Type u inst\u271d\u00b2 : CommRing K inst\u271d\u00b9 : IsDomain K L : Type u_1 inst\u271d : CommGroupWithZero L \u03c6 : K[X] \u2192*\u2080 L h\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070 p q : K[X] \u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) = \u2191\u03c6 p / \u2191\u03c6 q ** rcases eq_or_ne q 0 with (rfl | hq) ** case inr K : Type u inst\u271d\u00b2 : CommRing K inst\u271d\u00b9 : IsDomain K L : Type u_1 inst\u271d : CommGroupWithZero L \u03c6 : K[X] \u2192*\u2080 L h\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070 p q : K[X] hq : q \u2260 0 \u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) q) = \u2191\u03c6 p / \u2191\u03c6 q ** simp only [\u2190 mk_eq_div, mk_eq_localization_mk _ hq,\n liftMonoidWithZeroHom_apply_ofFractionRing_mk] ** case inl K : Type u inst\u271d\u00b2 : CommRing K inst\u271d\u00b9 : IsDomain K L : Type u_1 inst\u271d : CommGroupWithZero L \u03c6 : K[X] \u2192*\u2080 L h\u03c6 : K[X]\u2070 \u2264 Submonoid.comap \u03c6 L\u2070 p : K[X] \u22a2 \u2191(liftMonoidWithZeroHom \u03c6 h\u03c6) (\u2191(algebraMap K[X] (RatFunc K)) p / \u2191(algebraMap K[X] (RatFunc K)) 0) = \u2191\u03c6 p / \u2191\u03c6 0 ** simp only [div_zero, map_zero] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.mem_span_mul_finite_of_mem_span_mul ** \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') \u22a2 \u2203 T T', \u2191T \u2286 S \u2227 \u2191T' \u2286 S' \u2227 x \u2208 span R (\u2191T * \u2191T') ** obtain \u27e8U, h, hU\u27e9 := mem_span_finite_of_mem_span hx ** case intro.intro \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') U : Finset A h : \u2191U \u2286 S * S' hU : x \u2208 span R \u2191U \u22a2 \u2203 T T', \u2191T \u2286 S \u2227 \u2191T' \u2286 S' \u2227 x \u2208 span R (\u2191T * \u2191T') ** obtain \u27e8T, T', hS, hS', h\u27e9 := Finset.subset_mul h ** case intro.intro.intro.intro.intro.intro \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T\u271d : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') U : Finset A h\u271d : \u2191U \u2286 S * S' hU : x \u2208 span R \u2191U T T' : Finset A hS : \u2191T \u2286 S hS' : \u2191T' \u2286 S' h : U \u2286 T * T' \u22a2 \u2203 T T', \u2191T \u2286 S \u2227 \u2191T' \u2286 S' \u2227 x \u2208 span R (\u2191T * \u2191T') ** use T, T', hS, hS' ** case right \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T\u271d : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') U : Finset A h\u271d : \u2191U \u2286 S * S' hU : x \u2208 span R \u2191U T T' : Finset A hS : \u2191T \u2286 S hS' : \u2191T' \u2286 S' h : U \u2286 T * T' \u22a2 x \u2208 span R (\u2191T * \u2191T') ** have h' : (U : Set A) \u2286 T * T' := by assumption_mod_cast ** case right \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T\u271d : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') U : Finset A h\u271d : \u2191U \u2286 S * S' hU : x \u2208 span R \u2191U T T' : Finset A hS : \u2191T \u2286 S hS' : \u2191T' \u2286 S' h : U \u2286 T * T' h' : \u2191U \u2286 \u2191T * \u2191T' \u22a2 x \u2208 span R (\u2191T * \u2191T') ** have h'' := span_mono h' hU ** case right \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T\u271d : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') U : Finset A h\u271d : \u2191U \u2286 S * S' hU : x \u2208 span R \u2191U T T' : Finset A hS : \u2191T \u2286 S hS' : \u2191T' \u2286 S' h : U \u2286 T * T' h' : \u2191U \u2286 \u2191T * \u2191T' h'' : x \u2208 span R (\u2191T * \u2191T') \u22a2 x \u2208 span R (\u2191T * \u2191T') ** assumption ** \u03b9 : Sort u\u03b9 R\u271d : Type u inst\u271d\u2076 : CommSemiring R\u271d A\u271d : Type v inst\u271d\u2075 : Semiring A\u271d inst\u271d\u2074 : Algebra R\u271d A\u271d S\u271d T\u271d : Set A\u271d M N P Q : Submodule R\u271d A\u271d m n : A\u271d R : Type u_1 A : Type u_2 inst\u271d\u00b3 : Semiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Mul A inst\u271d : Module R A S S' : Set A x : A hx : x \u2208 span R (S * S') U : Finset A h\u271d : \u2191U \u2286 S * S' hU : x \u2208 span R \u2191U T T' : Finset A hS : \u2191T \u2286 S hS' : \u2191T' \u2286 S' h : U \u2286 T * T' \u22a2 \u2191U \u2286 \u2191T * \u2191T' ** assumption_mod_cast ** Qed", + "informal": "" + }, + { + "formal": "Fintype.card_le_one_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u03b2 n : \u2115 := card \u03b1 hn : n = card \u03b1 ha : 1 = card \u03b1 _h : card \u03b1 \u2264 1 a b : \u03b1 \u22a2 a = b ** let \u27e8x, hx\u27e9 := card_eq_one_iff.1 ha.symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u03b2 n : \u2115 := card \u03b1 hn : n = card \u03b1 ha : 1 = card \u03b1 _h : card \u03b1 \u2264 1 a b x : \u03b1 hx : \u2200 (y : \u03b1), y = x \u22a2 a = b ** rw [hx a, hx b] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u03b2 n\u271d : \u2115 := card \u03b1 hn : n\u271d = card \u03b1 n : \u2115 ha : n + 2 = card \u03b1 h : card \u03b1 \u2264 1 \u22a2 False ** rw [\u2190 ha] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u03b2 n\u271d : \u2115 := card \u03b1 hn : n\u271d = card \u03b1 n : \u2115 ha : n + 2 = card \u03b1 h : n + 2 \u2264 1 \u22a2 False ** cases h with | step h => cases h; done ** case step \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : Fintype \u03b2 n\u271d : \u2115 := card \u03b1 hn : n\u271d = card \u03b1 n : \u2115 ha : n + 2 = card \u03b1 h : Nat.le (n + 2) 0 \u22a2 False ** cases h ** Qed", + "informal": "" + }, + { + "formal": "Set.mulIndicator_comp_right ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s\u271d t : Set \u03b1 f\u271d g\u271d : \u03b1 \u2192 M a : \u03b1 s : Set \u03b1 f : \u03b2 \u2192 \u03b1 g : \u03b1 \u2192 M x : \u03b2 \u22a2 mulIndicator (f \u207b\u00b9' s) (g \u2218 f) x = mulIndicator s g (f x) ** simp only [mulIndicator, Function.comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s\u271d t : Set \u03b1 f\u271d g\u271d : \u03b1 \u2192 M a : \u03b1 s : Set \u03b1 f : \u03b2 \u2192 \u03b1 g : \u03b1 \u2192 M x : \u03b2 \u22a2 (if x \u2208 f \u207b\u00b9' s then g (f x) else 1) = if f x \u2208 s then g (f x) else 1 ** split_ifs with h h' h'' <;> first | rfl | contradiction ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s\u271d t : Set \u03b1 f\u271d g\u271d : \u03b1 \u2192 M a : \u03b1 s : Set \u03b1 f : \u03b2 \u2192 \u03b1 g : \u03b1 \u2192 M x : \u03b2 h : \u00acx \u2208 f \u207b\u00b9' s h'' : \u00acf x \u2208 s \u22a2 1 = 1 ** rfl ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b9 : One M inst\u271d : One N s\u271d t : Set \u03b1 f\u271d g\u271d : \u03b1 \u2192 M a : \u03b1 s : Set \u03b1 f : \u03b2 \u2192 \u03b1 g : \u03b1 \u2192 M x : \u03b2 h : \u00acx \u2208 f \u207b\u00b9' s h'' : f x \u2208 s \u22a2 1 = g (f x) ** contradiction ** Qed", + "informal": "" + }, + { + "formal": "Set.Finite.pi ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 t : (d : \u03b4) \u2192 Set (\u03ba d) ht : \u2200 (d : \u03b4), Set.Finite (t d) \u22a2 Set.Finite (Set.pi univ t) ** cases _root_.nonempty_fintype \u03b4 ** case intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 t : (d : \u03b4) \u2192 Set (\u03ba d) ht : \u2200 (d : \u03b4), Set.Finite (t d) val\u271d : Fintype \u03b4 \u22a2 Set.Finite (Set.pi univ t) ** lift t to \u2200 d, Finset (\u03ba d) using ht ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 val\u271d : Fintype \u03b4 t : (i : \u03b4) \u2192 Finset (\u03ba i) \u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i)) ** classical\n rw [\u2190 Fintype.coe_piFinset]\n apply Finset.finite_toSet ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 val\u271d : Fintype \u03b4 t : (i : \u03b4) \u2192 Finset (\u03ba i) \u22a2 Set.Finite (Set.pi univ fun i => \u2191(t i)) ** rw [\u2190 Fintype.coe_piFinset] ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x \u03b4 : Type u_1 inst\u271d : Finite \u03b4 \u03ba : \u03b4 \u2192 Type u_2 val\u271d : Fintype \u03b4 t : (i : \u03b4) \u2192 Finset (\u03ba i) \u22a2 Set.Finite \u2191(Fintype.piFinset fun i => t i) ** apply Finset.finite_toSet ** Qed", + "informal": "" + }, + { + "formal": "Complex.lim_re ** f : CauSeq \u2102 \u2191abs \u22a2 CauSeq.lim (cauSeqRe f) = (CauSeq.lim f).re ** rw [lim_eq_lim_im_add_lim_re] ** f : CauSeq \u2102 \u2191abs \u22a2 CauSeq.lim (cauSeqRe f) = (\u2191(CauSeq.lim (cauSeqRe f)) + \u2191(CauSeq.lim (cauSeqIm f)) * I).re ** simp [ofReal'] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.PushoutCocone.epi_inr_of_is_pushout_of_epi ** C : Type u inst\u271d\u00b2 : Category.{v, u} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D W\u271d X Y Z : C f : X \u27f6 Y g : X \u27f6 Z t : PushoutCocone f g ht : IsColimit t inst\u271d : Epi f W : C h k : t.pt \u27f6 W i : inr t \u226b h = inr t \u226b k \u22a2 inl t \u226b h = inl t \u226b k ** simp [\u2190 cancel_epi f, t.condition_assoc, i] ** Qed", + "informal": "" + }, + { + "formal": "List.last_ofFn ** \u03b1 : Type u n : \u2115 f : Fin n \u2192 \u03b1 h : ofFn f \u2260 [] hn : optParam (n - 1 < n) (_ : pred (Nat.sub n 0) < Nat.sub n 0) \u22a2 getLast (ofFn f) h = f { val := n - 1, isLt := hn } ** simp [getLast_eq_get] ** Qed", + "informal": "" + }, + { + "formal": "Metric.cthickening_max_zero ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d : PseudoEMetricSpace \u03b1 \u03b4\u271d \u03b5 : \u211d s t : Set \u03b1 x : \u03b1 \u03b4 : \u211d E : Set \u03b1 \u22a2 cthickening (max 0 \u03b4) E = cthickening \u03b4 E ** cases le_total \u03b4 0 <;> simp [cthickening_of_nonpos, *] ** Qed", + "informal": "" + }, + { + "formal": "Finset.update_piecewise_of_mem ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f g : (i : \u03b1) \u2192 \u03b4 i inst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d : DecidableEq \u03b1 i : \u03b1 hi : i \u2208 s v : \u03b4 i \u22a2 update (piecewise s f g) i v = piecewise s (update f i v) g ** rw [update_piecewise] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f g : (i : \u03b1) \u2192 \u03b4 i inst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d : DecidableEq \u03b1 i : \u03b1 hi : i \u2208 s v : \u03b4 i \u22a2 piecewise s (update f i v) (update g i v) = piecewise s (update f i v) g ** refine' s.piecewise_congr (fun _ _ => rfl) fun j hj => update_noteq _ _ _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : \u03b1 \u2192 Sort u_4 s : Finset \u03b1 f g : (i : \u03b1) \u2192 \u03b4 i inst\u271d\u00b9 : (j : \u03b1) \u2192 Decidable (j \u2208 s) inst\u271d : DecidableEq \u03b1 i : \u03b1 hi : i \u2208 s v : \u03b4 i j : \u03b1 hj : \u00acj \u2208 s \u22a2 j \u2260 i ** exact fun h => hj (h.symm \u25b8 hi) ** Qed", + "informal": "" + }, + { + "formal": "exists_extension_norm_eq ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** letI : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** letI : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower _ _ _ ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** letI : NormedSpace \u211d F := NormedSpace.restrictScalars _ \ud835\udd5c _ ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** let fr := reClm.comp (f.restrictScalars \u211d) ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** rcases Real.exists_extension_norm_eq (p.restrictScalars \u211d) fr with \u27e8g, \u27e8hextends, hnormeq\u27e9\u27e9 ** case intro.intro \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 \u22a2 \u2203 g, (\u2200 (x : { x // x \u2208 p }), \u2191g \u2191x = \u2191f x) \u2227 \u2016g\u2016 = \u2016f\u2016 ** refine' \u27e8g.extendTo\ud835\udd5c, _\u27e9 ** case intro.intro \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 h : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x \u22a2 (\u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x) \u2227 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 = \u2016f\u2016 ** refine' \u27e8h, le_antisymm _ _\u27e9 ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 \u22a2 \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x ** intro x ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } \u22a2 \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x ** erw [ContinuousLinearMap.extendTo\ud835\udd5c_apply, \u2190 Submodule.coe_smul, hextends, hextends] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } \u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191f x ** have : (fr x : \ud835\udd5c) - I * \u2191(fr (I \u2022 x)) = (re (f x) : \ud835\udd5c) - (I : \ud835\udd5c) * re (f ((I : \ud835\udd5c) \u2022 x)) := by\n rfl ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } this : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) \u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191f x ** erw [this] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } this : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) \u22a2 \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) = \u2191f x ** apply ext ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } \u22a2 \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) ** rfl ** case hre \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } this : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) \u22a2 \u2191re (\u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))) = \u2191re (\u2191f x) ** simp only [add_zero, Algebra.id.smul_eq_mul, I_re, ofReal_im, AddMonoidHom.map_add, zero_sub,\n I_im', zero_mul, ofReal_re, eq_self_iff_true, sub_zero, mul_neg, ofReal_neg,\n mul_re, mul_zero, sub_neg_eq_add, ContinuousLinearMap.map_smul] ** case him \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b2 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d\u00b9 : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this\u271d : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 x : { x // x \u2208 p } this : \u2191(\u2191fr x) - I * \u2191(\u2191fr (I \u2022 x)) = \u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x))) \u22a2 \u2191im (\u2191(\u2191re (\u2191f x)) - I * \u2191(\u2191re (\u2191f (I \u2022 x)))) = \u2191im (\u2191f x) ** simp only [Algebra.id.smul_eq_mul, I_re, ofReal_im, AddMonoidHom.map_add, zero_sub, I_im',\n zero_mul, ofReal_re, mul_neg, mul_im, zero_add, ofReal_neg, mul_re,\n sub_neg_eq_add, ContinuousLinearMap.map_smul] ** case intro.intro.refine'_1 \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 h : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x \u22a2 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 \u2264 \u2016f\u2016 ** calc\n \u2016g.extendTo\ud835\udd5c\u2016 = \u2016g\u2016 := g.norm_extendTo\ud835\udd5c\n _ = \u2016fr\u2016 := hnormeq\n _ \u2264 \u2016reClm\u2016 * \u2016f\u2016 := (ContinuousLinearMap.op_norm_comp_le _ _)\n _ = \u2016f\u2016 := by rw [reClm_norm, one_mul] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 h : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x \u22a2 \u2016reClm\u2016 * \u2016f\u2016 = \u2016f\u2016 ** rw [reClm_norm, one_mul] ** case intro.intro.refine'_2 \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c F : Type u_2 inst\u271d\u00b9 : SeminormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : Subspace \ud835\udd5c F f : { x // x \u2208 p } \u2192L[\ud835\udd5c] \ud835\udd5c this\u271d\u00b9 : Module \u211d F := RestrictScalars.module \u211d \ud835\udd5c F this\u271d : IsScalarTower \u211d \ud835\udd5c F := RestrictScalars.isScalarTower \u211d \ud835\udd5c F this : NormedSpace \u211d F := NormedSpace.restrictScalars \u211d \ud835\udd5c F fr : { x // x \u2208 p } \u2192L[\u211d] \u211d := ContinuousLinearMap.comp reClm (ContinuousLinearMap.restrictScalars \u211d f) g : F \u2192L[\u211d] \u211d hextends : \u2200 (x : { x // x \u2208 Submodule.restrictScalars \u211d p }), \u2191g \u2191x = \u2191fr x hnormeq : \u2016g\u2016 = \u2016fr\u2016 h : \u2200 (x : { x // x \u2208 p }), \u2191(ContinuousLinearMap.extendTo\ud835\udd5c g) \u2191x = \u2191f x \u22a2 \u2016f\u2016 \u2264 \u2016ContinuousLinearMap.extendTo\ud835\udd5c g\u2016 ** exact f.op_norm_le_bound g.extendTo\ud835\udd5c.op_norm_nonneg fun x => h x \u25b8 g.extendTo\ud835\udd5c.le_op_norm x ** Qed", + "informal": "" + }, + { + "formal": "RingHom.StableUnderBaseChange.mk ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom \u22a2 StableUnderBaseChange P ** introv R h H ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) \u22a2 P (algebraMap R' S') ** let e := h.symm.1.equiv ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) \u22a2 P (algebraMap R' S') ** let f' :=\n Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S')\n (IsScalarTower.toAlgHom R S S') ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') \u22a2 P (algebraMap R' S') ** have : \u2200 x, e x = f' x := by\n intro x\n change e.toLinearMap.restrictScalars R x = f'.toLinearMap x\n congr 1\n apply TensorProduct.ext'\n intro x y\n simp [IsBaseChange.equiv_tmul, Algebra.smul_def] ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x \u22a2 P (algebraMap R' S') ** convert h\u2081.1 (_ : R' \u2192+* TensorProduct R R' S) (_ : TensorProduct R R' S \u2243+* S')\n (h\u2082 H : P (_ : R' \u2192+* TensorProduct R R' S)) ** case h.e'_5 P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x \u22a2 algebraMap R' S' = comp (RingEquiv.toRingHom ?m.222691) Algebra.TensorProduct.includeLeftRingHom P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x \u22a2 TensorProduct R R' S \u2243+* S' ** swap ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') \u22a2 \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x ** intro x ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') x : TensorProduct R R' S \u22a2 \u2191e x = \u2191f' x ** change e.toLinearMap.restrictScalars R x = f'.toLinearMap x ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') x : TensorProduct R R' S \u22a2 \u2191(\u2191R \u2191e) x = \u2191(AlgHom.toLinearMap f') x ** congr 1 ** case e_a P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') x : TensorProduct R R' S \u22a2 \u2191R \u2191e = AlgHom.toLinearMap f' ** apply TensorProduct.ext' ** case e_a.H P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') x : TensorProduct R R' S \u22a2 \u2200 (x : R') (y : S), \u2191(\u2191R \u2191e) (x \u2297\u209c[R] y) = \u2191(AlgHom.toLinearMap f') (x \u2297\u209c[R] y) ** intro x y ** case e_a.H P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') x\u271d : TensorProduct R R' S x : R' y : S \u22a2 \u2191(\u2191R \u2191e) (x \u2297\u209c[R] y) = \u2191(AlgHom.toLinearMap f') (x \u2297\u209c[R] y) ** simp [IsBaseChange.equiv_tmul, Algebra.smul_def] ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x \u22a2 TensorProduct R R' S \u2243+* S' ** refine' { e with map_mul' := fun x y => _ } ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x x y : TensorProduct R R' S \u22a2 Equiv.toFun { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) } (x * y) = Equiv.toFun { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) } x * Equiv.toFun { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) } y ** change e (x * y) = e x * e y ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x x y : TensorProduct R R' S \u22a2 \u2191e (x * y) = \u2191e x * \u2191e y ** simp_rw [this] ** P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x x y : TensorProduct R R' S \u22a2 \u2191(Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')) (x * y) = \u2191(Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')) x * \u2191(Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S')) y ** exact map_mul f' _ _ ** case h.e'_5 P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x \u22a2 algebraMap R' S' = comp (RingEquiv.toRingHom { toEquiv := { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) }, map_mul' := (_ : \u2200 (x y : TensorProduct R R' S), \u2191e (x * y) = \u2191e x * \u2191e y), map_add' := (_ : \u2200 (x y : TensorProduct R R' S), AddHom.toFun e.toAddHom (x + y) = AddHom.toFun e.toAddHom x + AddHom.toFun e.toAddHom y) }) Algebra.TensorProduct.includeLeftRingHom ** ext x ** case h.e'_5.a P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x x : R' \u22a2 \u2191(algebraMap R' S') x = \u2191(comp (RingEquiv.toRingHom { toEquiv := { toFun := e.toFun, invFun := e.invFun, left_inv := (_ : Function.LeftInverse e.invFun e.toFun), right_inv := (_ : Function.RightInverse e.invFun e.toFun) }, map_mul' := (_ : \u2200 (x y : TensorProduct R R' S), \u2191e (x * y) = \u2191e x * \u2191e y), map_add' := (_ : \u2200 (x y : TensorProduct R R' S), AddHom.toFun e.toAddHom (x + y) = AddHom.toFun e.toAddHom x + AddHom.toFun e.toAddHom y) }) Algebra.TensorProduct.includeLeftRingHom) x ** change _ = e (x \u2297\u209c[R] 1) ** case h.e'_5.a P : {R S : Type u} \u2192 [inst : CommRing R] \u2192 [inst_1 : CommRing S] \u2192 (R \u2192+* S) \u2192 Prop h\u2081 : RespectsIso P h\u2082 : \u2200 \u2983R S T : Type u\u2984 [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : CommRing T] [inst_3 : Algebra R S] [inst_4 : Algebra R T], P (algebraMap R T) \u2192 P Algebra.TensorProduct.includeLeftRingHom R S R' S' : Type u inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing S inst\u271d\u2078 : CommRing R' inst\u271d\u2077 : CommRing S' inst\u271d\u2076 : Algebra R S inst\u271d\u2075 : Algebra R R' inst\u271d\u2074 : Algebra R S' inst\u271d\u00b3 : Algebra S S' inst\u271d\u00b2 : Algebra R' S' inst\u271d\u00b9 : IsScalarTower R S S' inst\u271d : IsScalarTower R R' S' h : Algebra.IsPushout R S R' S' H : P (algebraMap R S) e : TensorProduct R R' S \u2243\u209b\u2097[id R'] S' := IsBaseChange.equiv (_ : IsBaseChange R' (AlgHom.toLinearMap (IsScalarTower.toAlgHom R S S'))) f' : TensorProduct R R' S \u2192\u2090[R] S' := Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S') (IsScalarTower.toAlgHom R S S') this : \u2200 (x : TensorProduct R R' S), \u2191e x = \u2191f' x x : R' \u22a2 \u2191(algebraMap R' S') x = \u2191e (x \u2297\u209c[R] 1) ** rw [h.symm.1.equiv_tmul, Algebra.smul_def, AlgHom.toLinearMap_apply, map_one, mul_one] ** Qed", + "informal": "" + }, + { + "formal": "ENat.coe_toNat_eq_self ** m n : \u2115\u221e \u22a2 \u2191(\u2191toNat \u22a4) = \u22a4 \u2194 \u22a4 \u2260 \u22a4 ** simp ** m n : \u2115\u221e x\u271d : \u2115 \u22a2 \u2191(\u2191toNat \u2191x\u271d) = \u2191x\u271d \u2194 \u2191x\u271d \u2260 \u22a4 ** simp [toNat_coe] ** Qed", + "informal": "" + }, + { + "formal": "AffineSubspace.direction_sup ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 s1 hp2 : p2 \u2208 s2 \u22a2 direction (s1 \u2294 s2) = direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} ** refine' le_antisymm _ _ ** case refine'_1 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 s1 hp2 : p2 \u2208 s2 \u22a2 direction (s1 \u2294 s2) \u2264 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} ** change (affineSpan k ((s1 : Set P) \u222a s2)).direction \u2264 _ ** case refine'_1 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 s1 hp2 : p2 \u2208 s2 \u22a2 direction (affineSpan k (\u2191s1 \u222a \u2191s2)) \u2264 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} ** rw [\u2190 mem_coe] at hp1 ** case refine'_1 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 \u22a2 direction (affineSpan k (\u2191s1 \u222a \u2191s2)) \u2264 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} ** rw [direction_affineSpan, vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ hp1),\n Submodule.span_le] ** case refine'_1 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 \u22a2 (fun x => x -\u1d65 p1) '' (\u2191s1 \u222a \u2191s2) \u2286 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}) ** rintro v \u27e8p3, hp3, rfl\u27e9 ** case refine'_1.intro.intro k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s1 \u222a \u2191s2 \u22a2 (fun x => x -\u1d65 p1) p3 \u2208 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}) ** cases' hp3 with hp3 hp3 ** case refine'_1.intro.intro.inl k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s1 \u22a2 (fun x => x -\u1d65 p1) p3 \u2208 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}) ** rw [sup_assoc, sup_comm, SetLike.mem_coe, Submodule.mem_sup] ** case refine'_1.intro.intro.inl k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s1 \u22a2 \u2203 y, y \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2227 \u2203 z, z \u2208 direction s1 \u2227 y + z = (fun x => x -\u1d65 p1) p3 ** use 0, Submodule.zero_mem _, p3 -\u1d65 p1, vsub_mem_direction hp3 hp1 ** case right k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s1 \u22a2 0 + (p3 -\u1d65 p1) = (fun x => x -\u1d65 p1) p3 ** rw [zero_add] ** case refine'_1.intro.intro.inr k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s2 \u22a2 (fun x => x -\u1d65 p1) p3 \u2208 \u2191(direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1}) ** rw [sup_assoc, SetLike.mem_coe, Submodule.mem_sup] ** case refine'_1.intro.intro.inr k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s2 \u22a2 \u2203 y, y \u2208 direction s1 \u2227 \u2203 z, z \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2227 y + z = (fun x => x -\u1d65 p1) p3 ** use 0, Submodule.zero_mem _, p3 -\u1d65 p1 ** case h k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s2 \u22a2 p3 -\u1d65 p1 \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2227 0 + (p3 -\u1d65 p1) = (fun x => x -\u1d65 p1) p3 ** rw [and_comm, zero_add] ** case h k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s2 \u22a2 p3 -\u1d65 p1 = (fun x => x -\u1d65 p1) p3 \u2227 p3 -\u1d65 p1 \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} ** use rfl ** case right k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s2 \u22a2 p3 -\u1d65 p1 \u2208 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} ** rw [\u2190 vsub_add_vsub_cancel p3 p2 p1, Submodule.mem_sup] ** case right k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 \u2191s1 hp2 : p2 \u2208 s2 p3 : P hp3 : p3 \u2208 \u2191s2 \u22a2 \u2203 y, y \u2208 direction s2 \u2227 \u2203 z, z \u2208 Submodule.span k {p2 -\u1d65 p1} \u2227 y + z = p3 -\u1d65 p2 + (p2 -\u1d65 p1) ** use p3 -\u1d65 p2, vsub_mem_direction hp3 hp2, p2 -\u1d65 p1, Submodule.mem_span_singleton_self _ ** case refine'_2 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 s1 hp2 : p2 \u2208 s2 \u22a2 direction s1 \u2294 direction s2 \u2294 Submodule.span k {p2 -\u1d65 p1} \u2264 direction (s1 \u2294 s2) ** refine' sup_le (sup_direction_le _ _) _ ** case refine'_2 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 s1 hp2 : p2 \u2208 s2 \u22a2 Submodule.span k {p2 -\u1d65 p1} \u2264 direction (s1 \u2294 s2) ** rw [direction_eq_vectorSpan, vectorSpan_def] ** case refine'_2 k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s1 s2 : AffineSubspace k P p1 p2 : P hp1 : p1 \u2208 s1 hp2 : p2 \u2208 s2 \u22a2 Submodule.span k {p2 -\u1d65 p1} \u2264 Submodule.span k (\u2191(s1 \u2294 s2) -\u1d65 \u2191(s1 \u2294 s2)) ** exact\n sInf_le_sInf fun p hp =>\n Set.Subset.trans\n (Set.singleton_subset_iff.2\n (vsub_mem_vsub (mem_spanPoints k p2 _ (Set.mem_union_right _ hp2))\n (mem_spanPoints k p1 _ (Set.mem_union_left _ hp1))))\n hp ** Qed", + "informal": "" + }, + { + "formal": "clusterPt_iff_not_disjoint ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a : \u03b1 s s\u2081 s\u2082 t : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 x : \u03b1 F : Filter \u03b1 \u22a2 ClusterPt x F \u2194 \u00acDisjoint (\ud835\udcdd x) F ** rw [disjoint_iff, ClusterPt, neBot_iff] ** Qed", + "informal": "" + }, + { + "formal": "Subalgebra.inv_mem_of_root_of_coeff_zero_ne_zero ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 \u22a2 (\u2191x)\u207b\u00b9 \u2208 A ** suffices (x\u207b\u00b9 : L) = (-p.coeff 0)\u207b\u00b9 \u2022 aeval x (divX p) by\n rw [this]\n exact A.smul_mem (aeval x _).2 _ ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 \u22a2 (\u2191x)\u207b\u00b9 = \u2191((-coeff p 0)\u207b\u00b9 \u2022 \u2191(aeval x) (divX p)) ** have : aeval (x : L) p = 0 := by rw [Subalgebra.aeval_coe, aeval_eq, Subalgebra.coe_zero] ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 this : \u2191(aeval \u2191x) p = 0 \u22a2 (\u2191x)\u207b\u00b9 = \u2191((-coeff p 0)\u207b\u00b9 \u2022 \u2191(aeval x) (divX p)) ** rw [inv_eq_of_root_of_coeff_zero_ne_zero this coeff_zero_ne, div_eq_inv_mul, Algebra.smul_def] ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 this : \u2191(aeval \u2191x) p = 0 \u22a2 -((\u2191(algebraMap K L) (coeff p 0))\u207b\u00b9 * \u2191(aeval \u2191x) (divX p)) = \u2191(\u2191(algebraMap K ((fun x => { x // x \u2208 A }) (divX p))) (-coeff p 0)\u207b\u00b9 * \u2191(aeval x) (divX p)) ** simp only [aeval_coe, Submonoid.coe_mul, Subsemiring.coe_toSubmonoid, coe_toSubsemiring,\n coe_algebraMap] ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 this : \u2191(aeval \u2191x) p = 0 \u22a2 -((\u2191(algebraMap K L) (coeff p 0))\u207b\u00b9 * \u2191(\u2191(aeval x) (divX p))) = \u2191(algebraMap K L) (-coeff p 0)\u207b\u00b9 * \u2191(\u2191(aeval x) (divX p)) ** rw [map_inv\u2080, map_neg, inv_neg, neg_mul] ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 this : (\u2191x)\u207b\u00b9 = \u2191((-coeff p 0)\u207b\u00b9 \u2022 \u2191(aeval x) (divX p)) \u22a2 (\u2191x)\u207b\u00b9 \u2208 A ** rw [this] ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 this : (\u2191x)\u207b\u00b9 = \u2191((-coeff p 0)\u207b\u00b9 \u2022 \u2191(aeval x) (divX p)) \u22a2 \u2191((-coeff p 0)\u207b\u00b9 \u2022 \u2191(aeval x) (divX p)) \u2208 A ** exact A.smul_mem (aeval x _).2 _ ** R : Type u_1 S : Type u_2 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R inst\u271d\u00b3 : CommRing S K : Type u_3 L : Type u_4 inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Algebra K L A : Subalgebra K L x : { x // x \u2208 A } p : K[X] aeval_eq : \u2191(aeval x) p = 0 coeff_zero_ne : coeff p 0 \u2260 0 \u22a2 \u2191(aeval \u2191x) p = 0 ** rw [Subalgebra.aeval_coe, aeval_eq, Subalgebra.coe_zero] ** Qed", + "informal": "" + }, + { + "formal": "UpperHalfPlane.center_zero ** z\u271d w : \u210d r R : \u211d z : \u210d \u22a2 im (center z 0) = im z ** rw [center_im, Real.cosh_zero, mul_one] ** Qed", + "informal": "" + }, + { + "formal": "Homeomorph.mulLeft_symm ** \u03b1 : Type u \u03b2 : Type v G : Type w H : Type x inst\u271d\u00b2 : TopologicalSpace G inst\u271d\u00b9 : Group G inst\u271d : ContinuousMul G a : G \u22a2 Homeomorph.symm (Homeomorph.mulLeft a) = Homeomorph.mulLeft a\u207b\u00b9 ** ext ** case H \u03b1 : Type u \u03b2 : Type v G : Type w H : Type x inst\u271d\u00b2 : TopologicalSpace G inst\u271d\u00b9 : Group G inst\u271d : ContinuousMul G a x\u271d : G \u22a2 \u2191(Homeomorph.symm (Homeomorph.mulLeft a)) x\u271d = \u2191(Homeomorph.mulLeft a\u207b\u00b9) x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Set.sumDiffSubset_symm_apply_of_mem ** \u03b1\u271d : Sort u \u03b2 : Sort v \u03b3 : Sort w \u03b1 : Type u_1 s t : Set \u03b1 h : s \u2286 t inst\u271d : DecidablePred fun x => x \u2208 s x : \u2191t hx : \u2191x \u2208 s \u22a2 \u2191(Set.sumDiffSubset h).symm x = Sum.inl { val := \u2191x, property := hx } ** apply (Equiv.Set.sumDiffSubset h).injective ** case a \u03b1\u271d : Sort u \u03b2 : Sort v \u03b3 : Sort w \u03b1 : Type u_1 s t : Set \u03b1 h : s \u2286 t inst\u271d : DecidablePred fun x => x \u2208 s x : \u2191t hx : \u2191x \u2208 s \u22a2 \u2191(Set.sumDiffSubset h) (\u2191(Set.sumDiffSubset h).symm x) = \u2191(Set.sumDiffSubset h) (Sum.inl { val := \u2191x, property := hx }) ** simp only [apply_symm_apply, sumDiffSubset_apply_inl] ** case a \u03b1\u271d : Sort u \u03b2 : Sort v \u03b3 : Sort w \u03b1 : Type u_1 s t : Set \u03b1 h : s \u2286 t inst\u271d : DecidablePred fun x => x \u2208 s x : \u2191t hx : \u2191x \u2208 s \u22a2 x = inclusion h { val := \u2191x, property := hx } ** exact Subtype.eq rfl ** Qed", + "informal": "" + }, + { + "formal": "toIocMod_zero_sub_comm ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c : \u03b1 n : \u2124 a b : \u03b1 \u22a2 toIocMod hp 0 (a - b) = p - toIcoMod hp 0 (b - a) ** rw [\u2190 neg_sub, toIocMod_neg, neg_zero] ** Qed", + "informal": "" + }, + { + "formal": "tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s' : Set \u03b1 x : \u03b1 p : Filter \u03b9 p' : Filter \u03b1 g : \u03b9 \u2192 \u03b1 inst\u271d : TopologicalSpace \u03b1 \u22a2 TendstoLocallyUniformlyOn F f p s \u2194 TendstoLocallyUniformly (fun i x => F i \u2191x) (f \u2218 Subtype.val) p ** simp only [tendstoLocallyUniformly_iff_forall_tendsto, Subtype.forall', tendsto_map'_iff,\n tendstoLocallyUniformlyOn_iff_forall_tendsto, \u2190 map_nhds_subtype_val, prod_map_right] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Type x inst\u271d\u00b9 : UniformSpace \u03b2 F : \u03b9 \u2192 \u03b1 \u2192 \u03b2 f : \u03b1 \u2192 \u03b2 s s' : Set \u03b1 x : \u03b1 p : Filter \u03b9 p' : Filter \u03b1 g : \u03b9 \u2192 \u03b1 inst\u271d : TopologicalSpace \u03b1 \u22a2 (\u2200 (x : { a // a \u2208 s }), Tendsto ((fun y => (f y.2, F y.1 y.2)) \u2218 Prod.map id Subtype.val) (p \u00d7\u02e2 \ud835\udcdd x) (\ud835\udce4 \u03b2)) \u2194 \u2200 (x : \u2191s), Tendsto (fun y => ((f \u2218 Subtype.val) y.2, F y.1 \u2191y.2)) (p \u00d7\u02e2 \ud835\udcdd x) (\ud835\udce4 \u03b2) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.count_apply_finset' ** \u03b1 : Type u_1 \u03b2 : Type ?u.3930 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s\u271d : Set \u03b1 s : Finset \u03b1 s_mble : MeasurableSet \u2191s \u22a2 \u2211 i in s, 1 = \u2191(Finset.card s) ** simp ** Qed", + "informal": "" + }, + { + "formal": "MonoidAlgebra.induction_on ** k : Type u\u2081 G : Type u\u2082 H : Type u_1 R : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : Monoid G p : MonoidAlgebra k G \u2192 Prop f : MonoidAlgebra k G hM : \u2200 (g : G), p (\u2191(of k G) g) hadd : \u2200 (f g : MonoidAlgebra k G), p f \u2192 p g \u2192 p (f + g) hsmul : \u2200 (r : k) (f : MonoidAlgebra k G), p f \u2192 p (r \u2022 f) \u22a2 p f ** refine' Finsupp.induction_linear f _ (fun f g hf hg => hadd f g hf hg) fun g r => _ ** case refine'_1 k : Type u\u2081 G : Type u\u2082 H : Type u_1 R : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : Monoid G p : MonoidAlgebra k G \u2192 Prop f : MonoidAlgebra k G hM : \u2200 (g : G), p (\u2191(of k G) g) hadd : \u2200 (f g : MonoidAlgebra k G), p f \u2192 p g \u2192 p (f + g) hsmul : \u2200 (r : k) (f : MonoidAlgebra k G), p f \u2192 p (r \u2022 f) \u22a2 p 0 ** simpa using hsmul 0 (of k G 1) (hM 1) ** case refine'_2 k : Type u\u2081 G : Type u\u2082 H : Type u_1 R : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : Monoid G p : MonoidAlgebra k G \u2192 Prop f : MonoidAlgebra k G hM : \u2200 (g : G), p (\u2191(of k G) g) hadd : \u2200 (f g : MonoidAlgebra k G), p f \u2192 p g \u2192 p (f + g) hsmul : \u2200 (r : k) (f : MonoidAlgebra k G), p f \u2192 p (r \u2022 f) g : G r : k \u22a2 p fun\u2080 | g => r ** convert hsmul r (of k G g) (hM g) ** case h.e'_1 k : Type u\u2081 G : Type u\u2082 H : Type u_1 R : Type u_2 inst\u271d\u00b9 : Semiring k inst\u271d : Monoid G p : MonoidAlgebra k G \u2192 Prop f : MonoidAlgebra k G hM : \u2200 (g : G), p (\u2191(of k G) g) hadd : \u2200 (f g : MonoidAlgebra k G), p f \u2192 p g \u2192 p (f + g) hsmul : \u2200 (r : k) (f : MonoidAlgebra k G), p f \u2192 p (r \u2022 f) g : G r : k \u22a2 (fun\u2080 | g => r) = r \u2022 \u2191(of k G) g ** rw [of_apply, smul_single', mul_one] ** Qed", + "informal": "" + }, + { + "formal": "Complex.cos_sub_sin_I ** x y : \u2102 \u22a2 cos x - sin x * I = cexp (-x * I) ** rw [neg_mul, \u2190 cosh_sub_sinh, sinh_mul_I, cosh_mul_I] ** Qed", + "informal": "" + }, + { + "formal": "List.prod_range_succ' ** \u03b1\u271d \u03b1 : Type u inst\u271d : Monoid \u03b1 f : \u2115 \u2192 \u03b1 n : \u2115 \u22a2 1 * f 0 = f 0 * 1 ** rw [one_mul, mul_one] ** \u03b1\u271d \u03b1 : Type u inst\u271d : Monoid \u03b1 f : \u2115 \u2192 \u03b1 n x\u271d : \u2115 hd : prod (map f (range (succ x\u271d))) = f 0 * prod (map (fun i => f (succ i)) (range x\u271d)) \u22a2 prod (map f (range (succ (succ x\u271d)))) = f 0 * prod (map (fun i => f (succ i)) (range (succ x\u271d))) ** rw [List.prod_range_succ, hd, mul_assoc, \u2190 List.prod_range_succ] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.integrable_toReal_condexpKernel ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : NormedAddCommGroup F f : \u03a9 \u2192 F s : Set \u03a9 hs : MeasurableSet s \u22a2 Integrable fun \u03c9 => ENNReal.toReal (\u2191\u2191(\u2191(condexpKernel \u03bc m) \u03c9) s) ** rw [condexpKernel] ** \u03a9 : Type u_1 F : Type u_2 inst\u271d\u2075 : TopologicalSpace \u03a9 m m\u03a9 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : NormedAddCommGroup F f : \u03a9 \u2192 F s : Set \u03a9 hs : MeasurableSet s \u22a2 Integrable fun \u03c9 => ENNReal.toReal (\u2191\u2191(\u2191(kernel.comap (condDistrib id id \u03bc) id (_ : Measurable id)) \u03c9) s) ** exact integrable_toReal_condDistrib (aemeasurable_id'' \u03bc (inf_le_right : m \u2293 m\u03a9 \u2264 m\u03a9)) hs ** Qed", + "informal": "" + }, + { + "formal": "Commute.div_add_div ** \u03b1 : Type u_1 \u03b2 : Type u_2 K : Type u_3 inst\u271d : DivisionSemiring \u03b1 a b c d : \u03b1 hbc : Commute b c hbd : Commute b d hb : b \u2260 0 hd : d \u2260 0 \u22a2 a / b + c / d = (a * d + b * c) / (b * d) ** rw [add_div, mul_div_mul_right _ b hd, hbc.eq, hbd.eq, mul_div_mul_right c d hb] ** Qed", + "informal": "" + }, + { + "formal": "Sym2.filter_image_quotient_mk''_isDiag ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 \u22a2 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) = image Quotient.mk'' (Finset.diag s) ** ext z ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 z : Sym2 \u03b1 \u22a2 z \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194 z \u2208 image Quotient.mk'' (Finset.diag s) ** induction' z using Sym2.inductionOn ** case a.hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d : \u03b1 \u22a2 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 filter IsDiag (image Quotient.mk'' (s \u00d7\u02e2 s)) \u2194 Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u2208 image Quotient.mk'' (Finset.diag s) ** simp only [mem_image, mem_diag, exists_prop, mem_filter, Prod.exists, mem_product] ** case a.hf \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d : \u03b1 \u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227 IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2194 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) ** constructor ** case a.hf.mp \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d : \u03b1 \u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227 IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) ** rintro \u27e8\u27e8a, b, \u27e8ha, hb\u27e9, (h : Quotient.mk _ _ = _)\u27e9, hab\u27e9 ** case a.hf.mp.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d : \u03b1 hab : IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) a b : \u03b1 h : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) ha : a \u2208 s hb : b \u2208 s \u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) ** rw [\u2190 h, Sym2.mk''_isDiag_iff] at hab ** case a.hf.mp.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d a b : \u03b1 hab : a = b h : Quotient.mk (Rel.setoid \u03b1) (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) ha : a \u2208 s hb : b \u2208 s \u22a2 \u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) ** exact \u27e8a, b, \u27e8ha, hab\u27e9, h\u27e9 ** case a.hf.mpr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d : \u03b1 \u22a2 (\u2203 a b, (a \u2208 s \u2227 a = b) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2192 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227 IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) ** rintro \u27e8a, b, \u27e8ha, rfl\u27e9, h\u27e9 ** case a.hf.mpr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d a : \u03b1 ha : a \u2208 s h : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u22a2 (\u2203 a b, (a \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a, b) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) \u2227 IsDiag (Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d)) ** rw [\u2190 h] ** case a.hf.mpr.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 x\u271d y\u271d a : \u03b1 ha : a \u2208 s h : Quotient.mk'' (a, a) = Quotient.mk (Rel.setoid \u03b1) (x\u271d, y\u271d) \u22a2 (\u2203 a_1 b, (a_1 \u2208 s \u2227 b \u2208 s) \u2227 Quotient.mk'' (a_1, b) = Quotient.mk'' (a, a)) \u2227 IsDiag (Quotient.mk'' (a, a)) ** exact \u27e8\u27e8a, a, \u27e8ha, ha\u27e9, rfl\u27e9, rfl\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Std.RBNode.Stream.toList_cons ** \u03b1 : Type u_1 x : \u03b1 r : RBNode \u03b1 s : RBNode.Stream \u03b1 \u22a2 toList (cons x r s) = x :: RBNode.toList r ++ toList s ** rw [toList, toList, foldr, RBNode.foldr_cons] ** \u03b1 : Type u_1 x : \u03b1 r : RBNode \u03b1 s : RBNode.Stream \u03b1 \u22a2 x :: (RBNode.toList r ++ foldr (fun x x_1 => x :: x_1) s []) = x :: RBNode.toList r ++ foldr (fun x x_1 => x :: x_1) s [] ** rfl ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.StructureSheaf.comapFunIsLocallyFraction ** R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s \u22a2 PrelocalPredicate.pred (isLocallyFraction S).toPrelocalPredicate (comapFun f U V hUV s) ** rintro \u27e8p, hpV\u27e9 ** case mk R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V \u22a2 \u2203 V_1 x i, PrelocalPredicate.pred (isFractionPrelocal S) fun x => comapFun f U V hUV s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x) ** rcases hs \u27e8PrimeSpectrum.comap f p, hUV hpV\u27e9 with \u27e8W, m, iWU, a, b, h_frac\u27e9 ** case mk.intro.intro.intro.intro.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R h_frac : \u2200 (x : { x // x \u2208 W }), \u00acb \u2208 (\u2191x).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) x * \u2191(algebraMap R (Localizations R \u2191x)) b = \u2191(algebraMap R (Localizations R \u2191x)) a \u22a2 \u2203 V_1 x i, PrelocalPredicate.pred (isFractionPrelocal S) fun x => comapFun f U V hUV s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x) ** refine' \u27e8Opens.comap (PrimeSpectrum.comap f) W \u2293 V, \u27e8m, hpV\u27e9, Opens.infLERight _ _, f a, f b, _\u27e9 ** case mk.intro.intro.intro.intro.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R h_frac : \u2200 (x : { x // x \u2208 W }), \u00acb \u2208 (\u2191x).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) x * \u2191(algebraMap R (Localizations R \u2191x)) b = \u2191(algebraMap R (Localizations R \u2191x)) a \u22a2 \u2200 (x : { x // x \u2208 \u2191(Opens.comap (PrimeSpectrum.comap f)) W \u2293 V }), \u00ac\u2191f b \u2208 (\u2191x).asIdeal \u2227 (fun x => comapFun f U V hUV s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) x * \u2191(algebraMap S (Localizations S \u2191x)) (\u2191f b) = \u2191(algebraMap S (Localizations S \u2191x)) (\u2191f a) ** rintro \u27e8q, \u27e8hqW, hqV\u27e9\u27e9 ** case mk.intro.intro.intro.intro.intro.mk.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R h_frac : \u2200 (x : { x // x \u2208 W }), \u00acb \u2208 (\u2191x).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) x * \u2191(algebraMap R (Localizations R \u2191x)) b = \u2191(algebraMap R (Localizations R \u2191x)) a q : \u2191(PrimeSpectrum.Top S) hqW : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) hqV : q \u2208 \u2191V \u22a2 \u00ac\u2191f b \u2208 (\u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) }).asIdeal \u2227 (fun x => comapFun f U V hUV s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) { val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) } * \u2191(algebraMap S (Localizations S \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) })) (\u2191f b) = \u2191(algebraMap S (Localizations S \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) })) (\u2191f a) ** specialize h_frac \u27e8PrimeSpectrum.comap f q, hqW\u27e9 ** case mk.intro.intro.intro.intro.intro.mk.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R q : \u2191(PrimeSpectrum.Top S) hqW : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) hqV : q \u2208 \u2191V h_frac : \u00acb \u2208 (\u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW }).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) { val := \u2191(PrimeSpectrum.comap f) q, property := hqW } * \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) b = \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) a \u22a2 \u00ac\u2191f b \u2208 (\u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) }).asIdeal \u2227 (fun x => comapFun f U V hUV s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) { val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) } * \u2191(algebraMap S (Localizations S \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) })) (\u2191f b) = \u2191(algebraMap S (Localizations S \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) })) (\u2191f a) ** refine' \u27e8h_frac.1, _\u27e9 ** case mk.intro.intro.intro.intro.intro.mk.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R q : \u2191(PrimeSpectrum.Top S) hqW : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) hqV : q \u2208 \u2191V h_frac : \u00acb \u2208 (\u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW }).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) { val := \u2191(PrimeSpectrum.comap f) q, property := hqW } * \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) b = \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) a \u22a2 (fun x => comapFun f U V hUV s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191V) }) x)) { val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) } * \u2191(algebraMap S (Localizations S \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) })) (\u2191f b) = \u2191(algebraMap S (Localizations S \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) })) (\u2191f a) ** dsimp only [comapFun] ** case mk.intro.intro.intro.intro.intro.mk.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R q : \u2191(PrimeSpectrum.Top S) hqW : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) hqV : q \u2208 \u2191V h_frac : \u00acb \u2208 (\u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW }).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) { val := \u2191(PrimeSpectrum.comap f) q, property := hqW } * \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) b = \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) a \u22a2 \u2191(Localization.localRingHom (\u2191(PrimeSpectrum.comap f) q).asIdeal q.asIdeal f (_ : (\u2191(PrimeSpectrum.comap f) \u2191{ val := q, property := (_ : \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) } \u2208 \u2191V) }).asIdeal = (\u2191(PrimeSpectrum.comap f) \u2191{ val := q, property := (_ : \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) } \u2208 \u2191V) }).asIdeal)) (s { val := \u2191(PrimeSpectrum.comap f) q, property := (_ : \u2191{ val := q, property := (_ : \u2191{ val := q, property := (_ : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) \u2227 q \u2208 \u2191V) } \u2208 \u2191V) } \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) }) * \u2191(algebraMap S (Localizations S q)) (\u2191f b) = \u2191(algebraMap S (Localizations S q)) (\u2191f a) ** erw [\u2190 Localization.localRingHom_to_map (PrimeSpectrum.comap f q).asIdeal, \u2190 RingHom.map_mul,\n h_frac.2, Localization.localRingHom_to_map] ** case mk.intro.intro.intro.intro.intro.mk.intro R : Type u inst\u271d\u00b2 : CommRing R S : Type u inst\u271d\u00b9 : CommRing S P : Type u inst\u271d : CommRing P f : R \u2192+* S U : Opens \u2191(PrimeSpectrum.Top R) V : Opens \u2191(PrimeSpectrum.Top S) hUV : V.carrier \u2286 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier s : (x : { x // x \u2208 U }) \u2192 Localizations R \u2191x hs : PrelocalPredicate.pred (isLocallyFraction R).toPrelocalPredicate s p : \u2191(PrimeSpectrum.Top S) hpV : p \u2208 V W : Opens \u2191(PrimeSpectrum.Top R) m : \u2191{ val := \u2191(PrimeSpectrum.comap f) p, property := (_ : p \u2208 \u2191(PrimeSpectrum.comap f) \u207b\u00b9' U.carrier) } \u2208 W iWU : W \u27f6 U a b : R q : \u2191(PrimeSpectrum.Top S) hqW : q \u2208 \u2191(\u2191(Opens.comap (PrimeSpectrum.comap f)) W) hqV : q \u2208 \u2191V h_frac : \u00acb \u2208 (\u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW }).asIdeal \u2227 (fun x => s ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) x)) { val := \u2191(PrimeSpectrum.comap f) q, property := hqW } * \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) b = \u2191(algebraMap R (Localizations R \u2191{ val := \u2191(PrimeSpectrum.comap f) q, property := hqW })) a \u22a2 \u2191(algebraMap ((fun x => S) a) (Localization.AtPrime q.asIdeal)) (\u2191f a) = \u2191(algebraMap S (Localizations S q)) (\u2191f a) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Nat.Iio_eq_range ** a b c : \u2115 \u22a2 Iio = range ** ext b x ** case h.a a b\u271d c b x : \u2115 \u22a2 x \u2208 Iio b \u2194 x \u2208 range b ** rw [mem_Iio, mem_range] ** Qed", + "informal": "" + }, + { + "formal": "mul_basis_toMatrix ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03ba : Type u_3 \u03ba' : Type u_4 R : Type u_5 M : Type u_6 inst\u271d\u00b9\u00b3 : CommSemiring R inst\u271d\u00b9\u00b2 : AddCommMonoid M inst\u271d\u00b9\u00b9 : Module R M R\u2082 : Type u_7 M\u2082 : Type u_8 inst\u271d\u00b9\u2070 : CommRing R\u2082 inst\u271d\u2079 : AddCommGroup M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 e : Basis \u03b9 R M v : \u03b9' \u2192 M i : \u03b9 j : \u03b9' N : Type u_9 inst\u271d\u2077 : AddCommMonoid N inst\u271d\u2076 : Module R N b : Basis \u03b9 R M b' : Basis \u03b9' R M c : Basis \u03ba R N c' : Basis \u03ba' R N f : M \u2192\u2097[R] N inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : Fintype \u03ba inst\u271d\u00b3 : Fintype \u03ba' inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : DecidableEq \u03b9' b\u2081 : Basis \u03b9 R M b\u2082 : Basis \u03b9' R M b\u2083 : Basis \u03ba R N A : Matrix \u03ba \u03b9 R \u22a2 A * Basis.toMatrix b\u2081 \u2191b\u2082 = \u2191(toMatrix b\u2082 b\u2083) (\u2191(toLin b\u2081 b\u2083) A) ** have := linearMap_toMatrix_mul_basis_toMatrix b\u2082 b\u2081 b\u2083 (Matrix.toLin b\u2081 b\u2083 A) ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03ba : Type u_3 \u03ba' : Type u_4 R : Type u_5 M : Type u_6 inst\u271d\u00b9\u00b3 : CommSemiring R inst\u271d\u00b9\u00b2 : AddCommMonoid M inst\u271d\u00b9\u00b9 : Module R M R\u2082 : Type u_7 M\u2082 : Type u_8 inst\u271d\u00b9\u2070 : CommRing R\u2082 inst\u271d\u2079 : AddCommGroup M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 e : Basis \u03b9 R M v : \u03b9' \u2192 M i : \u03b9 j : \u03b9' N : Type u_9 inst\u271d\u2077 : AddCommMonoid N inst\u271d\u2076 : Module R N b : Basis \u03b9 R M b' : Basis \u03b9' R M c : Basis \u03ba R N c' : Basis \u03ba' R N f : M \u2192\u2097[R] N inst\u271d\u2075 : Fintype \u03b9' inst\u271d\u2074 : Fintype \u03ba inst\u271d\u00b3 : Fintype \u03ba' inst\u271d\u00b2 : Fintype \u03b9 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : DecidableEq \u03b9' b\u2081 : Basis \u03b9 R M b\u2082 : Basis \u03b9' R M b\u2083 : Basis \u03ba R N A : Matrix \u03ba \u03b9 R this : \u2191(toMatrix b\u2081 b\u2083) (\u2191(toLin b\u2081 b\u2083) A) * Basis.toMatrix b\u2081 \u2191b\u2082 = \u2191(toMatrix b\u2082 b\u2083) (\u2191(toLin b\u2081 b\u2083) A) \u22a2 A * Basis.toMatrix b\u2081 \u2191b\u2082 = \u2191(toMatrix b\u2082 b\u2083) (\u2191(toLin b\u2081 b\u2083) A) ** rwa [LinearMap.toMatrix_toLin] at this ** Qed", + "informal": "" + }, + { + "formal": "Submodule.mem_ideal_smul_span_iff_exists_sum ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M \u22a2 x \u2208 I \u2022 span R (Set.range f) \u2194 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x ** constructor ** case mp R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M \u22a2 x \u2208 I \u2022 span R (Set.range f) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x case mpr R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M \u22a2 (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 x \u2208 I \u2022 span R (Set.range f) ** swap ** case mp R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M \u22a2 x \u2208 I \u2022 span R (Set.range f) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x ** refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ ** case mpr R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M \u22a2 (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 x \u2208 I \u2022 span R (Set.range f) ** rintro \u27e8a, ha, rfl\u27e9 ** case mpr.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M a : \u03b9 \u2192\u2080 R ha : \u2200 (i : \u03b9), \u2191a i \u2208 I \u22a2 (Finsupp.sum a fun i c => c \u2022 f i) \u2208 I \u2022 span R (Set.range f) ** exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ ** case mp.refine'_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) \u22a2 \u2200 (x : M), x \u2208 \u22c3 a \u2208 I, \u22c3 b \u2208 Set.range f, {a \u2022 b} \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x ** simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] ** case mp.refine'_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) \u22a2 \u2200 (x : M), (\u2203 i h i_1 h, x = i \u2022 i_1) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x ** rintro x \u27e8y, hy, x, \u27e8i, rfl\u27e9, rfl\u27e9 ** case mp.refine'_1.intro.intro.intro.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i : \u03b9 \u22a2 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = y \u2022 f i ** refine' \u27e8Finsupp.single i y, fun j => _, _\u27e9 ** case mp.refine'_1.intro.intro.intro.intro.intro.refine'_2 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i : \u03b9 \u22a2 (Finsupp.sum (fun\u2080 | i => y) fun i c => c \u2022 f i) = y \u2022 f i ** refine' @Finsupp.sum_single_index \u03b9 R M _ _ i _ (fun i y => y \u2022 f i) _ ** case mp.refine'_1.intro.intro.intro.intro.intro.refine'_2 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i : \u03b9 \u22a2 (fun i y => y \u2022 f i) i 0 = 0 ** simp ** case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i j : \u03b9 \u22a2 (\u2191fun\u2080 | i => y) j \u2208 I ** letI := Classical.decEq \u03b9 ** case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i j : \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 \u22a2 (\u2191fun\u2080 | i => y) j \u2208 I ** rw [Finsupp.single_apply] ** case mp.refine'_1.intro.intro.intro.intro.intro.refine'_1 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i j : \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 \u22a2 (if i = j then y else 0) \u2208 I ** split_ifs ** case pos R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i j : \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 h\u271d : i = j \u22a2 y \u2208 I ** assumption ** case neg R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) y : R hy : y \u2208 I i j : \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 h\u271d : \u00aci = j \u22a2 0 \u2208 I ** exact I.zero_mem ** case mp.refine'_2 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) \u22a2 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = 0 ** exact \u27e80, fun _ => I.zero_mem, Finsupp.sum_zero_index\u27e9 ** case mp.refine'_3 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) \u22a2 \u2200 (x y : M), (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 (\u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = y) \u2192 \u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x + y ** rintro x y \u27e8ax, hax, rfl\u27e9 \u27e8ay, hay, rfl\u27e9 ** case mp.refine'_3.intro.intro.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) ax : \u03b9 \u2192\u2080 R hax : \u2200 (i : \u03b9), \u2191ax i \u2208 I ay : \u03b9 \u2192\u2080 R hay : \u2200 (i : \u03b9), \u2191ay i \u2208 I \u22a2 \u2203 a x, (Finsupp.sum a fun i c => c \u2022 f i) = (Finsupp.sum ax fun i c => c \u2022 f i) + Finsupp.sum ay fun i c => c \u2022 f i ** refine' \u27e8ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _\u27e9 <;> intros <;>\n simp only [zero_smul, add_smul] ** case mp.refine'_4 R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) \u22a2 \u2200 (a : R) (x : M), (\u2203 a x_1, (Finsupp.sum a fun i c => c \u2022 f i) = x) \u2192 \u2203 a_2 x_1, (Finsupp.sum a_2 fun i c => c \u2022 f i) = a \u2022 x ** rintro c x \u27e8a, ha, rfl\u27e9 ** case mp.refine'_4.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) c : R a : \u03b9 \u2192\u2080 R ha : \u2200 (i : \u03b9), \u2191a i \u2208 I \u22a2 \u2203 a_1 x, (Finsupp.sum a_1 fun i c => c \u2022 f i) = c \u2022 Finsupp.sum a fun i c => c \u2022 f i ** refine' \u27e8c \u2022 a, fun i => I.mul_mem_left c (ha i), _\u27e9 ** case mp.refine'_4.intro.intro R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' \u03b9 : Type u_3 f : \u03b9 \u2192 M x : M hx : x \u2208 I \u2022 span R (Set.range f) c : R a : \u03b9 \u2192\u2080 R ha : \u2200 (i : \u03b9), \u2191a i \u2208 I \u22a2 (Finsupp.sum (c \u2022 a) fun i c => c \u2022 f i) = c \u2022 Finsupp.sum a fun i c => c \u2022 f i ** rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] ** Qed", + "informal": "" + }, + { + "formal": "Nat.choose_le_succ_of_lt_half_left ** r n : \u2115 h : r < n / 2 \u22a2 choose n r \u2264 choose n (r + 1) ** refine' le_of_mul_le_mul_right _ (lt_tsub_iff_left.mpr (lt_of_lt_of_le h (n.div_le_self 2))) ** r n : \u2115 h : r < n / 2 \u22a2 choose n r * (n - r) \u2264 choose n (r + 1) * (n - r) ** rw [\u2190 choose_succ_right_eq] ** r n : \u2115 h : r < n / 2 \u22a2 choose n (r + 1) * (r + 1) \u2264 choose n (r + 1) * (n - r) ** apply Nat.mul_le_mul_left ** case h r n : \u2115 h : r < n / 2 \u22a2 r + 1 \u2264 n - r ** rw [\u2190 Nat.lt_iff_add_one_le, lt_tsub_iff_left, \u2190 mul_two] ** case h r n : \u2115 h : r < n / 2 \u22a2 r * 2 < n ** exact lt_of_lt_of_le (mul_lt_mul_of_pos_right h zero_lt_two) (n.div_mul_le_self 2) ** Qed", + "informal": "" + }, + { + "formal": "collinear_insert_insert_insert_left_of_mem_affineSpan_pair ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : DivisionRing k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P p\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P h\u2081 : p\u2081 \u2208 affineSpan k {p\u2084, p\u2085} h\u2082 : p\u2082 \u2208 affineSpan k {p\u2084, p\u2085} h\u2083 : p\u2083 \u2208 affineSpan k {p\u2084, p\u2085} \u22a2 Collinear k {p\u2081, p\u2082, p\u2083, p\u2084} ** refine' (collinear_insert_insert_insert_of_mem_affineSpan_pair h\u2081 h\u2082 h\u2083).subset _ ** k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : DivisionRing k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P p\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P h\u2081 : p\u2081 \u2208 affineSpan k {p\u2084, p\u2085} h\u2082 : p\u2082 \u2208 affineSpan k {p\u2084, p\u2085} h\u2083 : p\u2083 \u2208 affineSpan k {p\u2084, p\u2085} \u22a2 {p\u2081, p\u2082, p\u2083, p\u2084} \u2286 {p\u2081, p\u2082, p\u2083, p\u2084, p\u2085} ** repeat apply Set.insert_subset_insert ** case h.h.h k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : DivisionRing k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P p\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P h\u2081 : p\u2081 \u2208 affineSpan k {p\u2084, p\u2085} h\u2082 : p\u2082 \u2208 affineSpan k {p\u2084, p\u2085} h\u2083 : p\u2083 \u2208 affineSpan k {p\u2084, p\u2085} \u22a2 {p\u2084} \u2286 {p\u2084, p\u2085} ** simp ** case h.h k : Type u_1 V : Type u_2 P : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b3 : DivisionRing k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P p\u2081 p\u2082 p\u2083 p\u2084 p\u2085 : P h\u2081 : p\u2081 \u2208 affineSpan k {p\u2084, p\u2085} h\u2082 : p\u2082 \u2208 affineSpan k {p\u2084, p\u2085} h\u2083 : p\u2083 \u2208 affineSpan k {p\u2084, p\u2085} \u22a2 {p\u2083, p\u2084} \u2286 {p\u2083, p\u2084, p\u2085} ** apply Set.insert_subset_insert ** Qed", + "informal": "" + }, + { + "formal": "String.Iterator.Valid.remainingToString ** l r : List Char it : Iterator h : ValidFor l r it \u22a2 Iterator.remainingToString it = { data := r } ** cases h.out ** case refl l r : List Char h : ValidFor l r { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } \u22a2 Iterator.remainingToString { s := { data := List.reverseAux l r }, i := { byteIdx := utf8Len l } } = { data := r } ** simpa [Iterator.remainingToString, List.reverseAux_eq] using extract_of_valid l.reverse r [] ** Qed", + "informal": "" + }, + { + "formal": "ChainComplex.next_nat_zero ** \u03b9 : Type u_1 V : Type u inst\u271d\u00b9 : Category.{v, u} V inst\u271d : HasZeroMorphisms V \u22a2 ComplexShape.next (ComplexShape.down \u2115) 0 = 0 ** classical\n refine' dif_neg _\n push_neg\n intro\n apply Nat.noConfusion ** \u03b9 : Type u_1 V : Type u inst\u271d\u00b9 : Category.{v, u} V inst\u271d : HasZeroMorphisms V \u22a2 ComplexShape.next (ComplexShape.down \u2115) 0 = 0 ** refine' dif_neg _ ** \u03b9 : Type u_1 V : Type u inst\u271d\u00b9 : Category.{v, u} V inst\u271d : HasZeroMorphisms V \u22a2 \u00ac\u2203 j, ComplexShape.Rel (ComplexShape.down \u2115) 0 j ** push_neg ** \u03b9 : Type u_1 V : Type u inst\u271d\u00b9 : Category.{v, u} V inst\u271d : HasZeroMorphisms V \u22a2 \u2200 (j : \u2115), \u00acComplexShape.Rel (ComplexShape.down \u2115) 0 j ** intro ** \u03b9 : Type u_1 V : Type u inst\u271d\u00b9 : Category.{v, u} V inst\u271d : HasZeroMorphisms V j\u271d : \u2115 \u22a2 \u00acComplexShape.Rel (ComplexShape.down \u2115) 0 j\u271d ** apply Nat.noConfusion ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.lift_le_continuum ** c : Cardinal.{u} \u22a2 lift.{v, u} c \u2264 \ud835\udd20 \u2194 c \u2264 \ud835\udd20 ** rw [\u2190 lift_continuum.{u,v}, lift_le] ** Qed", + "informal": "" + }, + { + "formal": "HasFPowerSeriesOnBall.sub ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \ud835\udd5c G f g : E \u2192 F p pf pg : FormalMultilinearSeries \ud835\udd5c E F x : E r r' : \u211d\u22650\u221e hf : HasFPowerSeriesOnBall f pf x r hg : HasFPowerSeriesOnBall g pg x r \u22a2 HasFPowerSeriesOnBall (f - g) (pf - pg) x r ** simpa only [sub_eq_add_neg] using hf.add hg.neg ** Qed", + "informal": "" + }, + { + "formal": "Real.exp_eq_exp_\u211d ** \u22a2 rexp = _root_.exp \u211d ** ext x ** case h x : \u211d \u22a2 rexp x = _root_.exp \u211d x ** exact_mod_cast congr_fun Complex.exp_eq_exp_\u2102 x ** Qed", + "informal": "" + }, + { + "formal": "jacobiSym.legendreSym.to_jacobiSym ** p : \u2115 fp : Fact (Nat.Prime p) a : \u2124 \u22a2 legendreSym p a = J(a | p) ** simp only [jacobiSym, factors_prime fp.1, List.prod_cons, List.prod_nil, mul_one, List.pmap] ** Qed", + "informal": "" + }, + { + "formal": "Finset.subset_union_elim ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 \u22a2 filter (fun x => x \u2208 t\u2081) s \u222a filter (fun x => \u00acx \u2208 t\u2081) s = s ** simp [filter_union_right, em] ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 \u22a2 \u2191(filter (fun x => x \u2208 t\u2081) s) \u2286 t\u2081 ** intro x ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 x : \u03b1 \u22a2 x \u2208 \u2191(filter (fun x => x \u2208 t\u2081) s) \u2192 x \u2208 t\u2081 ** simp ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 \u22a2 \u2191(filter (fun x => \u00acx \u2208 t\u2081) s) \u2286 t\u2082 \\ t\u2081 ** intro x ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 x : \u03b1 \u22a2 x \u2208 \u2191(filter (fun x => \u00acx \u2208 t\u2081) s) \u2192 x \u2208 t\u2082 \\ t\u2081 ** simp only [not_not, coe_filter, Set.mem_setOf_eq, Set.mem_diff, and_imp] ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 x : \u03b1 \u22a2 x \u2208 s \u2192 \u00acx \u2208 t\u2081 \u2192 x \u2208 t\u2082 \u2227 \u00acx \u2208 t\u2081 ** intro hx hx\u2082 ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b2 : DecidablePred p inst\u271d\u00b9 : DecidablePred q s\u271d : Finset \u03b1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 t\u2081 t\u2082 : Set \u03b1 h : \u2191s \u2286 t\u2081 \u222a t\u2082 x : \u03b1 hx : x \u2208 s hx\u2082 : \u00acx \u2208 t\u2081 \u22a2 x \u2208 t\u2082 \u2227 \u00acx \u2208 t\u2081 ** refine' \u27e8Or.resolve_left (h hx) hx\u2082, hx\u2082\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "List.nthLe_take' ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 L : List \u03b1 i j : \u2115 hi : i < length (take j L) \u22a2 length (take j L) \u2264 length L ** simp [le_refl] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicTopology.DoldKan.HigherFacesVanish.induction ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 \u22a2 HigherFacesVanish (q + 1) (\u03c6 \u226b HomologicalComplex.Hom.f (\ud835\udfd9 (AlternatingFaceMapComplex.obj X) + H\u03c3 q) (n + 1)) ** intro j hj\u2081 ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) \u22a2 (\u03c6 \u226b HomologicalComplex.Hom.f (\ud835\udfd9 (AlternatingFaceMapComplex.obj X) + H\u03c3 q) (n + 1)) \u226b \u03b4 X (Fin.succ j) = 0 ** dsimp ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) \u22a2 (\u03c6 \u226b (\ud835\udfd9 (X.obj (Opposite.op [n + 1])) + HomologicalComplex.Hom.f (H\u03c3 q) (n + 1))) \u226b \u03b4 X (Fin.succ j) = 0 ** simp only [comp_add, add_comp, comp_id] ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) + (\u03c6 \u226b HomologicalComplex.Hom.f (H\u03c3 q) (n + 1)) \u226b \u03b4 X (Fin.succ j) = 0 ** by_cases hqn : n < q ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) hqn : \u00acn < q \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) + (\u03c6 \u226b HomologicalComplex.Hom.f (H\u03c3 q) (n + 1)) \u226b \u03b4 X (Fin.succ j) = 0 ** cases' Nat.le.dest (not_lt.mp hqn) with a ha ** case neg.intro C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) hqn : \u00acn < q a : \u2115 ha : q + a = n \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) + (\u03c6 \u226b HomologicalComplex.Hom.f (H\u03c3 q) (n + 1)) \u226b \u03b4 X (Fin.succ j) = 0 ** rw [v.comp_H\u03c3_eq (show n = a + q by linarith), neg_comp, add_neg_eq_zero, assoc, assoc] ** case neg.intro C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) hqn : \u00acn < q a : \u2115 ha : q + a = n \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (n + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) } \u226b \u03b4 X (Fin.succ j) ** cases' n with m hm ** case neg.intro.succ C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } \u226b \u03b4 X (Fin.succ j) ** by_cases hj\u2082 : a = (j : \u2115) ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } \u226b \u03b4 X (Fin.succ j) ** have haj : a < j := (Ne.le_iff_lt hj\u2082).mp (by linarith) ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } \u226b \u03b4 X (Fin.succ j) ** have hj\u2083 := j.is_lt ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } \u226b \u03b4 X (Fin.succ j) ** have ham : a \u2264 m := by\n by_contra h\n rw [not_le, \u2190 Nat.succ_le_iff] at h\n linarith ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } \u226b \u03b4 X (Fin.succ j) ** rw [X.\u03b4_comp_\u03c3_of_gt', j.pred_succ] ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03b4 X j \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 Fin.succ { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < Fin.succ j case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 Fin.succ { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < Fin.succ j ** swap ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03b4 X j \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) ** obtain _ | ham'' := ham.lt_or_eq ** case pos C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) hqn : n < q \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) + (\u03c6 \u226b HomologicalComplex.Hom.f (H\u03c3 q) (n + 1)) \u226b \u03b4 X (Fin.succ j) = 0 ** rw [v.comp_H\u03c3_eq_zero hqn, zero_comp, add_zero, v j (by linarith)] ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) hqn : n < q \u22a2 n + 1 \u2264 \u2191j + q ** linarith ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C n q : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [n + 1]) v : HigherFacesVanish q \u03c6 j : Fin (n + 1) hj\u2081 : n + 1 \u2264 \u2191j + (q + 1) hqn : \u00acn < q a : \u2115 ha : q + a = n \u22a2 n = a + q ** linarith ** case neg.intro.zero C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.zero + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.zero + 1) hj\u2081 : Nat.zero + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.zero < q ha : q + a = Nat.zero \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.zero + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ Nat.zero) } \u226b \u03b4 X (Fin.succ j) ** simp only [Nat.eq_zero_of_add_eq_zero_left ha, Fin.eq_zero j, Fin.mk_zero, Fin.mk_one,\n \u03b4_comp_\u03c3_succ, comp_id] ** case neg.intro.zero C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.zero + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.zero + 1) hj\u2081 : Nat.zero + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.zero < q ha : q + a = Nat.zero \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ 0) = \u03c6 \u226b \u03b4 X { val := 0 + 1, isLt := (_ : 0 + 1 < Nat.zero + 2) } ** rfl ** case pos C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : a = \u2191j \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03c3 X { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } \u226b \u03b4 X (Fin.succ j) ** simp only [hj\u2082, Fin.eta, \u03b4_comp_\u03c3_succ, comp_id] ** case pos C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : a = \u2191j \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := \u2191j + 1, isLt := (_ : \u2191j + 1 < Nat.succ m + 2) } ** rfl ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j \u22a2 a \u2264 \u2191j ** linarith ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 \u22a2 a \u2264 m ** by_contra h ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 h : \u00aca \u2264 m \u22a2 False ** rw [not_le, \u2190 Nat.succ_le_iff] at h ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 h : Nat.succ m \u2264 a \u22a2 False ** linarith ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 Fin.succ { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < Fin.succ j ** rw [Fin.lt_iff_val_lt_val] ** case neg C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m \u22a2 \u2191(Fin.succ { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) }) < \u2191(Fin.succ j) ** simpa only [Fin.val_mk, Fin.val_succ, add_lt_add_iff_right] using haj ** case neg.inl C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03b4 X j \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) ** rw [\u2190 X.\u03b4_comp_\u03b4''_assoc] ** case neg.inl C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X (Fin.succ j) \u226b \u03b4 X (Fin.castLT { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } (_ : \u2191{ val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } < m + 2)) \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) case neg.inl.H C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u2264 Fin.castSucc j ** swap ** case neg.inl C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X (Fin.succ j) \u226b \u03b4 X (Fin.castLT { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } (_ : \u2191{ val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } < m + 2)) \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) ** simp only [\u2190 assoc, v j (by linarith), zero_comp] ** case neg.inl.H C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u2264 Fin.castSucc j ** rw [Fin.le_iff_val_le_val] ** case neg.inl.H C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 \u2191{ val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u2264 \u2191(Fin.castSucc j) ** dsimp ** case neg.inl.H C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 a + 1 \u2264 \u2191j ** linarith ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m h\u271d : a < m \u22a2 Nat.succ m + 1 \u2264 \u2191j + q ** linarith ** case neg.inr C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } \u226b \u03b4 X j \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) ** rw [X.\u03b4_comp_\u03b4_self'_assoc] ** case neg.inr C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X (Fin.succ j) \u226b \u03b4 X j \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) case neg.inr.H C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } = Fin.castSucc j ** swap ** case neg.inr C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 \u03c6 \u226b \u03b4 X (Fin.succ j) = \u03c6 \u226b \u03b4 X (Fin.succ j) \u226b \u03b4 X j \u226b \u03c3 X (Fin.castLT { val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } (_ : \u2191{ val := a, isLt := (_ : a < Nat.succ (Nat.succ m)) } < m + 1)) ** simp only [\u2190 assoc, v j (by linarith), zero_comp] ** case neg.inr.H C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 { val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } = Fin.castSucc j ** ext ** case neg.inr.H.h C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 \u2191{ val := a + 1, isLt := (_ : Nat.succ a < Nat.succ (Nat.succ m + 1)) } = \u2191(Fin.castSucc j) ** dsimp ** case neg.inr.H.h C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 a + 1 = \u2191j ** linarith ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C Y : C q a m : \u2115 \u03c6 : Y \u27f6 X.obj (Opposite.op [Nat.succ m + 1]) v : HigherFacesVanish q \u03c6 j : Fin (Nat.succ m + 1) hj\u2081 : Nat.succ m + 1 \u2264 \u2191j + (q + 1) hqn : \u00acNat.succ m < q ha : q + a = Nat.succ m hj\u2082 : \u00aca = \u2191j haj : a < \u2191j hj\u2083 : \u2191j < Nat.succ m + 1 ham : a \u2264 m ham'' : a = m \u22a2 Nat.succ m + 1 \u2264 \u2191j + q ** linarith ** Qed", + "informal": "" + }, + { + "formal": "Beatty.hit_or_miss' ** r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 \u22a2 j \u2208 {x | \u2203 k, beattySeq' r k = x} \u2228 \u2203 k, \u2191k \u2264 \u2191j / r \u2227 (\u2191j + 1) / r < \u2191k + 1 ** cases le_or_gt (\u230a(j + 1) / r\u230b * r) j ** case inl r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 h\u271d : \u2191\u230a(\u2191j + 1) / r\u230b * r \u2264 \u2191j \u22a2 j \u2208 {x | \u2203 k, beattySeq' r k = x} \u2228 \u2203 k, \u2191k \u2264 \u2191j / r \u2227 (\u2191j + 1) / r < \u2191k + 1 ** exact Or.inr \u27e8\u230a(j + 1) / r\u230b, (le_div_iff h).2 \u2039_\u203a, Int.lt_floor_add_one _\u27e9 ** case inr r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 h\u271d : \u2191\u230a(\u2191j + 1) / r\u230b * r > \u2191j \u22a2 j \u2208 {x | \u2203 k, beattySeq' r k = x} \u2228 \u2203 k, \u2191k \u2264 \u2191j / r \u2227 (\u2191j + 1) / r < \u2191k + 1 ** refine Or.inl \u27e8\u230a(j + 1) / r\u230b, ?_\u27e9 ** case inr r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 h\u271d : \u2191\u230a(\u2191j + 1) / r\u230b * r > \u2191j \u22a2 beattySeq' r \u230a(\u2191j + 1) / r\u230b = j ** rw [beattySeq', sub_eq_iff_eq_add, Int.ceil_eq_iff, Int.cast_add, Int.cast_one] ** case inr r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 h\u271d : \u2191\u230a(\u2191j + 1) / r\u230b * r > \u2191j \u22a2 \u2191j + 1 - 1 < \u2191\u230a(\u2191j + 1) / r\u230b * r \u2227 \u2191\u230a(\u2191j + 1) / r\u230b * r \u2264 \u2191j + 1 ** constructor ** case inr.right r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 h\u271d : \u2191\u230a(\u2191j + 1) / r\u230b * r > \u2191j \u22a2 \u2191\u230a(\u2191j + 1) / r\u230b * r \u2264 \u2191j + 1 ** exact sub_nonneg.1 (Int.sub_floor_div_mul_nonneg (j + 1 : \u211d) h) ** case inr.left r s : \u211d hrs : Real.IsConjugateExponent r s j k : \u2124 h : r > 0 h\u271d : \u2191\u230a(\u2191j + 1) / r\u230b * r > \u2191j \u22a2 \u2191j + 1 - 1 < \u2191\u230a(\u2191j + 1) / r\u230b * r ** rwa [add_sub_cancel] ** Qed", + "informal": "" + }, + { + "formal": "star_exp ** \ud835\udd42 : Type u_1 \ud835\udd38 : Type u_2 inst\u271d\u2077 : Field \ud835\udd42 inst\u271d\u2076 : Ring \ud835\udd38 inst\u271d\u2075 : Algebra \ud835\udd42 \ud835\udd38 inst\u271d\u2074 : TopologicalSpace \ud835\udd38 inst\u271d\u00b3 : TopologicalRing \ud835\udd38 inst\u271d\u00b2 : T2Space \ud835\udd38 inst\u271d\u00b9 : StarRing \ud835\udd38 inst\u271d : ContinuousStar \ud835\udd38 x : \ud835\udd38 \u22a2 star (exp \ud835\udd42 x) = exp \ud835\udd42 (star x) ** simp_rw [exp_eq_tsum, \u2190 star_pow, \u2190 star_inv_nat_cast_smul, \u2190 tsum_star] ** Qed", + "informal": "" + }, + { + "formal": "derivedSeries_le_map_derivedSeries ** G : Type u_1 G' : Type u_2 inst\u271d\u00b9 : Group G inst\u271d : Group G' f : G \u2192* G' hf : Function.Surjective \u2191f n : \u2115 \u22a2 derivedSeries G' n \u2264 map f (derivedSeries G n) ** induction' n with n ih ** case zero G : Type u_1 G' : Type u_2 inst\u271d\u00b9 : Group G inst\u271d : Group G' f : G \u2192* G' hf : Function.Surjective \u2191f \u22a2 derivedSeries G' Nat.zero \u2264 map f (derivedSeries G Nat.zero) ** exact (map_top_of_surjective f hf).ge ** case succ G : Type u_1 G' : Type u_2 inst\u271d\u00b9 : Group G inst\u271d : Group G' f : G \u2192* G' hf : Function.Surjective \u2191f n : \u2115 ih : derivedSeries G' n \u2264 map f (derivedSeries G n) \u22a2 derivedSeries G' (Nat.succ n) \u2264 map f (derivedSeries G (Nat.succ n)) ** exact commutator_le_map_commutator ih ih ** Qed", + "informal": "" + }, + { + "formal": "Function.mulSupport_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d\u2074 : One M inst\u271d\u00b3 : One N inst\u271d\u00b2 : One P inst\u271d\u00b9 : ConditionallyCompleteLattice M inst\u271d : Nonempty \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 M \u22a2 (mulSupport fun x => \u2a06 i, f i x) \u2286 \u22c3 i, mulSupport (f i) ** rw [mulSupport_subset_iff'] ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d\u2074 : One M inst\u271d\u00b3 : One N inst\u271d\u00b2 : One P inst\u271d\u00b9 : ConditionallyCompleteLattice M inst\u271d : Nonempty \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 M \u22a2 \u2200 (x : \u03b1), \u00acx \u2208 \u22c3 i, mulSupport (f i) \u2192 \u2a06 i, f i x = 1 ** simp only [mem_iUnion, not_exists, nmem_mulSupport] ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d\u2074 : One M inst\u271d\u00b3 : One N inst\u271d\u00b2 : One P inst\u271d\u00b9 : ConditionallyCompleteLattice M inst\u271d : Nonempty \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 M \u22a2 \u2200 (x : \u03b1), (\u2200 (x_1 : \u03b9), f x_1 x = 1) \u2192 \u2a06 i, f i x = 1 ** intro x hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d\u2074 : One M inst\u271d\u00b3 : One N inst\u271d\u00b2 : One P inst\u271d\u00b9 : ConditionallyCompleteLattice M inst\u271d : Nonempty \u03b9 f : \u03b9 \u2192 \u03b1 \u2192 M x : \u03b1 hx : \u2200 (x_1 : \u03b9), f x_1 x = 1 \u22a2 \u2a06 i, f i x = 1 ** simp only [hx, ciSup_const] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Adjunction.homEquiv_symm_id ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj : F \u22a3 G X' X\u271d : C Y Y' X : D \u22a2 \u2191(homEquiv adj (G.obj X) X).symm (\ud835\udfd9 (G.obj X)) = adj.counit.app X ** simp ** Qed", + "informal": "" + }, + { + "formal": "Colex.forall_lt_of_colex_lt_of_forall_lt ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2081 : toColex A < toColex B h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t \u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 x < t ** rw [Colex.lt_def] at h\u2081 ** \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2081 : \u2203 k, (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t \u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 x < t ** rcases h\u2081 with \u27e8k, z, _, _\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t k : \u03b1 z : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B) left\u271d : \u00ack \u2208 A right\u271d : k \u2208 B \u22a2 \u2200 (x : \u03b1), x \u2208 A \u2192 x < t ** intro x hx ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t k : \u03b1 z : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B) left\u271d : \u00ack \u2208 A right\u271d : k \u2208 B x : \u03b1 hx : x \u2208 A \u22a2 x < t ** apply lt_of_not_ge ** case intro.intro.intro.h \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t k : \u03b1 z : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B) left\u271d : \u00ack \u2208 A right\u271d : k \u2208 B x : \u03b1 hx : x \u2208 A \u22a2 \u00acx \u2265 t ** intro a ** case intro.intro.intro.h \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t k : \u03b1 z : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B) left\u271d : \u00ack \u2208 A right\u271d : k \u2208 B x : \u03b1 hx : x \u2208 A a : x \u2265 t \u22a2 False ** refine' not_lt_of_ge a (h\u2082 x _) ** case intro.intro.intro.h \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t k : \u03b1 z : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B) left\u271d : \u00ack \u2208 A right\u271d : k \u2208 B x : \u03b1 hx : x \u2208 A a : x \u2265 t \u22a2 x \u2208 B ** rwa [\u2190 z] ** case intro.intro.intro.h \u03b1 : Type u_1 inst\u271d : LinearOrder \u03b1 A B : Finset \u03b1 t : \u03b1 h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t k : \u03b1 z : \u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B) left\u271d : \u00ack \u2208 A right\u271d : k \u2208 B x : \u03b1 hx : x \u2208 A a : x \u2265 t \u22a2 k < x ** apply lt_of_lt_of_le (h\u2082 k \u2039_\u203a) a ** Qed", + "informal": "" + }, + { + "formal": "Complex.equivRealProd_symm_apply ** p : \u211d \u00d7 \u211d \u22a2 \u2191equivRealProd.symm p = \u2191p.1 + \u2191p.2 * I ** ext <;> simp [Complex.equivRealProd, ofReal'] ** Qed", + "informal": "" + }, + { + "formal": "countable_image_lt_image_Ioi ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** nontriviality \u03b2 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** have : Nonempty \u03b1 := Nonempty.map f (by infer_instance) ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** let s := {x | \u2203 z, f x < z \u2227 \u2200 y, x < y \u2192 z \u2264 f y} ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** have : \u2200 x, x \u2208 s \u2192 \u2203 z, f x < z \u2227 \u2200 y, x < y \u2192 z \u2264 f y := fun x hx \u21a6 hx ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this\u271d : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} this : \u2200 (x : \u03b2), x \u2208 s \u2192 \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** choose! z hz using this ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** have I : InjOn f s := by\n apply StrictMonoOn.injOn\n intro x hx y _ hxy\n calc\n f x < z x := (hz x hx).1\n _ \u2264 f y := (hz x hx).2 y hxy ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s fs_count : Set.Countable (f '' s) \u22a2 Set.Countable {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} ** exact MapsTo.countable_of_injOn (mapsTo_image f s) I fs_count ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 \u22a2 Nonempty \u03b2 ** infer_instance ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y \u22a2 InjOn f s ** apply StrictMonoOn.injOn ** case H \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y \u22a2 StrictMonoOn f s ** intro x hx y _ hxy ** case H \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y x : \u03b2 hx : x \u2208 s y : \u03b2 x\u271d : y \u2208 s hxy : x < y \u22a2 f x < f y ** calc\n f x < z x := (hz x hx).1\n _ \u2264 f y := (hz x hx).2 y hxy ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s A : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x)) \u22a2 Set.Countable (f '' s) ** apply Set.PairwiseDisjoint.countable_of_Ioo A ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s A : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x)) \u22a2 \u2200 (x : \u03b1), x \u2208 f '' s \u2192 x < z (invFunOn f s x) ** rintro _ \u27e8y, ys, rfl\u27e9 ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s A : PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x)) y : \u03b2 ys : y \u2208 s \u22a2 f y < z (invFunOn f s (f y)) ** simpa only [I.leftInvOn_invFunOn ys] using (hz y ys).1 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s \u22a2 PairwiseDisjoint (f '' s) fun x => Ioo x (z (invFunOn f s x)) ** rintro _ \u27e8u, us, rfl\u27e9 _ \u27e8v, vs, rfl\u27e9 huv ** case intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v \u22a2 (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v) ** wlog hle : u \u2264 v generalizing u v ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v hle : u \u2264 v \u22a2 (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v) ** have hlt : u < v := hle.lt_of_ne (ne_of_apply_ne _ huv) ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v hle : u \u2264 v hlt : u < v \u22a2 (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v) ** apply disjoint_iff_forall_ne.2 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v hle : u \u2264 v hlt : u < v \u22a2 \u2200 \u2983a : \u03b1\u2984, a \u2208 (fun x => Ioo x (z (invFunOn f s x))) (f u) \u2192 \u2200 \u2983b : \u03b1\u2984, b \u2208 (fun x => Ioo x (z (invFunOn f s x))) (f v) \u2192 a \u2260 b ** rintro a ha b hb rfl ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v hle : u \u2264 v hlt : u < v a : \u03b1 ha : a \u2208 (fun x => Ioo x (z (invFunOn f s x))) (f u) hb : a \u2208 (fun x => Ioo x (z (invFunOn f s x))) (f v) \u22a2 False ** simp only [I.leftInvOn_invFunOn us, I.leftInvOn_invFunOn vs] at ha hb ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v hle : u \u2264 v hlt : u < v a : \u03b1 ha : a \u2208 Ioo (f u) (z u) hb : a \u2208 Ioo (f v) (z v) \u22a2 False ** exact lt_irrefl _ ((ha.2.trans_le ((hz u us).2 v hlt)).trans hb.1) ** case intro.intro.intro.intro.inr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2074 : TopologicalSpace \u03b1 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : OrderTopology \u03b1 inst\u271d\u00b9 : LinearOrder \u03b2 f : \u03b2 \u2192 \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u271d : Nontrivial \u03b2 this\u271d : Nonempty \u03b1 s : Set \u03b2 := {x | \u2203 z, f x < z \u2227 \u2200 (y : \u03b2), x < y \u2192 z \u2264 f y} z : \u03b2 \u2192 \u03b1 hz : \u2200 (x : \u03b2), x \u2208 s \u2192 f x < z x \u2227 \u2200 (y : \u03b2), x < y \u2192 z x \u2264 f y I : InjOn f s u : \u03b2 us : u \u2208 s v : \u03b2 vs : v \u2208 s huv : f u \u2260 f v this : \u2200 (u : \u03b2), u \u2208 s \u2192 \u2200 (v : \u03b2), v \u2208 s \u2192 f u \u2260 f v \u2192 u \u2264 v \u2192 (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v) hle : \u00acu \u2264 v \u22a2 (Disjoint on fun x => Ioo x (z (invFunOn f s x))) (f u) (f v) ** exact (this v vs u us huv.symm (le_of_not_le hle)).symm ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.vars_C ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R \u22a2 vars (\u2191C r) = \u2205 ** classical rw [vars_def, degrees_C, Multiset.toFinset_zero] ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R \u22a2 vars (\u2191C r) = \u2205 ** rw [vars_def, degrees_C, Multiset.toFinset_zero] ** Qed", + "informal": "" + }, + { + "formal": "PFun.comp_id ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b5 : Type u_5 \u03b9 : Type u_6 f\u271d f : \u03b1 \u2192. \u03b2 x\u271d\u00b9 : \u03b1 x\u271d : \u03b2 \u22a2 x\u271d \u2208 comp f (PFun.id \u03b1) x\u271d\u00b9 \u2194 x\u271d \u2208 f x\u271d\u00b9 ** simp ** Qed", + "informal": "" + }, + { + "formal": "WittVector.RecursionBase.solution_nonzero ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k a\u2081 a\u2082 : \ud835\udd4e k ha\u2081 : coeff a\u2081 0 \u2260 0 ha\u2082 : coeff a\u2082 0 \u2260 0 \u22a2 solution p a\u2081 a\u2082 \u2260 0 ** intro h ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k a\u2081 a\u2082 : \ud835\udd4e k ha\u2081 : coeff a\u2081 0 \u2260 0 ha\u2082 : coeff a\u2082 0 \u2260 0 h : solution p a\u2081 a\u2082 = 0 \u22a2 False ** have := solution_spec p a\u2081 a\u2082 ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k a\u2081 a\u2082 : \ud835\udd4e k ha\u2081 : coeff a\u2081 0 \u2260 0 ha\u2082 : coeff a\u2082 0 \u2260 0 h : solution p a\u2081 a\u2082 = 0 this : solution p a\u2081 a\u2082 ^ (p - 1) = coeff a\u2082 0 / coeff a\u2081 0 \u22a2 False ** rw [h, zero_pow] at this ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k a\u2081 a\u2082 : \ud835\udd4e k ha\u2081 : coeff a\u2081 0 \u2260 0 ha\u2082 : coeff a\u2082 0 \u2260 0 h : solution p a\u2081 a\u2082 = 0 this : 0 = coeff a\u2082 0 / coeff a\u2081 0 \u22a2 False ** simpa [ha\u2081, ha\u2082] using _root_.div_eq_zero_iff.mp this.symm ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k a\u2081 a\u2082 : \ud835\udd4e k ha\u2081 : coeff a\u2081 0 \u2260 0 ha\u2082 : coeff a\u2082 0 \u2260 0 h : solution p a\u2081 a\u2082 = 0 this : 0 ^ (p - 1) = coeff a\u2082 0 / coeff a\u2081 0 \u22a2 0 < p - 1 ** exact tsub_pos_of_lt hp.out.one_lt ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Adjunction.leftAdjointUniq_refl ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G \u22a2 (leftAdjointUniq adj1 adj1).hom = \ud835\udfd9 F ** ext ** case w.h C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G x\u271d : C \u22a2 (leftAdjointUniq adj1 adj1).hom.app x\u271d = (\ud835\udfd9 F).app x\u271d ** apply Quiver.Hom.op_inj ** case w.h.a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G x\u271d : C \u22a2 ((leftAdjointUniq adj1 adj1).hom.app x\u271d).op = ((\ud835\udfd9 F).app x\u271d).op ** apply coyoneda.map_injective ** case w.h.a.a C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G x\u271d : C \u22a2 coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x\u271d).op = coyoneda.map ((\ud835\udfd9 F).app x\u271d).op ** ext ** case w.h.a.a.w.h.h C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G x\u271d\u00b9 : C x\u271d : D a\u271d : (coyoneda.obj (Opposite.op (F.obj x\u271d\u00b9))).obj x\u271d \u22a2 (coyoneda.map ((leftAdjointUniq adj1 adj1).hom.app x\u271d\u00b9).op).app x\u271d a\u271d = (coyoneda.map ((\ud835\udfd9 F).app x\u271d\u00b9).op).app x\u271d a\u271d ** simp [leftAdjointsCoyonedaEquiv, leftAdjointUniq] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.inner_nonneg_of_dist_le_radius ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P s : Sphere P p\u2081 p\u2082 : P hp\u2081 : p\u2081 \u2208 s hp\u2082 : dist p\u2082 s.center \u2264 s.radius \u22a2 0 \u2264 inner (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 s.center) ** rcases inner_pos_or_eq_of_dist_le_radius hp\u2081 hp\u2082 with (h | rfl) ** case inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P s : Sphere P p\u2081 p\u2082 : P hp\u2081 : p\u2081 \u2208 s hp\u2082 : dist p\u2082 s.center \u2264 s.radius h : 0 < inner (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 s.center) \u22a2 0 \u2264 inner (p\u2081 -\u1d65 p\u2082) (p\u2081 -\u1d65 s.center) ** exact h.le ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P s : Sphere P p\u2081 : P hp\u2081 : p\u2081 \u2208 s hp\u2082 : dist p\u2081 s.center \u2264 s.radius \u22a2 0 \u2264 inner (p\u2081 -\u1d65 p\u2081) (p\u2081 -\u1d65 s.center) ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.set_lintegral_empty ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e \u22a2 \u222b\u207b (x : \u03b1) in \u2205, f x \u2202\u03bc = 0 ** rw [Measure.restrict_empty, lintegral_zero_measure] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.VectorMeasure.zero_le_restrict_not_measurable ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M v w : VectorMeasure \u03b1 M i j : Set \u03b1 hi : \u00acMeasurableSet i \u22a2 restrict 0 i \u2264 restrict v i ** rw [restrict_zero, restrict_not_measurable _ hi] ** Qed", + "informal": "" + }, + { + "formal": "volume_setOf_liouville ** \u22a2 \u2191\u2191volume {x | Liouville x} = 0 ** simpa only [ae_iff, Classical.not_not] using ae_not_liouville ** Qed", + "informal": "" + }, + { + "formal": "Seminorm.sub_mem_ball ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b2 : SeminormedRing \ud835\udd5c inst\u271d\u00b9 : AddCommGroup E inst\u271d : SMul \ud835\udd5c E p\u271d : Seminorm \ud835\udd5c E x y\u271d : E r\u271d : \u211d p : Seminorm \ud835\udd5c E x\u2081 x\u2082 y : E r : \u211d \u22a2 x\u2081 - x\u2082 \u2208 ball p y r \u2194 x\u2081 \u2208 ball p (x\u2082 + y) r ** simp_rw [mem_ball, sub_sub] ** Qed", + "informal": "" + }, + { + "formal": "Irreducible.isUnit_gcd_iff ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : GCDMonoid \u03b1 x y : \u03b1 hx : Irreducible x \u22a2 IsUnit (gcd x y) \u2194 \u00acx \u2223 y ** rw [hx.isUnit_iff_not_associated_of_dvd (gcd_dvd_left x y), not_iff_not, associated_gcd_left_iff] ** Qed", + "informal": "" + }, + { + "formal": "Fin.pred_mk_succ ** n i : Nat h : i < n + 1 \u22a2 pred { val := i + 1, isLt := (_ : i + 1 < n + 1 + 1) } (_ : \u00ac{ val := i + 1, isLt := (_ : i + 1 < n + 1 + 1) } = 0) = { val := i, isLt := h } ** simp only [ext_iff, coe_pred, Nat.add_sub_cancel] ** Qed", + "informal": "" + }, + { + "formal": "padicValInt.self ** p : \u2115 hp : 1 < p \u22a2 padicValInt p \u2191p = 1 ** simp [padicValNat.self hp] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.mem\u2112p_re_im_iff ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F K : Type u_6 inst\u271d : IsROrC K f : \u03b1 \u2192 K \u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p \u2227 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p \u2194 Mem\u2112p f p ** refine' \u27e8_, fun hf => \u27e8hf.re, hf.im\u27e9\u27e9 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F K : Type u_6 inst\u271d : IsROrC K f : \u03b1 \u2192 K \u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p \u2227 Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p \u2192 Mem\u2112p f p ** rintro \u27e8hre, him\u27e9 ** case intro \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F K : Type u_6 inst\u271d : IsROrC K f : \u03b1 \u2192 K hre : Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p him : Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p \u22a2 Mem\u2112p f p ** convert MeasureTheory.Mem\u2112p.add (E := K) hre.ofReal (him.ofReal.const_mul IsROrC.I) ** case h.e'_5 \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F K : Type u_6 inst\u271d : IsROrC K f : \u03b1 \u2192 K hre : Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p him : Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p \u22a2 f = (fun x => \u2191(\u2191IsROrC.re (f x))) + fun x => IsROrC.I * \u2191(\u2191IsROrC.im (f x)) ** ext1 x ** case h.e'_5.h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedAddCommGroup G g : E \u2192 F c : \u211d\u22650 \ud835\udd5c : Type u_5 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedSpace \ud835\udd5c F K : Type u_6 inst\u271d : IsROrC K f : \u03b1 \u2192 K hre : Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p him : Mem\u2112p (fun x => \u2191IsROrC.im (f x)) p x : \u03b1 \u22a2 f x = ((fun x => \u2191(\u2191IsROrC.re (f x))) + fun x => IsROrC.I * \u2191(\u2191IsROrC.im (f x))) x ** rw [Pi.add_apply, mul_comm, IsROrC.re_add_im] ** Qed", + "informal": "" + }, + { + "formal": "padicValInt.mul ** p : \u2115 hp : Fact (Nat.Prime p) a b : \u2124 ha : a \u2260 0 hb : b \u2260 0 \u22a2 padicValInt p (a * b) = padicValInt p a + padicValInt p b ** simp_rw [padicValInt] ** p : \u2115 hp : Fact (Nat.Prime p) a b : \u2124 ha : a \u2260 0 hb : b \u2260 0 \u22a2 padicValNat p (Int.natAbs (a * b)) = padicValNat p (Int.natAbs a) + padicValNat p (Int.natAbs b) ** rw [Int.natAbs_mul, padicValNat.mul] <;> rwa [Int.natAbs_ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.add_iSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d : Set \u211d\u22650\u221e \u03b9 : Sort u_4 s : \u03b9 \u2192 \u211d\u22650\u221e inst\u271d : Nonempty \u03b9 \u22a2 a + iSup s = \u2a06 b, a + s b ** rw [add_comm, iSup_add] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s\u271d : Set \u211d\u22650\u221e \u03b9 : Sort u_4 s : \u03b9 \u2192 \u211d\u22650\u221e inst\u271d : Nonempty \u03b9 \u22a2 \u2a06 b, s b + a = \u2a06 b, a + s b ** simp [add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.trace_kroneckerMapBilinear ** R : Type u_1 \u03b1 : Type u_2 \u03b1' : Type u_3 \u03b2 : Type u_4 \u03b2' : Type u_5 \u03b3 : Type u_6 \u03b3' : Type u_7 l : Type u_8 m : Type u_9 n : Type u_10 p : Type u_11 q : Type u_12 r : Type u_13 l' : Type u_14 m' : Type u_15 n' : Type u_16 p' : Type u_17 inst\u271d\u2078 : CommSemiring R inst\u271d\u2077 : Fintype m inst\u271d\u2076 : Fintype n inst\u271d\u2075 : AddCommMonoid \u03b1 inst\u271d\u2074 : AddCommMonoid \u03b2 inst\u271d\u00b3 : AddCommMonoid \u03b3 inst\u271d\u00b2 : Module R \u03b1 inst\u271d\u00b9 : Module R \u03b2 inst\u271d : Module R \u03b3 f : \u03b1 \u2192\u2097[R] \u03b2 \u2192\u2097[R] \u03b3 A : Matrix m m \u03b1 B : Matrix n n \u03b2 \u22a2 trace (\u2191(\u2191(kroneckerMapBilinear f) A) B) = \u2191(\u2191f (trace A)) (trace B) ** simp_rw [Matrix.trace, Matrix.diag, kroneckerMapBilinear_apply_apply, LinearMap.map_sum\u2082,\n map_sum, \u2190 Finset.univ_product_univ, Finset.sum_product, kroneckerMap_apply] ** Qed", + "informal": "" + }, + { + "formal": "Convex.lipschitzOnWith_of_nnnorm_hasDerivWithin_le ** E : Type u_1 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \u211d E F : Type u_2 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \u211d F \ud835\udd5c : Type u_3 G : Type u_4 inst\u271d\u00b3 : IsROrC \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \ud835\udd5c G f f' : \ud835\udd5c \u2192 G s : Set \ud835\udd5c x\u271d y : \ud835\udd5c C : \u211d\u22650 hs : Convex \u211d s hf : \u2200 (x : \ud835\udd5c), x \u2208 s \u2192 HasDerivWithinAt f (f' x) s x bound : \u2200 (x : \ud835\udd5c), x \u2208 s \u2192 \u2016f' x\u2016\u208a \u2264 C x : \ud835\udd5c hx : x \u2208 s \u22a2 \u2016smulRight 1 (f' x)\u2016\u208a \u2264 \u2016f' x\u2016\u208a ** simp ** Qed", + "informal": "" + }, + { + "formal": "Filter.eventually_smallSets ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Sort u_3 l l' la : Filter \u03b1 lb : Filter \u03b2 p : Set \u03b1 \u2192 Prop \u22a2 (\u2200\u1da0 (s : Set \u03b1) in smallSets l, p s) \u2194 \u2203 s, s \u2208 l \u2227 \u2200 (t : Set \u03b1), t \u2286 s \u2192 p t ** rw [smallSets, eventually_lift'_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Sort u_3 l l' la : Filter \u03b1 lb : Filter \u03b2 p : Set \u03b1 \u2192 Prop \u22a2 (\u2203 t, t \u2208 l \u2227 \u2200 (y : Set \u03b1), y \u2208 \ud835\udcab t \u2192 p y) \u2194 \u2203 s, s \u2208 l \u2227 \u2200 (t : Set \u03b1), t \u2286 s \u2192 p t case hh \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Sort u_3 l l' la : Filter \u03b1 lb : Filter \u03b2 p : Set \u03b1 \u2192 Prop \u22a2 Monotone powerset ** rfl ** case hh \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Sort u_3 l l' la : Filter \u03b1 lb : Filter \u03b2 p : Set \u03b1 \u2192 Prop \u22a2 Monotone powerset ** exact monotone_powerset ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Subobject.inf_factors ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X\u271d Y\u271d Z : C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D inst\u271d : HasPullbacks C A B : C X Y : Subobject B f : A \u27f6 B \u22a2 Factors X f \u2227 Factors Y f \u2192 Factors (X \u2293 Y) f ** revert X Y ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X Y Z : C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D inst\u271d : HasPullbacks C A B : C f : A \u27f6 B \u22a2 \u2200 {X Y : Subobject B}, Factors X f \u2227 Factors Y f \u2192 Factors (X \u2293 Y) f ** apply Quotient.ind\u2082' ** case h C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X Y Z : C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D inst\u271d : HasPullbacks C A B : C f : A \u27f6 B \u22a2 \u2200 (a\u2081 a\u2082 : MonoOver B), Factors (Quotient.mk'' a\u2081) f \u2227 Factors (Quotient.mk'' a\u2082) f \u2192 Factors (Quotient.mk'' a\u2081 \u2293 Quotient.mk'' a\u2082) f ** rintro X Y \u27e8\u27e8g\u2081, rfl\u27e9, \u27e8g\u2082, hg\u2082\u27e9\u27e9 ** case h.intro.intro.intro C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X\u271d Y\u271d Z : C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D inst\u271d : HasPullbacks C A B : C X Y : MonoOver B g\u2081 : A \u27f6 X.obj.left g\u2082 : A \u27f6 Y.obj.left hg\u2082 : g\u2082 \u226b MonoOver.arrow Y = g\u2081 \u226b MonoOver.arrow X \u22a2 Factors (Quotient.mk'' X \u2293 Quotient.mk'' Y) (g\u2081 \u226b MonoOver.arrow X) ** exact \u27e8_, pullback.lift_snd_assoc _ _ hg\u2082 _\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Set.image_const_sub_Ioc ** \u03b1 : Type u_1 inst\u271d : OrderedAddCommGroup \u03b1 a b c : \u03b1 \u22a2 (fun x => a - x) '' Ioc b c = Ico (a - c) (a - b) ** have := image_comp (fun x => a + x) fun x => -x ** \u03b1 : Type u_1 inst\u271d : OrderedAddCommGroup \u03b1 a b c : \u03b1 this : \u2200 (a_1 : Set \u03b1), ((fun x => a + x) \u2218 fun x => -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1) \u22a2 (fun x => a - x) '' Ioc b c = Ico (a - c) (a - b) ** dsimp [Function.comp] at this ** \u03b1 : Type u_1 inst\u271d : OrderedAddCommGroup \u03b1 a b c : \u03b1 this : \u2200 (a_1 : Set \u03b1), (fun x => a + -x) '' a_1 = (fun x => a + x) '' ((fun x => -x) '' a_1) \u22a2 (fun x => a - x) '' Ioc b c = Ico (a - c) (a - b) ** simp [sub_eq_add_neg, this, add_comm] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.mul_dist_le_mul_dist_add_mul_dist ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** rcases eq_or_ne b a with (rfl | hb) ** case inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb : b \u2260 a \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** rcases eq_or_ne c a with (rfl | hc) ** case inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb : b \u2260 a hc : c \u2260 a \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** rcases eq_or_ne d a with (rfl | hd) ** case inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb : b \u2260 a hc : c \u2260 a hd : d \u2260 a \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** have H := dist_triangle (inversion a 1 b) (inversion a 1 c) (inversion a 1 d) ** case inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb : b \u2260 a hc : c \u2260 a hd : d \u2260 a H : dist (inversion a 1 b) (inversion a 1 d) \u2264 dist (inversion a 1 b) (inversion a 1 c) + dist (inversion a 1 c) (inversion a 1 d) \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** rw [dist_inversion_inversion hb hd, dist_inversion_inversion hb hc,\n dist_inversion_inversion hc hd, one_pow] at H ** case inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb : b \u2260 a hc : c \u2260 a hd : d \u2260 a H : 1 / (dist b a * dist d a) * dist b d \u2264 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** rw [\u2190 dist_pos] at hb hc hd ** case inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb\u271d : b \u2260 a hb : 0 < dist b a hc\u271d : c \u2260 a hc : 0 < dist c a hd\u271d : d \u2260 a hd : 0 < dist d a H : 1 / (dist b a * dist d a) * dist b d \u2264 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d \u22a2 dist a c * dist b d \u2264 dist a b * dist c d + dist b c * dist a d ** rw [\u2190 div_le_div_right (mul_pos hb (mul_pos hc hd))] ** case inr.inr.inr V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb\u271d : b \u2260 a hb : 0 < dist b a hc\u271d : c \u2260 a hc : 0 < dist c a hd\u271d : d \u2260 a hd : 0 < dist d a H : 1 / (dist b a * dist d a) * dist b d \u2264 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d \u22a2 dist a c * dist b d / (dist b a * (dist c a * dist d a)) \u2264 (dist a b * dist c d + dist b c * dist a d) / (dist b a * (dist c a * dist d a)) ** convert H using 1 <;> (field_simp [hb.ne', hc.ne', hd.ne', dist_comm a]; ring) ** case inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b\u271d c\u271d d\u271d x y z : P r R : \u211d b c d : P \u22a2 dist b c * dist b d \u2264 dist b b * dist c d + dist b c * dist b d ** rw [dist_self, zero_mul, zero_add] ** case inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b\u271d c\u271d d\u271d x y z : P r R : \u211d b c d : P hb : b \u2260 c \u22a2 dist c c * dist b d \u2264 dist c b * dist c d + dist b c * dist c d ** rw [dist_self, zero_mul] ** case inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b\u271d c\u271d d\u271d x y z : P r R : \u211d b c d : P hb : b \u2260 c \u22a2 0 \u2264 dist c b * dist c d + dist b c * dist c d ** apply_rules [add_nonneg, mul_nonneg, dist_nonneg] ** case inr.inr.inl V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a b\u271d c\u271d d\u271d x y z : P r R : \u211d b c d : P hb : b \u2260 d hc : c \u2260 d \u22a2 dist d c * dist b d \u2264 dist d b * dist c d + dist b c * dist d d ** rw [dist_self, mul_zero, add_zero, dist_comm d, dist_comm d, mul_comm] ** case h.e'_4 V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb\u271d : b \u2260 a hb : 0 < dist b a hc\u271d : c \u2260 a hc : 0 < dist c a hd\u271d : d \u2260 a hd : 0 < dist d a H : 1 / (dist b a * dist d a) * dist b d \u2264 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d \u22a2 (dist a b * dist c d + dist b c * dist a d) / (dist b a * (dist c a * dist d a)) = 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d ** field_simp [hb.ne', hc.ne', hd.ne', dist_comm a] ** case h.e'_4 V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P a\u271d b\u271d c\u271d d\u271d x y z : P r R : \u211d a b c d : P hb\u271d : b \u2260 a hb : 0 < dist b a hc\u271d : c \u2260 a hc : 0 < dist c a hd\u271d : d \u2260 a hd : 0 < dist d a H : 1 / (dist b a * dist d a) * dist b d \u2264 1 / (dist b a * dist c a) * dist b c + 1 / (dist c a * dist d a) * dist c d \u22a2 (dist b a * dist c d + dist b c * dist d a) * (dist b a * dist c a * (dist c a * dist d a)) = (dist b c * (dist c a * dist d a) + dist c d * (dist b a * dist c a)) * (dist b a * (dist c a * dist d a)) ** ring ** Qed", + "informal": "" + }, + { + "formal": "starConvex_iff_forall_pos ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y z : E s : Set E hx : x \u2208 s \u22a2 StarConvex \ud835\udd5c x s \u2194 \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s ** refine' \u27e8fun h y hy a b ha hb hab => h hy ha.le hb.le hab, _\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y z : E s : Set E hx : x \u2208 s \u22a2 (\u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s) \u2192 StarConvex \ud835\udd5c x s ** intro h y hy a b ha hb hab ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s a b : \ud835\udd5c ha : 0 \u2264 a hb : 0 \u2264 b hab : a + b = 1 \u22a2 a \u2022 x + b \u2022 y \u2208 s ** obtain rfl | ha := ha.eq_or_lt ** case inr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s a b : \ud835\udd5c ha\u271d : 0 \u2264 a hb : 0 \u2264 b hab : a + b = 1 ha : 0 < a \u22a2 a \u2022 x + b \u2022 y \u2208 s ** obtain rfl | hb := hb.eq_or_lt ** case inr.inr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s a b : \ud835\udd5c ha\u271d : 0 \u2264 a hb\u271d : 0 \u2264 b hab : a + b = 1 ha : 0 < a hb : 0 < b \u22a2 a \u2022 x + b \u2022 y \u2208 s ** exact h hy ha hb hab ** case inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s b : \ud835\udd5c hb : 0 \u2264 b ha : 0 \u2264 0 hab : 0 + b = 1 \u22a2 0 \u2022 x + b \u2022 y \u2208 s ** rw [zero_add] at hab ** case inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s b : \ud835\udd5c hb : 0 \u2264 b ha : 0 \u2264 0 hab : b = 1 \u22a2 0 \u2022 x + b \u2022 y \u2208 s ** rwa [hab, one_smul, zero_smul, zero_add] ** case inr.inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s a : \ud835\udd5c ha\u271d : 0 \u2264 a ha : 0 < a hb : 0 \u2264 0 hab : a + 0 = 1 \u22a2 a \u2022 x + 0 \u2022 y \u2208 s ** rw [add_zero] at hab ** case inr.inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y\u271d z : E s : Set E hx : x \u2208 s h : \u2200 \u2983y : E\u2984, y \u2208 s \u2192 \u2200 \u2983a b : \ud835\udd5c\u2984, 0 < a \u2192 0 < b \u2192 a + b = 1 \u2192 a \u2022 x + b \u2022 y \u2208 s y : E hy : y \u2208 s a : \ud835\udd5c ha\u271d : 0 \u2264 a ha : 0 < a hb : 0 \u2264 0 hab : a = 1 \u22a2 a \u2022 x + 0 \u2022 y \u2208 s ** rwa [hab, one_smul, zero_smul, add_zero] ** Qed", + "informal": "" + }, + { + "formal": "Set.Iio_subset_Iic_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1 inst\u271d : DenselyOrdered \u03b1 \u22a2 Iio a \u2286 Iic b \u2194 a \u2264 b ** rw [\u2190 diff_eq_empty, Iio_diff_Iic, Ioo_eq_empty_iff, not_lt] ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.mul_le_max_of_aleph0_le_right ** a b : Cardinal.{u_1} h : \u2135\u2080 \u2264 b \u22a2 a * b \u2264 max a b ** simpa only [mul_comm b, max_comm b] using mul_le_max_of_aleph0_le_left h ** Qed", + "informal": "" + }, + { + "formal": "Matrix.sub_mulVec ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 m' : o \u2192 Type u_5 n' : o \u2192 Type u_6 R : Type u_7 S : Type u_8 \u03b1 : Type v \u03b2 : Type w \u03b3 : Type u_9 inst\u271d\u00b9 : NonUnitalNonAssocRing \u03b1 inst\u271d : Fintype n A B : Matrix m n \u03b1 x : n \u2192 \u03b1 \u22a2 mulVec (A - B) x = mulVec A x - mulVec B x ** simp [sub_eq_add_neg, add_mulVec, neg_mulVec] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Presheaf.isSheaf_iff_isLimit_pretopology ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C A : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} A J : GrothendieckTopology C P : C\u1d52\u1d56 \u2964 A X : C S : Sieve X R : Presieve X E : A\u1d52\u1d56 x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows hx : SieveCompatible x inst\u271d : HasPullbacks C K : Pretopology C \u22a2 IsSheaf (Pretopology.toGrothendieck C K) P \u2194 \u2200 \u2983X : C\u2984 (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows)))) ** dsimp [IsSheaf] ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C A : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} A J : GrothendieckTopology C P : C\u1d52\u1d56 \u2964 A X : C S : Sieve X R : Presieve X E : A\u1d52\u1d56 x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows hx : SieveCompatible x inst\u271d : HasPullbacks C K : Pretopology C \u22a2 (\u2200 (E : A), Presieve.IsSheaf (Pretopology.toGrothendieck C K) (P \u22d9 coyoneda.obj (op E))) \u2194 \u2200 \u2983X : C\u2984 (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows)))) ** simp_rw [isSheaf_pretopology] ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C A : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} A J : GrothendieckTopology C P : C\u1d52\u1d56 \u2964 A X : C S : Sieve X R : Presieve X E : A\u1d52\u1d56 x : FamilyOfElements (P \u22d9 coyoneda.obj E) S.arrows hx : SieveCompatible x inst\u271d : HasPullbacks C K : Pretopology C \u22a2 (\u2200 (E : A) {X : C} (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 IsSheafFor (P \u22d9 coyoneda.obj (op E)) R) \u2194 \u2200 \u2983X : C\u2984 (R : Presieve X), R \u2208 Pretopology.coverings K X \u2192 Nonempty (IsLimit (P.mapCone (Cocone.op (cocone (generate R).arrows)))) ** exact\n \u27e8fun h X R hR => (isLimit_iff_isSheafFor_presieve P R).2 fun E => h E.unop R hR,\n fun h E X R hR => (isLimit_iff_isSheafFor_presieve P R).1 (h R hR) (op E)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Adapted.isStoppingTime_crossing ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f \u22a2 IsStoppingTime \u2131 (upperCrossingTime a b f N n) \u2227 IsStoppingTime \u2131 (lowerCrossingTime a b f N n) ** induction' n with k ih ** case zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f \u22a2 IsStoppingTime \u2131 (upperCrossingTime a b f N Nat.zero) \u2227 IsStoppingTime \u2131 (lowerCrossingTime a b f N Nat.zero) ** refine' \u27e8isStoppingTime_const _ 0, _\u27e9 ** case zero \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f \u22a2 IsStoppingTime \u2131 (lowerCrossingTime a b f N Nat.zero) ** simp [hitting_isStoppingTime hf measurableSet_Iic] ** case succ \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 ih : IsStoppingTime \u2131 (upperCrossingTime a b f N k) \u2227 IsStoppingTime \u2131 (lowerCrossingTime a b f N k) \u22a2 IsStoppingTime \u2131 (upperCrossingTime a b f N (Nat.succ k)) \u2227 IsStoppingTime \u2131 (lowerCrossingTime a b f N (Nat.succ k)) ** obtain \u27e8_, ih\u2082\u27e9 := ih ** case succ.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) \u22a2 IsStoppingTime \u2131 (upperCrossingTime a b f N (Nat.succ k)) \u2227 IsStoppingTime \u2131 (lowerCrossingTime a b f N (Nat.succ k)) ** have : IsStoppingTime \u2131 (upperCrossingTime a b f N (k + 1)) := by\n intro n\n simp_rw [upperCrossingTime_succ_eq]\n exact isStoppingTime_hitting_isStoppingTime ih\u2082 (fun _ => lowerCrossingTime_le)\n measurableSet_Ici hf _ ** case succ.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) this : IsStoppingTime \u2131 (upperCrossingTime a b f N (k + 1)) \u22a2 IsStoppingTime \u2131 (upperCrossingTime a b f N (Nat.succ k)) \u2227 IsStoppingTime \u2131 (lowerCrossingTime a b f N (Nat.succ k)) ** refine' \u27e8this, _\u27e9 ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) \u22a2 IsStoppingTime \u2131 (upperCrossingTime a b f N (k + 1)) ** intro n ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n\u271d m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) n : \u2115 \u22a2 MeasurableSet {\u03c9 | upperCrossingTime a b f N (k + 1) \u03c9 \u2264 n} ** simp_rw [upperCrossingTime_succ_eq] ** \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n\u271d m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) n : \u2115 \u22a2 MeasurableSet {\u03c9 | hitting f (Set.Ici b) (lowerCrossingTime a b f N k \u03c9) N \u03c9 \u2264 n} ** exact isStoppingTime_hitting_isStoppingTime ih\u2082 (fun _ => lowerCrossingTime_le)\n measurableSet_Ici hf _ ** case succ.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) this : IsStoppingTime \u2131 (upperCrossingTime a b f N (k + 1)) \u22a2 IsStoppingTime \u2131 (lowerCrossingTime a b f N (Nat.succ k)) ** intro n ** case succ.intro \u03a9 : Type u_1 \u03b9 : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 a b : \u211d f : \u2115 \u2192 \u03a9 \u2192 \u211d N n\u271d m : \u2115 \u03c9 : \u03a9 \u2131 : Filtration \u2115 m0 hf : Adapted \u2131 f k : \u2115 left\u271d : IsStoppingTime \u2131 (upperCrossingTime a b f N k) ih\u2082 : IsStoppingTime \u2131 (lowerCrossingTime a b f N k) this : IsStoppingTime \u2131 (upperCrossingTime a b f N (k + 1)) n : \u2115 \u22a2 MeasurableSet {\u03c9 | lowerCrossingTime a b f N (Nat.succ k) \u03c9 \u2264 n} ** exact isStoppingTime_hitting_isStoppingTime this (fun _ => upperCrossingTime_le)\n measurableSet_Iic hf _ ** Qed", + "informal": "" + }, + { + "formal": "Ordnode.Valid'.balanceR ** \u03b1 : Type u_1 inst\u271d : Preorder \u03b1 l : Ordnode \u03b1 x : \u03b1 r : Ordnode \u03b1 o\u2081 : WithBot \u03b1 o\u2082 : WithTop \u03b1 hl : Valid' o\u2081 l \u2191x hr : Valid' (\u2191x) r o\u2082 H : (\u2203 l', Raised (size l) l' \u2227 BalancedSz l' (size r)) \u2228 \u2203 r', Raised r' (size r) \u2227 BalancedSz (size l) r' \u22a2 Valid' o\u2081 (Ordnode.balanceR l x r) o\u2082 ** rw [Valid'.dual_iff, dual_balanceR] ** \u03b1 : Type u_1 inst\u271d : Preorder \u03b1 l : Ordnode \u03b1 x : \u03b1 r : Ordnode \u03b1 o\u2081 : WithBot \u03b1 o\u2082 : WithTop \u03b1 hl : Valid' o\u2081 l \u2191x hr : Valid' (\u2191x) r o\u2082 H : (\u2203 l', Raised (size l) l' \u2227 BalancedSz l' (size r)) \u2228 \u2203 r', Raised r' (size r) \u2227 BalancedSz (size l) r' \u22a2 Valid' o\u2082 (Ordnode.balanceL (Ordnode.dual r) x (Ordnode.dual l)) o\u2081 ** exact hr.dual.balanceL hl.dual (balance_sz_dual H) ** Qed", + "informal": "" + }, + { + "formal": "ModuleCat.Free.\u03bc_natural ** R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' \u22a2 ((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom = (\u03bc R X X').hom \u226b (free R).map (f \u2297 g) ** apply TensorProduct.ext ** case H R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' \u22a2 LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom) = LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g)) ** apply Finsupp.lhom_ext' ** case H.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' \u22a2 \u2200 (a : X), LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle a) = LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle a) ** intro x ** case H.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X \u22a2 LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x) = LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x) ** apply LinearMap.ext_ring ** case H.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X \u22a2 \u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x)) 1 = \u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x)) 1 ** apply Finsupp.lhom_ext' ** case H.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X \u22a2 \u2200 (a : X'), LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x)) 1) (Finsupp.lsingle a) = LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x)) 1) (Finsupp.lsingle a) ** intro x' ** case H.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X x' : X' \u22a2 LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x') = LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x') ** apply LinearMap.ext_ring ** case H.h.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X x' : X' \u22a2 \u2191(LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x')) 1 = \u2191(LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x')) 1 ** apply Finsupp.ext ** case H.h.h.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X x' : X' \u22a2 \u2200 (a : Y \u2297 Y'), \u2191(\u2191(LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x')) 1) a = \u2191(\u2191(LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x')) 1) a ** intro \u27e8y, y'\u27e9 ** case H.h.h.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X x' : X' y : Y y' : Y' \u22a2 \u2191(\u2191(LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) (((free R).map f \u2297 (free R).map g) \u226b (\u03bc R Y Y').hom)) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x')) 1) (y, y') = \u2191(\u2191(LinearMap.comp (\u2191(LinearMap.comp (LinearMap.compr\u2082 (TensorProduct.mk R \u2191((free R).obj X) \u2191((free R).obj X')) ((\u03bc R X X').hom \u226b (free R).map (f \u2297 g))) (Finsupp.lsingle x)) 1) (Finsupp.lsingle x')) 1) (y, y') ** change (finsuppTensorFinsupp' R Y Y')\n (Finsupp.mapDomain f (Finsupp.single x 1) \u2297\u209c[R] Finsupp.mapDomain g (Finsupp.single x' 1)) _\n = (Finsupp.mapDomain (f \u2297 g) (finsuppTensorFinsupp' R X X'\n (Finsupp.single x 1 \u2297\u209c[R] Finsupp.single x' 1))) _ ** case H.h.h.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X x' : X' y : Y y' : Y' \u22a2 \u2191(\u2191(finsuppTensorFinsupp' R Y Y') ((Finsupp.mapDomain f fun\u2080 | x => 1) \u2297\u209c[R] Finsupp.mapDomain g fun\u2080 | x' => 1)) (y, y') = \u2191(Finsupp.mapDomain (f \u2297 g) (\u2191(finsuppTensorFinsupp' R X X') ((fun\u2080 | x => 1) \u2297\u209c[R] fun\u2080 | x' => 1))) (y, y') ** simp_rw [Finsupp.mapDomain_single, finsuppTensorFinsupp'_single_tmul_single, mul_one,\n Finsupp.mapDomain_single, CategoryTheory.tensor_apply] ** case H.h.h.h.h.h R : Type u inst\u271d : CommRing R X Y X' Y' : Type u f : X \u27f6 Y g : X' \u27f6 Y' x : X x' : X' y : Y y' : Y' \u22a2 (\u2191fun\u2080 | (f x, g x') => 1) (y, y') = (\u2191fun\u2080 | (f x, g x') => 1) (y, y') ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Nat.count_le_card ** p : \u2115 \u2192 Prop inst\u271d : DecidablePred p hp : Set.Finite (setOf p) n : \u2115 \u22a2 count p n \u2264 card (Set.Finite.toFinset hp) ** rw [count_eq_card_filter_range] ** p : \u2115 \u2192 Prop inst\u271d : DecidablePred p hp : Set.Finite (setOf p) n : \u2115 \u22a2 card (filter p (range n)) \u2264 card (Set.Finite.toFinset hp) ** exact Finset.card_mono fun x hx \u21a6 hp.mem_toFinset.2 (mem_filter.1 hx).2 ** Qed", + "informal": "" + }, + { + "formal": "List.Func.get_set ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w a\u271d : \u03b1 as\u271d as1 as2 as3 : List \u03b1 inst\u271d\u00b9 : Inhabited \u03b1 inst\u271d : Inhabited \u03b2 a : \u03b1 as : List \u03b1 \u22a2 get 0 (as {0 \u21a6 a}) = a ** cases as <;> rfl ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w a\u271d : \u03b1 as\u271d as1 as2 as3 : List \u03b1 inst\u271d\u00b9 : Inhabited \u03b1 inst\u271d : Inhabited \u03b2 a : \u03b1 k : \u2115 as : List \u03b1 \u22a2 get (k + 1) (as {k + 1 \u21a6 a}) = a ** cases as <;> simp [get_set] ** Qed", + "informal": "" + }, + { + "formal": "Vector.tail_ofFn ** n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin (Nat.succ n) \u2192 \u03b1 \u22a2 ofFn (get (tail (ofFn f))) = ofFn fun i => f (Fin.succ i) ** congr ** case e_a n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin (Nat.succ n) \u2192 \u03b1 \u22a2 get (tail (ofFn f)) = fun i => f (Fin.succ i) ** funext i ** case e_a.h n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin (Nat.succ n) \u2192 \u03b1 i : Fin (Nat.succ n - 1) \u22a2 get (tail (ofFn f)) i = f (Fin.succ i) ** rw [get_tail, get_ofFn] ** case e_a.h n\u271d : \u2115 \u03b1 : Type u_1 n : \u2115 f : Fin (Nat.succ n) \u2192 \u03b1 i : Fin (Nat.succ n - 1) \u22a2 f { val := \u2191i + 1, isLt := (_ : \u2191i + 1 < Nat.succ n) } = f (Fin.succ i) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Bicategory.whiskerRight_id_symm ** B : Type u inst\u271d : Bicategory B a b c d e : B f g : a \u27f6 b \u03b7 : f \u27f6 g \u22a2 \u03b7 = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).hom ** simp ** Qed", + "informal": "" + }, + { + "formal": "Filter.EventuallyEq.derivWithin_eq ** \ud835\udd5c : Type u inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type w inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c hs : f\u2081 =\u1da0[\ud835\udcdd[s] x] f hx : f\u2081 x = f x \u22a2 derivWithin f\u2081 s x = derivWithin f s x ** unfold derivWithin ** \ud835\udd5c : Type u inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type w inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x : \ud835\udd5c s t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c hs : f\u2081 =\u1da0[\ud835\udcdd[s] x] f hx : f\u2081 x = f x \u22a2 \u2191(fderivWithin \ud835\udd5c f\u2081 s x) 1 = \u2191(fderivWithin \ud835\udd5c f s x) 1 ** rw [hs.fderivWithin_eq hx] ** Qed", + "informal": "" + }, + { + "formal": "Filter.pi_mem_pi ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 f f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i) s : (i : \u03b9) \u2192 Set (\u03b1 i) I : Set \u03b9 hI : Set.Finite I h : \u2200 (i : \u03b9), i \u2208 I \u2192 s i \u2208 f i \u22a2 Set.pi I s \u2208 pi f ** rw [pi_def, biInter_eq_iInter] ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 f f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i) s : (i : \u03b9) \u2192 Set (\u03b1 i) I : Set \u03b9 hI : Set.Finite I h : \u2200 (i : \u03b9), i \u2208 I \u2192 s i \u2208 f i \u22a2 \u22c2 x, eval \u2191x \u207b\u00b9' s \u2191x \u2208 pi f ** refine' mem_iInf_of_iInter hI (fun i => _) Subset.rfl ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 f f\u2081 f\u2082 : (i : \u03b9) \u2192 Filter (\u03b1 i) s : (i : \u03b9) \u2192 Set (\u03b1 i) I : Set \u03b9 hI : Set.Finite I h : \u2200 (i : \u03b9), i \u2208 I \u2192 s i \u2208 f i i : \u2191I \u22a2 eval \u2191i \u207b\u00b9' s \u2191i \u2208 comap (eval \u2191i) (f \u2191i) ** exact preimage_mem_comap (h i i.2) ** Qed", + "informal": "" + }, + { + "formal": "Ideal.FinrankQuotientMap.linearIndependent_of_nontrivial ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' hb' : LinearIndependent S (\u2191f' \u2218 b) \u22a2 LinearIndependent K (\u2191f \u2218 b) ** contrapose! hb' with hb ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' hb : \u00acLinearIndependent K (\u2191f \u2218 b) \u22a2 \u00acLinearIndependent S (\u2191f' \u2218 b) ** simp only [linearIndependent_iff', not_forall] at hb \u22a2 ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' hb : \u2203 x x_1 h x_2 x, \u00acx_1 x_2 = 0 \u22a2 \u2203 x x_1 h x_2 x, \u00acx_1 x_2 = 0 ** obtain \u27e8s, g, eq, j', hj's, hj'g\u27e9 := hb ** case intro.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 \u22a2 \u2203 x x_1 h x_2 x, \u00acx_1 x_2 = 0 ** use s ** case h R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** obtain \u27e8a, hag, j, hjs, hgI\u27e9 := Ideal.exist_integer_multiples_not_mem hRS s g hj's hj'g ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K hag : \u2200 (i : \u03b9), i \u2208 s \u2192 IsLocalization.IsInteger R (a * g i) j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** choose g'' hg'' using hag ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** letI := Classical.propDecidable ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** let g' i := if h : i \u2208 s then g'' i h else 0 ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** have hg' : \u2200 i \u2208 s, algebraMap _ _ (g' i) = a * g i := by\n intro i hi; exact (congr_arg _ (dif_pos hi)).trans (hg'' i hi) ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** have hgI : algebraMap R S (g' j) \u2260 0 := by\n simp only [FractionalIdeal.mem_coeIdeal, not_exists, not_and'] at hgI\n exact hgI _ (hg' j hjs) ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI\u271d : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2191(algebraMap R S) (g' j) \u2260 0 \u22a2 \u2203 x h x_1 x_2, \u00acx x_1 = 0 ** refine \u27e8fun i => algebraMap R S (g' i), ?_, j, hjs, hgI\u27e9 ** case h.intro.intro.intro.intro R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI\u271d : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2191(algebraMap R S) (g' j) \u2260 0 \u22a2 \u2211 i in s, (fun i => \u2191(algebraMap R S) (g' i)) i \u2022 (\u2191f' \u2218 b) i = 0 ** have eq : f (\u2211 i in s, g' i \u2022 b i) = 0 := by\n rw [map_sum, \u2190 smul_zero a, \u2190 eq, Finset.smul_sum]\n refine Finset.sum_congr rfl ?_\n intro i hi\n rw [LinearMap.map_smul, \u2190 IsScalarTower.algebraMap_smul K, hg' i hi, \u2190 smul_assoc,\n smul_eq_mul, Function.comp_apply] ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 \u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i ** intro i hi ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 i : \u03b9 hi : i \u2208 s \u22a2 \u2191(algebraMap R K) (g' i) = a * g i ** exact (congr_arg _ (dif_pos hi)).trans (hg'' i hi) ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i \u22a2 \u2191(algebraMap R S) (g' j) \u2260 0 ** simp only [FractionalIdeal.mem_coeIdeal, not_exists, not_and'] at hgI ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2200 (x : R), \u2191(algebraMap R K) x = a * g j \u2192 \u00acx \u2208 RingHom.ker (algebraMap R S) \u22a2 \u2191(algebraMap R S) (g' j) \u2260 0 ** exact hgI _ (hg' j hjs) ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI\u271d : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2191(algebraMap R S) (g' j) \u2260 0 \u22a2 \u2191f (\u2211 i in s, g' i \u2022 b i) = 0 ** rw [map_sum, \u2190 smul_zero a, \u2190 eq, Finset.smul_sum] ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI\u271d : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2191(algebraMap R S) (g' j) \u2260 0 \u22a2 \u2211 x in s, \u2191f (g' x \u2022 b x) = \u2211 x in s, a \u2022 g x \u2022 (\u2191f \u2218 b) x ** refine Finset.sum_congr rfl ?_ ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI\u271d : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2191(algebraMap R S) (g' j) \u2260 0 \u22a2 \u2200 (x : \u03b9), x \u2208 s \u2192 \u2191f (g' x \u2022 b x) = a \u2022 g x \u2022 (\u2191f \u2218 b) x ** intro i hi ** R : Type u inst\u271d\u00b9\u2079 : CommRing R S : Type v inst\u271d\u00b9\u2078 : CommRing S f\u271d : R \u2192+* S p : Ideal R P : Ideal S inst\u271d\u00b9\u2077 : Algebra R S K : Type u_1 inst\u271d\u00b9\u2076 : Field K inst\u271d\u00b9\u2075 : Algebra R K hRK : IsFractionRing R K L : Type u_2 inst\u271d\u00b9\u2074 : Field L inst\u271d\u00b9\u00b3 : Algebra S L inst\u271d\u00b9\u00b2 : IsFractionRing S L V : Type u_3 V' : Type u_4 V'' : Type u_5 inst\u271d\u00b9\u00b9 : AddCommGroup V inst\u271d\u00b9\u2070 : Module R V inst\u271d\u2079 : Module K V inst\u271d\u2078 : IsScalarTower R K V inst\u271d\u2077 : AddCommGroup V' inst\u271d\u2076 : Module R V' inst\u271d\u2075 : Module S V' inst\u271d\u2074 : IsScalarTower R S V' inst\u271d\u00b3 : AddCommGroup V'' inst\u271d\u00b2 : Module R V'' inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R hRS : RingHom.ker (algebraMap R S) \u2260 \u22a4 f : V'' \u2192\u2097[R] V hf : Function.Injective \u2191f f' : V'' \u2192\u2097[R] V' \u03b9 : Type u_6 b : \u03b9 \u2192 V'' s : Finset \u03b9 g : \u03b9 \u2192 K eq : \u2211 i in s, g i \u2022 (\u2191f \u2218 b) i = 0 j' : \u03b9 hj's : j' \u2208 s hj'g : \u00acg j' = 0 a : K j : \u03b9 hjs : j \u2208 s hgI\u271d : \u00aca * g j \u2208 \u2191(RingHom.ker (algebraMap R S)) g'' : (i : \u03b9) \u2192 i \u2208 s \u2192 R hg'' : \u2200 (i : \u03b9) (a_1 : i \u2208 s), \u2191(algebraMap R K) (g'' i a_1) = a * g i this : (a : Prop) \u2192 Decidable a := Classical.propDecidable g' : \u03b9 \u2192 R := fun i => if h : i \u2208 s then g'' i h else 0 hg' : \u2200 (i : \u03b9), i \u2208 s \u2192 \u2191(algebraMap R K) (g' i) = a * g i hgI : \u2191(algebraMap R S) (g' j) \u2260 0 i : \u03b9 hi : i \u2208 s \u22a2 \u2191f (g' i \u2022 b i) = a \u2022 g i \u2022 (\u2191f \u2218 b) i ** rw [LinearMap.map_smul, \u2190 IsScalarTower.algebraMap_smul K, hg' i hi, \u2190 smul_assoc,\n smul_eq_mul, Function.comp_apply] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.disjoint_singleton ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 l : Multiset \u03b1 a : \u03b1 \u22a2 Disjoint l {a} \u2194 \u00aca \u2208 l ** rw [disjoint_comm, singleton_disjoint] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.cons_vec_bit1_eq_alt1 ** \u03b1 : Type u m n o : \u2115 m' : Type u_1 n' : Type u_2 o' : Type u_3 x : \u03b1 u : Fin n \u2192 \u03b1 i : Fin (n + 1) \u22a2 vecCons x u (bit1 i) = vecAlt1 (_ : Nat.succ n + Nat.succ n = Nat.succ n + Nat.succ n) (vecAppend (_ : Nat.succ n + Nat.succ n = Nat.succ n + Nat.succ n) (vecCons x u) (vecCons x u)) i ** rw [vecAlt1_vecAppend] ** \u03b1 : Type u m n o : \u2115 m' : Type u_1 n' : Type u_2 o' : Type u_3 x : \u03b1 u : Fin n \u2192 \u03b1 i : Fin (n + 1) \u22a2 vecCons x u (bit1 i) = (vecCons x u \u2218 bit1) i ** rfl ** Qed", + "informal": "" + }, + { + "formal": "TensorPower.algebraMap\u2080_eq_smul_one ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M r : R \u22a2 \u2191algebraMap\u2080 r = r \u2022 GradedMonoid.GOne.one ** simp [algebraMap\u2080] ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M r : R \u22a2 r \u2022 \u2191(tprod R) isEmptyElim = r \u2022 GradedMonoid.GOne.one ** congr ** Qed", + "informal": "" + }, + { + "formal": "Orientation.inner_smul_rotation_pi_div_two_smul_left ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x : V r\u2081 r\u2082 : \u211d \u22a2 inner (r\u2081 \u2022 \u2191(rotation o \u2191(\u03c0 / 2)) x) (r\u2082 \u2022 x) = 0 ** rw [inner_smul_right, inner_smul_rotation_pi_div_two_left, mul_zero] ** Qed", + "informal": "" + }, + { + "formal": "Filter.hasBasis_iInf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) \u22a2 HasBasis (\u2a05 i, l i) (fun If => Set.Finite If.fst \u2227 \u2200 (i : \u2191If.fst), p (\u2191i) (Sigma.snd If i)) fun If => \u22c2 i, s (\u2191i) (Sigma.snd If i) ** refine' \u27e8fun t => \u27e8fun ht => _, _\u27e9\u27e9 ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t\u271d : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) t : Set \u03b1 ht : t \u2208 \u2a05 i, l i \u22a2 \u2203 i, (Set.Finite i.fst \u2227 \u2200 (i_1 : \u2191i.fst), p (\u2191i_1) (Sigma.snd i i_1)) \u2227 \u22c2 i_1, s (\u2191i_1) (Sigma.snd i i_1) \u2286 t ** rcases (hasBasis_iInf' hl).mem_iff.mp ht with \u27e8\u27e8I, f\u27e9, \u27e8hI, hf\u27e9, hsub\u27e9 ** case refine'_1.intro.mk.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t\u271d : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) t : Set \u03b1 ht : t \u2208 \u2a05 i, l i I : Set \u03b9 f : (i : \u03b9) \u2192 \u03b9' i hsub : \u22c2 i \u2208 (I, f).1, s i ((I, f).2 i) \u2286 t hI : Set.Finite (I, f).1 hf : \u2200 (i : \u03b9), i \u2208 (I, f).1 \u2192 p i ((I, f).2 i) \u22a2 \u2203 i, (Set.Finite i.fst \u2227 \u2200 (i_1 : \u2191i.fst), p (\u2191i_1) (Sigma.snd i i_1)) \u2227 \u22c2 i_1, s (\u2191i_1) (Sigma.snd i i_1) \u2286 t ** exact \u27e8\u27e8I, fun i => f i\u27e9, \u27e8hI, Subtype.forall.mpr hf\u27e9, trans (iInter_subtype _ _) hsub\u27e9 ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t\u271d : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) t : Set \u03b1 \u22a2 (\u2203 i, (Set.Finite i.fst \u2227 \u2200 (i_1 : \u2191i.fst), p (\u2191i_1) (Sigma.snd i i_1)) \u2227 \u22c2 i_1, s (\u2191i_1) (Sigma.snd i i_1) \u2286 t) \u2192 t \u2208 \u2a05 i, l i ** rintro \u27e8\u27e8I, f\u27e9, \u27e8hI, hf\u27e9, hsub\u27e9 ** case refine'_2.intro.mk.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t\u271d : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) t : Set \u03b1 I : Set \u03b9 f : (i : \u2191I) \u2192 \u03b9' \u2191i hsub : \u22c2 i, s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2286 t hI : Set.Finite { fst := I, snd := f }.fst hf : \u2200 (i : \u2191{ fst := I, snd := f }.fst), p (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u22a2 t \u2208 \u2a05 i, l i ** refine' mem_of_superset _ hsub ** case refine'_2.intro.mk.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t\u271d : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) t : Set \u03b1 I : Set \u03b9 f : (i : \u2191I) \u2192 \u03b9' \u2191i hsub : \u22c2 i, s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2286 t hI : Set.Finite { fst := I, snd := f }.fst hf : \u2200 (i : \u2191{ fst := I, snd := f }.fst), p (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u22a2 \u22c2 i, s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2208 \u2a05 i, l i ** cases hI.nonempty_fintype ** case refine'_2.intro.mk.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9\u271d : Sort u_4 \u03b9'\u271d : Sort u_5 l\u271d l' : Filter \u03b1 p\u271d : \u03b9\u271d \u2192 Prop s\u271d : \u03b9\u271d \u2192 Set \u03b1 t\u271d : Set \u03b1 i : \u03b9\u271d p' : \u03b9'\u271d \u2192 Prop s' : \u03b9'\u271d \u2192 Set \u03b1 i' : \u03b9'\u271d \u03b9 : Type u_6 \u03b9' : \u03b9 \u2192 Type u_7 l : \u03b9 \u2192 Filter \u03b1 p : (i : \u03b9) \u2192 \u03b9' i \u2192 Prop s : (i : \u03b9) \u2192 \u03b9' i \u2192 Set \u03b1 hl : \u2200 (i : \u03b9), HasBasis (l i) (p i) (s i) t : Set \u03b1 I : Set \u03b9 f : (i : \u2191I) \u2192 \u03b9' \u2191i hsub : \u22c2 i, s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2286 t hI : Set.Finite { fst := I, snd := f }.fst hf : \u2200 (i : \u2191{ fst := I, snd := f }.fst), p (\u2191i) (Sigma.snd { fst := I, snd := f } i) val\u271d : Fintype \u2191{ fst := I, snd := f }.fst \u22a2 \u22c2 i, s (\u2191i) (Sigma.snd { fst := I, snd := f } i) \u2208 \u2a05 i, l i ** exact iInter_mem.2 fun i => mem_iInf_of_mem \u2191i <| (hl i).mem_of_mem <| hf _ ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Endofunctor.Adjunction.Coalgebra.homEquiv_naturality_str_symm ** C : Type u inst\u271d : Category.{v, u} C F G : C \u2964 C adj : F \u22a3 G V\u2081 V\u2082 : Coalgebra G f : V\u2081 \u27f6 V\u2082 \u22a2 F.map f.f \u226b \u2191(Adjunction.homEquiv adj V\u2082.V V\u2082.V).symm V\u2082.str = \u2191(Adjunction.homEquiv adj V\u2081.V V\u2081.V).symm V\u2081.str \u226b f.f ** rw [\u2190 Adjunction.homEquiv_naturality_left_symm, \u2190 Adjunction.homEquiv_naturality_right_symm,\n f.h] ** Qed", + "informal": "" + }, + { + "formal": "Asymptotics.isLittleO_pow_sub_sub ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 G : Type u_5 E' : Type u_6 F' : Type u_7 G' : Type u_8 E'' : Type u_9 F'' : Type u_10 G'' : Type u_11 E''' : Type u_12 R : Type u_13 R' : Type u_14 \ud835\udd5c : Type u_15 \ud835\udd5c' : Type u_16 inst\u271d\u00b9\u00b3 : Norm E inst\u271d\u00b9\u00b2 : Norm F inst\u271d\u00b9\u00b9 : Norm G inst\u271d\u00b9\u2070 : SeminormedAddCommGroup E' inst\u271d\u2079 : SeminormedAddCommGroup F' inst\u271d\u2078 : SeminormedAddCommGroup G' inst\u271d\u2077 : NormedAddCommGroup E'' inst\u271d\u2076 : NormedAddCommGroup F'' inst\u271d\u2075 : NormedAddCommGroup G'' inst\u271d\u2074 : SeminormedRing R inst\u271d\u00b3 : SeminormedAddGroup E''' inst\u271d\u00b2 : SeminormedRing R' inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedField \ud835\udd5c' c c' c\u2081 c\u2082 : \u211d f : \u03b1 \u2192 E g : \u03b1 \u2192 F k : \u03b1 \u2192 G f' : \u03b1 \u2192 E' g' : \u03b1 \u2192 F' k' : \u03b1 \u2192 G' f'' : \u03b1 \u2192 E'' g'' : \u03b1 \u2192 F'' k'' : \u03b1 \u2192 G'' l l' : Filter \u03b1 x\u2080 : E' m : \u2115 h : 1 < m \u22a2 (fun x => \u2016x - x\u2080\u2016 ^ m) =o[\ud835\udcdd x\u2080] fun x => x - x\u2080 ** simpa only [isLittleO_norm_right, pow_one] using isLittleO_pow_sub_pow_sub x\u2080 h ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.exists_map_addHaar_eq_smul_addHaar ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L \u22a2 \u2203 c, 0 < c \u2227 map (\u2191L) \u03bc = c \u2022 \u03bd ** rcases L.exists_map_addHaar_eq_smul_addHaar' \u03bc \u03bd h with \u27e8c, c_pos, -, hc\u27e9 ** case intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L c : \u211d\u22650\u221e c_pos : 0 < c hc : map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd \u22a2 \u2203 c, 0 < c \u2227 map (\u2191L) \u03bc = c \u2022 \u03bd ** exact \u27e8_, by simp [c_pos, NeZero.ne addHaar], hc\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : CompleteSpace \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup E inst\u271d\u2079 : MeasurableSpace E inst\u271d\u2078 : BorelSpace E inst\u271d\u2077 : NormedSpace \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : MeasurableSpace F inst\u271d\u2074 : BorelSpace F inst\u271d\u00b3 : NormedSpace \ud835\udd5c F L : E \u2192\u2097[\ud835\udd5c] F \u03bc : Measure E \u03bd : Measure F inst\u271d\u00b2 : IsAddHaarMeasure \u03bc inst\u271d\u00b9 : IsAddHaarMeasure \u03bd inst\u271d : LocallyCompactSpace E h : Function.Surjective \u2191L c : \u211d\u22650\u221e c_pos : 0 < c hc : map (\u2191L) \u03bc = (c * \u2191\u2191addHaar univ) \u2022 \u03bd \u22a2 0 < c * \u2191\u2191addHaar univ ** simp [c_pos, NeZero.ne addHaar] ** Qed", + "informal": "" + }, + { + "formal": "Real.sinh_nonneg_iff ** x y z : \u211d \u22a2 0 \u2264 sinh x \u2194 0 \u2264 x ** simpa only [sinh_zero] using @sinh_le_sinh 0 x ** Qed", + "informal": "" + }, + { + "formal": "Turing.ToPartrec.Code.exists_code.comp ** m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v hg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) ** rsuffices \u27e8cg, hg\u27e9 :\n \u2203 c : Code, \u2200 v : Vector \u2115 m, c.eval v.1 = Subtype.val <$> Vector.mOfFn fun i => g i v ** m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v hg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v ** clear hf f ** m n : \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v ** induction' n with n IH ** case intro m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hf : \u2203 c, \u2200 (v : Vector \u2115 n), eval c \u2191v = pure <$> f v hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v cg : Code hg : \u2200 (v : Vector \u2115 m), eval cg \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) ** obtain \u27e8cf, hf\u27e9 := hf ** case intro.intro m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v cg : Code hg : \u2200 (v : Vector \u2115 m), eval cg \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> f v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) ** exact\n \u27e8cf.comp cg, fun v => by\n simp [hg, hf, map_bind, seq_bind_eq, Function.comp]\n rfl\u27e9 ** m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v cg : Code hg : \u2200 (v : Vector \u2115 m), eval cg \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> f v v : Vector \u2115 m \u22a2 eval (Code.comp cf cg) \u2191v = pure <$> ((Vector.mOfFn fun i => g i v) >>= f) ** simp [hg, hf, map_bind, seq_bind_eq, Function.comp] ** m n : \u2115 f : Vector \u2115 n \u2192. \u2115 g : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v cg : Code hg : \u2200 (v : Vector \u2115 m), eval cg \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v cf : Code hf : \u2200 (v : Vector \u2115 n), eval cf \u2191v = pure <$> f v v : Vector \u2115 m \u22a2 (do let x \u2190 Vector.mOfFn fun i => g i v pure <$> f x) = do let a \u2190 Vector.mOfFn fun i => g i v pure <$> f a ** rfl ** case zero m n : \u2115 g\u271d : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g\u271d i v g : Fin Nat.zero \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin Nat.zero), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v ** exact \u27e8nil, fun v => by simp [Vector.mOfFn, Bind.bind]; rfl\u27e9 ** m n : \u2115 g\u271d : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g\u271d i v g : Fin Nat.zero \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin Nat.zero), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v v : Vector \u2115 m \u22a2 eval nil \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v ** simp [Vector.mOfFn, Bind.bind] ** m n : \u2115 g\u271d : Fin n \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g\u271d i v g : Fin Nat.zero \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin Nat.zero), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v v : Vector \u2115 m \u22a2 Part.some [] = Subtype.val <$> Part.some Vector.nil ** rfl ** case succ m n\u271d : \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g\u271d i v n : \u2115 IH : \u2200 {g : Fin n \u2192 Vector \u2115 m \u2192. \u2115}, (\u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v) \u2192 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin (Nat.succ n)), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v ** obtain \u27e8cg, hg\u2081\u27e9 := hg 0 ** case succ.intro m n\u271d : \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g\u271d i v n : \u2115 IH : \u2200 {g : Fin n \u2192 Vector \u2115 m \u2192. \u2115}, (\u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v) \u2192 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin (Nat.succ n)), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v cg : Code hg\u2081 : \u2200 (v : Vector \u2115 m), eval cg \u2191v = pure <$> g 0 v \u22a2 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v ** obtain \u27e8cl, hl\u27e9 := IH fun i => hg i.succ ** m n\u271d : \u2115 g\u271d : Fin n\u271d \u2192 Vector \u2115 m \u2192. \u2115 hg\u271d : \u2200 (i : Fin n\u271d), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g\u271d i v n : \u2115 IH : \u2200 {g : Fin n \u2192 Vector \u2115 m \u2192. \u2115}, (\u2200 (i : Fin n), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v) \u2192 \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = Subtype.val <$> Vector.mOfFn fun i => g i v g : Fin (Nat.succ n) \u2192 Vector \u2115 m \u2192. \u2115 hg : \u2200 (i : Fin (Nat.succ n)), \u2203 c, \u2200 (v : Vector \u2115 m), eval c \u2191v = pure <$> g i v cg : Code hg\u2081 : \u2200 (v : Vector \u2115 m), eval cg \u2191v = pure <$> g 0 v cl : Code hl : \u2200 (v : Vector \u2115 m), eval cl \u2191v = Subtype.val <$> Vector.mOfFn fun i => g (Fin.succ i) v v : Vector \u2115 m \u22a2 (do let x \u2190 g 0 v let x_1 \u2190 Vector.mOfFn fun i => g (Fin.succ i) v Part.some (List.headI (pure x) :: \u2191x_1)) = do let a \u2190 g 0 v let a_1 \u2190 Vector.mOfFn fun i => g (Fin.succ i) v Subtype.val <$> Part.some (a ::\u1d65 a_1) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Prod.swap_iInf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d\u00b9 : InfSet \u03b1 inst\u271d : InfSet \u03b2 f : \u03b9 \u2192 \u03b1 \u00d7 \u03b2 \u22a2 swap (iInf f) = \u2a05 i, swap (f i) ** simp_rw [iInf, swap_sInf, \u2190range_comp, Function.comp] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.OuterMeasure.boundedBy_eq_ofFunction ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 \u22a2 \u2191(boundedBy m) s = \u2191(OuterMeasure.ofFunction m m_empty) s ** have : (fun s : Set \u03b1 => \u2a06 _ : s.Nonempty, m s) = m := by\n ext1 t\n cases' t.eq_empty_or_nonempty with h h <;> simp [h, Set.not_nonempty_empty, m_empty] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 this : (fun s => \u2a06 (_ : Set.Nonempty s), m s) = m \u22a2 \u2191(boundedBy m) s = \u2191(OuterMeasure.ofFunction m m_empty) s ** simp [boundedBy, this] ** \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s : Set \u03b1 \u22a2 (fun s => \u2a06 (_ : Set.Nonempty s), m s) = m ** ext1 t ** case h \u03b1 : Type u_1 m : Set \u03b1 \u2192 \u211d\u22650\u221e m_empty : m \u2205 = 0 s t : Set \u03b1 \u22a2 \u2a06 (_ : Set.Nonempty t), m t = m t ** cases' t.eq_empty_or_nonempty with h h <;> simp [h, Set.not_nonempty_empty, m_empty] ** Qed", + "informal": "" + }, + { + "formal": "IsFractionRing.isAlgebraic_iff' ** R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K \u22a2 Algebra.IsAlgebraic R S \u2194 Algebra.IsAlgebraic R K ** simp only [Algebra.IsAlgebraic] ** R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K \u22a2 (\u2200 (x : S), IsAlgebraic R x) \u2194 \u2200 (x : K), IsAlgebraic R x ** constructor ** case mp R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K \u22a2 (\u2200 (x : S), IsAlgebraic R x) \u2192 \u2200 (x : K), IsAlgebraic R x ** intro h x ** case mp R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x x : K \u22a2 IsAlgebraic R x ** letI := FractionRing.liftAlgebra R K ** case mp R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x x : K this : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K \u22a2 IsAlgebraic R x ** have := FractionRing.isScalarTower_liftAlgebra R K ** case mp R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x x : K this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K \u22a2 IsAlgebraic R x ** rw [IsFractionRing.isAlgebraic_iff R (FractionRing R) K, isAlgebraic_iff_isIntegral] ** case mp R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x x : K this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K \u22a2 IsIntegral (FractionRing R) x ** obtain \u27e8a : S, b, ha, rfl\u27e9 := @div_surjective S _ _ _ _ _ _ x ** case mp.intro.intro.intro R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S \u22a2 IsIntegral (FractionRing R) (\u2191(algebraMap S K) a / \u2191(algebraMap S K) b) ** obtain \u27e8f, hf\u2081, hf\u2082\u27e9 := h b ** case mp.intro.intro.intro.intro.intro R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsIntegral (FractionRing R) (\u2191(algebraMap S K) a / \u2191(algebraMap S K) b) ** rw [div_eq_mul_inv] ** case mp.intro.intro.intro.intro.intro R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsIntegral (FractionRing R) (\u2191(algebraMap S K) a * (\u2191(algebraMap S K) b)\u207b\u00b9) ** refine' isIntegral_mul _ _ ** case mp.intro.intro.intro.intro.intro.refine'_1 R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsIntegral (FractionRing R) (\u2191(algebraMap S K) a) ** rw [\u2190 isAlgebraic_iff_isIntegral] ** case mp.intro.intro.intro.intro.intro.refine'_1 R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsAlgebraic (FractionRing R) (\u2191(algebraMap S K) a) ** refine'\n _root_.isAlgebraic_of_larger_base_of_injective\n (NoZeroSMulDivisors.algebraMap_injective R (FractionRing R)) _ ** case mp.intro.intro.intro.intro.intro.refine'_1 R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsAlgebraic R (\u2191(algebraMap S K) a) ** exact isAlgebraic_algebraMap_of_isAlgebraic (h a) ** case mp.intro.intro.intro.intro.intro.refine'_2 R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsIntegral (FractionRing R) (\u2191(algebraMap S K) b)\u207b\u00b9 ** rw [\u2190 isAlgebraic_iff_isIntegral] ** case mp.intro.intro.intro.intro.intro.refine'_2 R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 IsAlgebraic (FractionRing R) (\u2191(algebraMap S K) b)\u207b\u00b9 ** use (f.map (algebraMap R (FractionRing R))).reverse ** case h R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 reverse (Polynomial.map (algebraMap R (FractionRing R)) f) \u2260 0 \u2227 \u2191(aeval (\u2191(algebraMap S K) b)\u207b\u00b9) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0 ** constructor ** case h.left R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 reverse (Polynomial.map (algebraMap R (FractionRing R)) f) \u2260 0 ** rwa [Ne.def, Polynomial.reverse_eq_zero, \u2190 Polynomial.degree_eq_bot,\n Polynomial.degree_map_eq_of_injective\n (NoZeroSMulDivisors.algebraMap_injective R (FractionRing R)),\n Polynomial.degree_eq_bot] ** case h.right R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 \u22a2 \u2191(aeval (\u2191(algebraMap S K) b)\u207b\u00b9) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0 ** have : Invertible (algebraMap S K b) :=\n IsUnit.invertible\n (isUnit_of_mem_nonZeroDivisors\n (mem_nonZeroDivisors_iff_ne_zero.2 fun h =>\n nonZeroDivisors.ne_zero ha\n ((injective_iff_map_eq_zero (algebraMap S K)).1\n (NoZeroSMulDivisors.algebraMap_injective _ _) b h))) ** case h.right R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : S), IsAlgebraic R x this\u271d\u00b9 : Algebra (FractionRing R) K := FractionRing.liftAlgebra R K this\u271d : IsScalarTower R (FractionRing R) K a b : S ha : b \u2208 nonZeroDivisors S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval b) f = 0 this : Invertible (\u2191(algebraMap S K) b) \u22a2 \u2191(aeval (\u2191(algebraMap S K) b)\u207b\u00b9) (reverse (Polynomial.map (algebraMap R (FractionRing R)) f)) = 0 ** rw [Polynomial.aeval_def, \u2190 invOf_eq_inv, Polynomial.eval\u2082_reverse_eq_zero_iff,\n Polynomial.eval\u2082_map, \u2190 IsScalarTower.algebraMap_eq, \u2190 Polynomial.aeval_def,\n Polynomial.aeval_algebraMap_apply, hf\u2082, RingHom.map_zero] ** case mpr R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K \u22a2 (\u2200 (x : K), IsAlgebraic R x) \u2192 \u2200 (x : S), IsAlgebraic R x ** intro h x ** case mpr R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : K), IsAlgebraic R x x : S \u22a2 IsAlgebraic R x ** obtain \u27e8f, hf\u2081, hf\u2082\u27e9 := h (algebraMap S K x) ** case mpr.intro.intro R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : K), IsAlgebraic R x x : S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval (\u2191(algebraMap S K) x)) f = 0 \u22a2 IsAlgebraic R x ** use f, hf\u2081 ** case right R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : K), IsAlgebraic R x x : S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(aeval (\u2191(algebraMap S K) x)) f = 0 \u22a2 \u2191(aeval x) f = 0 ** rw [Polynomial.aeval_algebraMap_apply] at hf\u2082 ** case right R : Type u_1 inst\u271d\u00b9\u00b3 : CommRing R M : Submonoid R S : Type u_2 inst\u271d\u00b9\u00b2 : CommRing S inst\u271d\u00b9\u00b9 : Algebra R S P : Type u_3 inst\u271d\u00b9\u2070 : CommRing P A : Type u_4 K : Type u_5 inst\u271d\u2079 : CommRing A inst\u271d\u2078 : IsDomain A inst\u271d\u2077 : Field K inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsDomain S inst\u271d\u2074 : Algebra R K inst\u271d\u00b3 : Algebra S K inst\u271d\u00b2 : NoZeroSMulDivisors R K inst\u271d\u00b9 : IsFractionRing S K inst\u271d : IsScalarTower R S K h : \u2200 (x : K), IsAlgebraic R x x : S f : R[X] hf\u2081 : f \u2260 0 hf\u2082 : \u2191(algebraMap S K) (\u2191(aeval x) f) = 0 \u22a2 \u2191(aeval x) f = 0 ** exact\n (injective_iff_map_eq_zero (algebraMap S K)).1 (NoZeroSMulDivisors.algebraMap_injective _ _) _\n hf\u2082 ** Qed", + "informal": "" + }, + { + "formal": "List.cyclicPermutations_eq_nil_iff ** \u03b1 : Type u l\u271d l' l : List \u03b1 \u22a2 cyclicPermutations l = [[]] \u2194 l = [] ** refine' \u27e8fun h => _, fun h => by simp [h]\u27e9 ** \u03b1 : Type u l\u271d l' l : List \u03b1 h : cyclicPermutations l = [[]] \u22a2 l = [] ** rw [eq_comm, \u2190 isRotated_nil_iff', \u2190 mem_cyclicPermutations_iff, h, mem_singleton] ** \u03b1 : Type u l\u271d l' l : List \u03b1 h : l = [] \u22a2 cyclicPermutations l = [[]] ** simp [h] ** Qed", + "informal": "" + }, + { + "formal": "normalizerCondition_of_isNilpotent ** G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G inst\u271d : Normal H h : Group.IsNilpotent G \u22a2 NormalizerCondition G ** rw [normalizerCondition_iff_only_full_group_self_normalizing] ** G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G inst\u271d : Normal H h : Group.IsNilpotent G \u22a2 \u2200 (H : Subgroup G), normalizer H = H \u2192 H = \u22a4 ** apply @nilpotent_center_quotient_ind _ G _ _ <;> clear! G ** case hbase \u22a2 \u2200 (G : Type u_1) [inst : Group G] [inst_1 : Subsingleton G] (H : Subgroup G), normalizer H = H \u2192 H = \u22a4 ** intro G _ _ H _ ** case hbase G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Subsingleton G H : Subgroup G a\u271d : normalizer H = H \u22a2 H = \u22a4 ** exact @Subsingleton.elim _ Unique.instSubsingleton _ _ ** case hstep \u22a2 \u2200 (G : Type u_1) [inst : Group G] [inst_1 : Group.IsNilpotent G], (\u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4) \u2192 \u2200 (H : Subgroup G), normalizer H = H \u2192 H = \u22a4 ** intro G _ _ ih H hH ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H \u22a2 H = \u22a4 ** have hch : center G \u2264 H := Subgroup.center_le_normalizer.trans (le_of_eq hH) ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H \u22a2 H = \u22a4 ** have hkh : (mk' (center G)).ker \u2264 H := by simpa using hch ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H \u22a2 H = \u22a4 ** have hsur : Function.Surjective (mk' (center G)) := surjective_quot_mk _ ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) \u22a2 H = \u22a4 ** let H' := H.map (mk' (center G)) ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) H' : Subgroup (G \u29f8 center G) := Subgroup.map (mk' (center G)) H \u22a2 H = \u22a4 ** have hH' : H'.normalizer = H' := by\n apply comap_injective hsur\n rw [comap_normalizer_eq_of_surjective _ hsur, comap_map_eq_self hkh]\n exact hH ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) H' : Subgroup (G \u29f8 center G) := Subgroup.map (mk' (center G)) H hH' : normalizer H' = H' \u22a2 H = \u22a4 ** apply map_injective_of_ker_le (mk' (center G)) hkh le_top ** case hstep G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) H' : Subgroup (G \u29f8 center G) := Subgroup.map (mk' (center G)) H hH' : normalizer H' = H' \u22a2 Subgroup.map (mk' (center G)) H = Subgroup.map (mk' (center G)) \u22a4 ** exact (ih H' hH').trans (symm (map_top_of_surjective _ hsur)) ** G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H \u22a2 MonoidHom.ker (mk' (center G)) \u2264 H ** simpa using hch ** G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) H' : Subgroup (G \u29f8 center G) := Subgroup.map (mk' (center G)) H \u22a2 normalizer H' = H' ** apply comap_injective hsur ** case a G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) H' : Subgroup (G \u29f8 center G) := Subgroup.map (mk' (center G)) H \u22a2 comap (mk' (center G)) (normalizer H') = comap (mk' (center G)) H' ** rw [comap_normalizer_eq_of_surjective _ hsur, comap_map_eq_self hkh] ** case a G : Type u_1 inst\u271d\u00b9 : Group G inst\u271d : Group.IsNilpotent G ih : \u2200 (H : Subgroup (G \u29f8 center G)), normalizer H = H \u2192 H = \u22a4 H : Subgroup G hH : normalizer H = H hch : center G \u2264 H hkh : MonoidHom.ker (mk' (center G)) \u2264 H hsur : Function.Surjective \u2191(mk' (center G)) H' : Subgroup (G \u29f8 center G) := Subgroup.map (mk' (center G)) H \u22a2 normalizer H = H ** exact hH ** Qed", + "informal": "" + }, + { + "formal": "IsUpperSet.mul_left ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 s t : Set \u03b1 a : \u03b1 ht : IsUpperSet t \u22a2 IsUpperSet (s * t) ** rw [\u2190 smul_eq_mul, \u2190 Set.iUnion_smul_set] ** \u03b1 : Type u_1 inst\u271d : OrderedCommGroup \u03b1 s t : Set \u03b1 a : \u03b1 ht : IsUpperSet t \u22a2 IsUpperSet (\u22c3 a \u2208 s, a \u2022 t) ** exact isUpperSet_iUnion\u2082 fun x _ \u21a6 ht.smul ** Qed", + "informal": "" + }, + { + "formal": "List.takeWhile_idem ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Bool l : List \u03b1 \u22a2 takeWhile p (takeWhile p l) = takeWhile p l ** simp_rw [takeWhile_takeWhile, and_self_iff, Bool.decide_coe] ** Qed", + "informal": "" + }, + { + "formal": "Finset.subset_smul_finset_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b2 inst\u271d\u00b9 : Group \u03b1 inst\u271d : MulAction \u03b1 \u03b2 s t : Finset \u03b2 a : \u03b1 b : \u03b2 \u22a2 s \u2286 a \u2022 t \u2194 a\u207b\u00b9 \u2022 s \u2286 t ** simp_rw [\u2190 coe_subset] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b2 inst\u271d\u00b9 : Group \u03b1 inst\u271d : MulAction \u03b1 \u03b2 s t : Finset \u03b2 a : \u03b1 b : \u03b2 \u22a2 \u2191s \u2286 \u2191(a \u2022 t) \u2194 \u2191(a\u207b\u00b9 \u2022 s) \u2286 \u2191t ** push_cast ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b2 inst\u271d\u00b9 : Group \u03b1 inst\u271d : MulAction \u03b1 \u03b2 s t : Finset \u03b2 a : \u03b1 b : \u03b2 \u22a2 \u2191s \u2286 a \u2022 \u2191t \u2194 a\u207b\u00b9 \u2022 \u2191s \u2286 \u2191t ** exact Set.subset_set_smul_iff ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_le_of_roots_le ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B \u22a2 \u2016coeff (map f p) i\u2016 \u2264 B ^ (natDegree p - i) * \u2191(Nat.choose (natDegree p) i) ** obtain hB | hB := lt_or_le B 0 ** case inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B \u22a2 \u2016coeff (map f p) i\u2016 \u2264 B ^ (natDegree p - i) * \u2191(Nat.choose (natDegree p) i) ** rw [\u2190 h1.natDegree_map f] ** case inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B \u22a2 \u2016coeff (map f p) i\u2016 \u2264 B ^ (natDegree (map f p) - i) * \u2191(Nat.choose (natDegree (map f p)) i) ** obtain hi | hi := lt_or_le (map f p).natDegree i ** case inr.inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) \u22a2 \u2016coeff (map f p) i\u2016 \u2264 B ^ (natDegree (map f p) - i) * \u2191(Nat.choose (natDegree (map f p)) i) ** rw [coeff_eq_esymm_roots_of_splits ((splits_id_iff_splits f).2 h2) hi, (h1.map _).leadingCoeff,\n one_mul, norm_mul, norm_pow, norm_neg, norm_one, one_pow, one_mul] ** case inr.inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) \u22a2 \u2016esymm (roots (map f p)) (natDegree (map f p) - i)\u2016 \u2264 B ^ (natDegree (map f p) - i) * \u2191(Nat.choose (natDegree (map f p)) i) ** apply ((norm_multiset_sum_le _).trans <| sum_le_card_nsmul _ _ fun r hr => _).trans ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) \u22a2 \u2200 (r : \u211d), r \u2208 Multiset.map (fun x => \u2016x\u2016) (Multiset.map prod (powersetCard (natDegree (map f p) - i) (roots (map f p)))) \u2192 r \u2264 B ^ (natDegree (map f p) - i) ** intro r hr ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) r : \u211d hr : r \u2208 Multiset.map (fun x => \u2016x\u2016) (Multiset.map prod (powersetCard (natDegree (map f p) - i) (roots (map f p)))) \u22a2 r \u2264 B ^ (natDegree (map f p) - i) ** simp_rw [Multiset.mem_map] at hr ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) r : \u211d hr : \u2203 a, (\u2203 a_1, a_1 \u2208 powersetCard (natDegree (map f p) - i) (roots (map f p)) \u2227 prod a_1 = a) \u2227 \u2016a\u2016 = r \u22a2 r \u2264 B ^ (natDegree (map f p) - i) ** obtain \u27e8_, \u27e8s, hs, rfl\u27e9, rfl\u27e9 := hr ** case intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2208 powersetCard (natDegree (map f p) - i) (roots (map f p)) \u22a2 \u2016prod s\u2016 \u2264 B ^ (natDegree (map f p) - i) ** rw [mem_powersetCard] at hs ** case intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2264 roots (map f p) \u2227 \u2191card s = natDegree (map f p) - i \u22a2 \u2016prod s\u2016 \u2264 B ^ (natDegree (map f p) - i) ** lift B to \u211d\u22650 using hB ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K i : \u2115 h1 : Monic p h2 : Splits f p hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2264 roots (map f p) \u2227 \u2191card s = natDegree (map f p) - i B : \u211d\u22650 h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 \u2191B \u22a2 \u2016prod s\u2016 \u2264 \u2191B ^ (natDegree (map f p) - i) ** rw [\u2190 coe_nnnorm, \u2190 NNReal.coe_pow, NNReal.coe_le_coe, \u2190 nnnormHom_apply, \u2190 MonoidHom.coe_coe,\n MonoidHom.map_multiset_prod] ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K i : \u2115 h1 : Monic p h2 : Splits f p hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2264 roots (map f p) \u2227 \u2191card s = natDegree (map f p) - i B : \u211d\u22650 h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 \u2191B \u22a2 prod (Multiset.map (\u2191\u2191nnnormHom) s) \u2264 B ^ (natDegree (map f p) - i) ** refine' (prod_le_pow_card _ B fun x hx => _).trans_eq (by rw [card_map, hs.2]) ** case intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K i : \u2115 h1 : Monic p h2 : Splits f p hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2264 roots (map f p) \u2227 \u2191card s = natDegree (map f p) - i B : \u211d\u22650 h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 \u2191B x : \u211d\u22650 hx : x \u2208 Multiset.map (\u2191\u2191nnnormHom) s \u22a2 x \u2264 B ** obtain \u27e8z, hz, rfl\u27e9 := Multiset.mem_map.1 hx ** case intro.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K i : \u2115 h1 : Monic p h2 : Splits f p hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2264 roots (map f p) \u2227 \u2191card s = natDegree (map f p) - i B : \u211d\u22650 h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 \u2191B z : K hz : z \u2208 s hx : \u2191\u2191nnnormHom z \u2208 Multiset.map (\u2191\u2191nnnormHom) s \u22a2 \u2191\u2191nnnormHom z \u2264 B ** exact h3 z (mem_of_le hs.1 hz) ** case inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : B < 0 \u22a2 \u2016coeff (map f p) i\u2016 \u2264 B ^ (natDegree p - i) * \u2191(Nat.choose (natDegree p) i) ** rw [eq_one_of_roots_le hB h1 h2 h3, Polynomial.map_one, natDegree_one, zero_tsub, pow_zero,\n one_mul, coeff_one] ** case inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : B < 0 \u22a2 \u2016if i = 0 then 1 else 0\u2016 \u2264 \u2191(Nat.choose 0 i) ** split_ifs with h <;> simp [h] ** case inr.inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : natDegree (map f p) < i \u22a2 \u2016coeff (map f p) i\u2016 \u2264 B ^ (natDegree (map f p) - i) * \u2191(Nat.choose (natDegree (map f p)) i) ** rw [coeff_eq_zero_of_natDegree_lt hi, norm_zero] ** case inr.inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : natDegree (map f p) < i \u22a2 0 \u2264 B ^ (natDegree (map f p) - i) * \u2191(Nat.choose (natDegree (map f p)) i) ** positivity ** case inr.inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K B : \u211d i : \u2115 h1 : Monic p h2 : Splits f p h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 B hB : 0 \u2264 B hi : i \u2264 natDegree (map f p) \u22a2 \u2191card (Multiset.map (fun x => \u2016x\u2016) (Multiset.map prod (powersetCard (natDegree (map f p) - i) (roots (map f p))))) \u2022 ?m.195583 \u2264 B ^ (natDegree (map f p) - i) * \u2191(Nat.choose (natDegree (map f p)) i) ** rw [Multiset.map_map, card_map, card_powersetCard, \u2190 natDegree_eq_card_roots' h2,\n Nat.choose_symm hi, mul_comm, nsmul_eq_mul] ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b3 : NormedRing R inst\u271d\u00b2 : IsAbsoluteValue norm F : Type u_3 K : Type u_4 inst\u271d\u00b9 : CommRing F inst\u271d : NormedField K p : F[X] f : F \u2192+* K i : \u2115 h1 : Monic p h2 : Splits f p hi : i \u2264 natDegree (map f p) s : Multiset K hs : s \u2264 roots (map f p) \u2227 \u2191card s = natDegree (map f p) - i B : \u211d\u22650 h3 : \u2200 (z : K), z \u2208 roots (map f p) \u2192 \u2016z\u2016 \u2264 \u2191B \u22a2 B ^ \u2191card (Multiset.map (\u2191\u2191nnnormHom) s) = B ^ (natDegree (map f p) - i) ** rw [card_map, hs.2] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.integrable_norm_rpow' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 p : \u211d\u22650\u221e hf : Mem\u2112p f p \u22a2 Integrable fun x => \u2016f x\u2016 ^ ENNReal.toReal p ** by_cases h_zero : p = 0 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 p : \u211d\u22650\u221e hf : Mem\u2112p f p h_zero : \u00acp = 0 \u22a2 Integrable fun x => \u2016f x\u2016 ^ ENNReal.toReal p ** by_cases h_top : p = \u221e ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 p : \u211d\u22650\u221e hf : Mem\u2112p f p h_zero : \u00acp = 0 h_top : \u00acp = \u22a4 \u22a2 Integrable fun x => \u2016f x\u2016 ^ ENNReal.toReal p ** exact hf.integrable_norm_rpow h_zero h_top ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 p : \u211d\u22650\u221e hf : Mem\u2112p f p h_zero : p = 0 \u22a2 Integrable fun x => \u2016f x\u2016 ^ ENNReal.toReal p ** simp [h_zero, integrable_const] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : NormedAddCommGroup \u03b2 inst\u271d\u00b9 : NormedAddCommGroup \u03b3 inst\u271d : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u03b2 p : \u211d\u22650\u221e hf : Mem\u2112p f p h_zero : \u00acp = 0 h_top : p = \u22a4 \u22a2 Integrable fun x => \u2016f x\u2016 ^ ENNReal.toReal p ** simp [h_top, integrable_const] ** Qed", + "informal": "" + }, + { + "formal": "BoxIntegral.Box.coe_splitLower ** \u03b9 : Type u_1 M : Type u_2 n : \u2115 I : Box \u03b9 i : \u03b9 x : \u211d y : \u03b9 \u2192 \u211d \u22a2 \u2191(splitLower I i x) = \u2191I \u2229 {y | y i \u2264 x} ** rw [splitLower, coe_mk'] ** \u03b9 : Type u_1 M : Type u_2 n : \u2115 I : Box \u03b9 i : \u03b9 x : \u211d y : \u03b9 \u2192 \u211d \u22a2 (Set.pi univ fun i_1 => Ioc (lower I i_1) (update I.upper i (min x (upper I i)) i_1)) = \u2191I \u2229 {y | y i \u2264 x} ** ext y ** case h \u03b9 : Type u_1 M : Type u_2 n : \u2115 I : Box \u03b9 i : \u03b9 x : \u211d y\u271d y : \u03b9 \u2192 \u211d \u22a2 (y \u2208 Set.pi univ fun i_1 => Ioc (lower I i_1) (update I.upper i (min x (upper I i)) i_1)) \u2194 y \u2208 \u2191I \u2229 {y | y i \u2264 x} ** simp only [mem_univ_pi, mem_Ioc, mem_inter_iff, mem_coe, mem_setOf_eq, forall_and, \u2190 Pi.le_def,\n le_update_iff, le_min_iff, and_assoc, and_forall_ne (p := fun j => y j \u2264 upper I j) i, mem_def] ** case h \u03b9 : Type u_1 M : Type u_2 n : \u2115 I : Box \u03b9 i : \u03b9 x : \u211d y\u271d y : \u03b9 \u2192 \u211d \u22a2 ((\u2200 (x : \u03b9), lower I x < y x) \u2227 y i \u2264 x \u2227 (fun i => y i) \u2264 fun i => upper I i) \u2194 (\u2200 (x : \u03b9), lower I x < y x) \u2227 ((fun i => y i) \u2264 fun i => upper I i) \u2227 y i \u2264 x ** rw [and_comm (a := y i \u2264 x), Pi.le_def] ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.supported_univ ** \u03c3 : Type u_1 \u03c4 : Type u_2 R : Type u S : Type v r : R e : \u2115 n m : \u03c3 inst\u271d : CommSemiring R p q : MvPolynomial \u03c3 R s t : Set \u03c3 \u22a2 supported R Set.univ = \u22a4 ** simp [Algebra.eq_top_iff, mem_supported] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.ae_le_of_ae_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x \u22a2 f \u2264\u1d50[\u03bc] g ** rw [Filter.EventuallyLE, ae_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, f x < g x \u22a2 \u2191\u2191\u03bc {a | \u00acf a \u2264 g a} = 0 ** rw [ae_iff] at h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h : \u2191\u2191\u03bc {a | \u00acf a < g a} = 0 \u22a2 \u2191\u2191\u03bc {a | \u00acf a \u2264 g a} = 0 ** refine' measure_mono_null (fun x hx => _) h ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 f g : \u03b1 \u2192 \u211d\u22650\u221e h : \u2191\u2191\u03bc {a | \u00acf a < g a} = 0 x : \u03b1 hx : x \u2208 {a | \u00acf a \u2264 g a} \u22a2 x \u2208 {a | \u00acf a < g a} ** exact not_lt.2 (le_of_lt (not_le.1 hx)) ** Qed", + "informal": "" + }, + { + "formal": "Submodule.linear_proj_add_linearProjOfIsCompl_eq_self ** R : Type u_1 inst\u271d\u2079 : Ring R E : Type u_2 inst\u271d\u2078 : AddCommGroup E inst\u271d\u2077 : Module R E F : Type u_3 inst\u271d\u2076 : AddCommGroup F inst\u271d\u2075 : Module R F G : Type u_4 inst\u271d\u2074 : AddCommGroup G inst\u271d\u00b3 : Module R G p q : Submodule R E S : Type u_5 inst\u271d\u00b2 : Semiring S M : Type u_6 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module S M m : Submodule S M hpq : IsCompl p q x : E \u22a2 \u2191(\u2191(linearProjOfIsCompl p q hpq) x) + \u2191(\u2191(linearProjOfIsCompl q p (_ : IsCompl q p)) x) = x ** dsimp only [linearProjOfIsCompl] ** R : Type u_1 inst\u271d\u2079 : Ring R E : Type u_2 inst\u271d\u2078 : AddCommGroup E inst\u271d\u2077 : Module R E F : Type u_3 inst\u271d\u2076 : AddCommGroup F inst\u271d\u2075 : Module R F G : Type u_4 inst\u271d\u2074 : AddCommGroup G inst\u271d\u00b3 : Module R G p q : Submodule R E S : Type u_5 inst\u271d\u00b2 : Semiring S M : Type u_6 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module S M m : Submodule S M hpq : IsCompl p q x : E \u22a2 \u2191(\u2191(comp (LinearMap.fst R { x // x \u2208 p } { x // x \u2208 q }) \u2191(LinearEquiv.symm (prodEquivOfIsCompl p q hpq))) x) + \u2191(\u2191(comp (LinearMap.fst R { x // x \u2208 q } { x // x \u2208 p }) \u2191(LinearEquiv.symm (prodEquivOfIsCompl q p (_ : IsCompl q p)))) x) = x ** rw [\u2190 prodComm_trans_prodEquivOfIsCompl _ _ hpq] ** R : Type u_1 inst\u271d\u2079 : Ring R E : Type u_2 inst\u271d\u2078 : AddCommGroup E inst\u271d\u2077 : Module R E F : Type u_3 inst\u271d\u2076 : AddCommGroup F inst\u271d\u2075 : Module R F G : Type u_4 inst\u271d\u2074 : AddCommGroup G inst\u271d\u00b3 : Module R G p q : Submodule R E S : Type u_5 inst\u271d\u00b2 : Semiring S M : Type u_6 inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module S M m : Submodule S M hpq : IsCompl p q x : E \u22a2 \u2191(\u2191(comp (LinearMap.fst R { x // x \u2208 p } { x // x \u2208 q }) \u2191(LinearEquiv.symm (prodEquivOfIsCompl p q hpq))) x) + \u2191(\u2191(comp (LinearMap.fst R { x // x \u2208 q } { x // x \u2208 p }) \u2191(LinearEquiv.symm (LinearEquiv.trans (LinearEquiv.prodComm R { x // x \u2208 q } { x // x \u2208 p }) (prodEquivOfIsCompl p q hpq)))) x) = x ** exact (prodEquivOfIsCompl _ _ hpq).apply_symm_apply x ** Qed", + "informal": "" + }, + { + "formal": "mem_rootsOfUnity_prime_pow_mul_iff' ** M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u2075 : CommMonoid M inst\u271d\u2074 : CommMonoid N inst\u271d\u00b3 : DivisionCommMonoid G k\u271d l : \u2115+ inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsReduced R p k : \u2115 m : \u2115+ hp : Fact (Nat.Prime p) inst\u271d : CharP R p \u03b6 : R\u02e3 \u22a2 \u03b6 ^ (p ^ k * \u2191m) = 1 \u2194 \u03b6 \u2208 rootsOfUnity m R ** rw [\u2190 PNat.mk_coe p hp.1.pos, \u2190 PNat.pow_coe, \u2190 PNat.mul_coe, \u2190 mem_rootsOfUnity,\n mem_rootsOfUnity_prime_pow_mul_iff] ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.sum_integral_adjacent_intervals ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a\u271d b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d a : \u2115 \u2192 \u211d n : \u2115 hint : \u2200 (k : \u2115), k < n \u2192 IntervalIntegrable f \u03bc (a k) (a (k + 1)) \u22a2 \u2211 k in Finset.range n, \u222b (x : \u211d) in a k..a (k + 1), f x \u2202\u03bc = \u222b (x : \u211d) in a 0 ..a n, f x \u2202\u03bc ** rw [\u2190 Nat.Ico_zero_eq_range] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E a\u271d b c d : \u211d f g : \u211d \u2192 E \u03bc : Measure \u211d a : \u2115 \u2192 \u211d n : \u2115 hint : \u2200 (k : \u2115), k < n \u2192 IntervalIntegrable f \u03bc (a k) (a (k + 1)) \u22a2 \u2211 k in Finset.Ico 0 n, \u222b (x : \u211d) in a k..a (k + 1), f x \u2202\u03bc = \u222b (x : \u211d) in a 0 ..a n, f x \u2202\u03bc ** exact sum_integral_adjacent_intervals_Ico (zero_le n) fun k hk => hint k hk.2 ** Qed", + "informal": "" + }, + { + "formal": "Int.isUnit_add_isUnit_eq_isUnit_add_isUnit ** a b c d : \u2124 ha : IsUnit a hb : IsUnit b hc : IsUnit c hd : IsUnit d \u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c ** rw [isUnit_iff] at ha hb hc hd ** a b c d : \u2124 ha : a = 1 \u2228 a = -1 hb : b = 1 \u2228 b = -1 hc : c = 1 \u2228 c = -1 hd : d = 1 \u2228 d = -1 \u22a2 a + b = c + d \u2194 a = c \u2227 b = d \u2228 a = d \u2227 b = c ** cases ha <;> cases hb <;> cases hc <;> cases hd <;>\n subst a <;> subst b <;> subst c <;> subst d <;>\n simp ** Qed", + "informal": "" + }, + { + "formal": "Complex.cpow_one ** x : \u2102 hx : x = 0 \u22a2 x ^ 1 = x ** simp [hx, cpow_def] ** x : \u2102 hx : \u00acx = 0 \u22a2 x ^ 1 = x ** rw [cpow_def, if_neg (one_ne_zero : (1 : \u2102) \u2260 0), if_neg hx, mul_one, exp_log hx] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Mem\u2112p.re ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p \u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p ** have : \u2200 x, \u2016IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 := by\n intro x\n rw [one_mul]\n exact IsROrC.norm_re_le_norm (f x) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p this : \u2200 (x : \u03b1), \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 \u22a2 Mem\u2112p (fun x => \u2191IsROrC.re (f x)) p ** refine' hf.of_le_mul _ (eventually_of_forall this) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p this : \u2200 (x : \u03b1), \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 \u22a2 AEStronglyMeasurable (fun x => \u2191IsROrC.re (f x)) \u03bc ** exact IsROrC.continuous_re.comp_aestronglyMeasurable hf.1 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p \u22a2 \u2200 (x : \u03b1), \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 ** intro x ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p x : \u03b1 \u22a2 \u2016\u2191IsROrC.re (f x)\u2016 \u2264 1 * \u2016f x\u2016 ** rw [one_mul] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \ud835\udd5c : Type u_5 inst\u271d : IsROrC \ud835\udd5c f : \u03b1 \u2192 \ud835\udd5c hf : Mem\u2112p f p x : \u03b1 \u22a2 \u2016\u2191IsROrC.re (f x)\u2016 \u2264 \u2016f x\u2016 ** exact IsROrC.norm_re_le_norm (f x) ** Qed", + "informal": "" + }, + { + "formal": "minpoly.coeff_zero_eq_zero ** A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x : B hx : IsIntegral A x \u22a2 coeff (minpoly A x) 0 = 0 \u2194 x = 0 ** constructor ** case mp A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x : B hx : IsIntegral A x \u22a2 coeff (minpoly A x) 0 = 0 \u2192 x = 0 ** intro h ** case mp A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x : B hx : IsIntegral A x h : coeff (minpoly A x) 0 = 0 \u22a2 x = 0 ** have zero_root := zero_isRoot_of_coeff_zero_eq_zero h ** case mp A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x : B hx : IsIntegral A x h : coeff (minpoly A x) 0 = 0 zero_root : IsRoot (minpoly A x) 0 \u22a2 x = 0 ** rw [\u2190 root hx zero_root] ** case mp A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x : B hx : IsIntegral A x h : coeff (minpoly A x) 0 = 0 zero_root : IsRoot (minpoly A x) 0 \u22a2 \u2191(algebraMap A B) 0 = 0 ** exact RingHom.map_zero _ ** case mpr A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B x : B hx : IsIntegral A x \u22a2 x = 0 \u2192 coeff (minpoly A x) 0 = 0 ** rintro rfl ** case mpr A : Type u_1 B : Type u_2 inst\u271d\u00b3 : Field A inst\u271d\u00b2 : Ring B inst\u271d\u00b9 : IsDomain B inst\u271d : Algebra A B hx : IsIntegral A 0 \u22a2 coeff (minpoly A 0) 0 = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "IsLUB.frequently_mem ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 a : \u03b1 s : Set \u03b1 ha : IsLUB s a hs : Set.Nonempty s \u22a2 \u2203\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a] a, x \u2208 s ** rcases hs with \u27e8a', ha'\u27e9 ** case intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 a : \u03b1 s : Set \u03b1 ha : IsLUB s a a' : \u03b1 ha' : a' \u2208 s \u22a2 \u2203\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a] a, x \u2208 s ** intro h ** case intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 a : \u03b1 s : Set \u03b1 ha : IsLUB s a a' : \u03b1 ha' : a' \u2208 s h : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a] a, \u00ac(fun x => x \u2208 s) x \u22a2 False ** rcases (ha.1 ha').eq_or_lt with (rfl | ha'a) ** case intro.inl \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 s : Set \u03b1 a' : \u03b1 ha' : a' \u2208 s ha : IsLUB s a' h : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a'] a', \u00ac(fun x => x \u2208 s) x \u22a2 False ** exact h.self_of_nhdsWithin le_rfl ha' ** case intro.inr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 a : \u03b1 s : Set \u03b1 ha : IsLUB s a a' : \u03b1 ha' : a' \u2208 s h : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a] a, \u00ac(fun x => x \u2208 s) x ha'a : a' < a \u22a2 False ** rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with \u27e8b, hba, hb\u27e9 ** case intro.inr.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 a : \u03b1 s : Set \u03b1 ha : IsLUB s a a' : \u03b1 ha' : a' \u2208 s h : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a] a, \u00ac(fun x => x \u2208 s) x ha'a : a' < a b : \u03b1 hba : b \u2208 Iio a hb : Ioc b a \u2286 {x | (fun x => \u00ac(fun x => x \u2208 s) x) x} \u22a2 False ** rcases ha.exists_between hba with \u27e8b', hb's, hb'\u27e9 ** case intro.inr.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : LinearOrder \u03b2 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : OrderTopology \u03b2 a : \u03b1 s : Set \u03b1 ha : IsLUB s a a' : \u03b1 ha' : a' \u2208 s h : \u2200\u1da0 (x : \u03b1) in \ud835\udcdd[Iic a] a, \u00ac(fun x => x \u2208 s) x ha'a : a' < a b : \u03b1 hba : b \u2208 Iio a hb : Ioc b a \u2286 {x | (fun x => \u00ac(fun x => x \u2208 s) x) x} b' : \u03b1 hb's : b' \u2208 s hb' : b < b' \u2227 b' \u2264 a \u22a2 False ** exact hb hb' hb's ** Qed", + "informal": "" + }, + { + "formal": "Basis.map_orientation_eq_det_inv_smul ** R : Type u_1 inst\u271d\u2075 : StrictOrderedCommRing R M : Type u_2 N : Type u_3 inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R N \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : Finite \u03b9 e : Basis \u03b9 R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M \u22a2 \u2191(Orientation.map \u03b9 f) x = (\u2191LinearEquiv.det f)\u207b\u00b9 \u2022 x ** cases nonempty_fintype \u03b9 ** case intro R : Type u_1 inst\u271d\u2075 : StrictOrderedCommRing R M : Type u_2 N : Type u_3 inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R N \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : Finite \u03b9 e : Basis \u03b9 R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M val\u271d : Fintype \u03b9 \u22a2 \u2191(Orientation.map \u03b9 f) x = (\u2191LinearEquiv.det f)\u207b\u00b9 \u2022 x ** letI := Classical.decEq \u03b9 ** case intro R : Type u_1 inst\u271d\u2075 : StrictOrderedCommRing R M : Type u_2 N : Type u_3 inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R N \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : Finite \u03b9 e : Basis \u03b9 R M x : Orientation R M \u03b9 f : M \u2243\u2097[R] M val\u271d : Fintype \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 \u22a2 \u2191(Orientation.map \u03b9 f) x = (\u2191LinearEquiv.det f)\u207b\u00b9 \u2022 x ** induction' x using Module.Ray.ind with g hg ** case intro.h R : Type u_1 inst\u271d\u2075 : StrictOrderedCommRing R M : Type u_2 N : Type u_3 inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R N \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : Finite \u03b9 e : Basis \u03b9 R M f : M \u2243\u2097[R] M val\u271d : Fintype \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 g : AlternatingMap R M R \u03b9 hg : g \u2260 0 \u22a2 \u2191(Orientation.map \u03b9 f) (rayOfNeZero R g hg) = (\u2191LinearEquiv.det f)\u207b\u00b9 \u2022 rayOfNeZero R g hg ** rw [Orientation.map_apply, smul_rayOfNeZero, ray_eq_iff, Units.smul_def,\n (g.compLinearMap f.symm).eq_smul_basis_det e, g.eq_smul_basis_det e,\n AlternatingMap.compLinearMap_apply, AlternatingMap.smul_apply,\n show (fun i \u21a6 (LinearEquiv.symm f).toLinearMap (e i)) = (LinearEquiv.symm f).toLinearMap \u2218 e\n by rfl, Basis.det_comp, Basis.det_self, mul_one, smul_eq_mul, mul_comm, mul_smul,\n LinearEquiv.coe_inv_det] ** R : Type u_1 inst\u271d\u2075 : StrictOrderedCommRing R M : Type u_2 N : Type u_3 inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R N \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : Finite \u03b9 e : Basis \u03b9 R M f : M \u2243\u2097[R] M val\u271d : Fintype \u03b9 this : DecidableEq \u03b9 := Classical.decEq \u03b9 g : AlternatingMap R M R \u03b9 hg : g \u2260 0 \u22a2 (fun i => \u2191\u2191(LinearEquiv.symm f) (\u2191e i)) = \u2191\u2191(LinearEquiv.symm f) \u2218 \u2191e ** rfl ** Qed", + "informal": "" + }, + { + "formal": "ContMDiffWithinAt.prod_mk_space ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f\u271d f\u2081 : M \u2192 M' s s\u2081 t : Set M x : M m n : \u2115\u221e f : M \u2192 E' g : M \u2192 F' hf : ContMDiffWithinAt I \ud835\udcd8(\ud835\udd5c, E') n f s x hg : ContMDiffWithinAt I \ud835\udcd8(\ud835\udd5c, F') n g s x \u22a2 ContMDiffWithinAt I \ud835\udcd8(\ud835\udd5c, E' \u00d7 F') n (fun x => (f x, g x)) s x ** rw [contMDiffWithinAt_iff] at * ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f\u271d f\u2081 : M \u2192 M' s s\u2081 t : Set M x : M m n : \u2115\u221e f : M \u2192 E' g : M \u2192 F' hf : ContinuousWithinAt f s x \u2227 ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, E') (f x)) \u2218 f \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x) hg : ContinuousWithinAt g s x \u2227 ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, F') (g x)) \u2218 g \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x) \u22a2 ContinuousWithinAt (fun x => (f x, g x)) s x \u2227 ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt \ud835\udcd8(\ud835\udd5c, E' \u00d7 F') (f x, g x)) \u2218 (fun x => (f x, g x)) \u2218 \u2191(LocalEquiv.symm (extChartAt I x))) (\u2191(LocalEquiv.symm (extChartAt I x)) \u207b\u00b9' s \u2229 range \u2191I) (\u2191(extChartAt I x) x) ** exact \u27e8hf.1.prod hg.1, hf.2.prod hg.2\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.C_dvd_iff_dvd_coeff ** R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191C r \u2223 \u03c6 \u2194 \u2200 (i : \u03c3 \u2192\u2080 \u2115), r \u2223 coeff i \u03c6 ** constructor ** case mp R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R \u22a2 \u2191C r \u2223 \u03c6 \u2192 \u2200 (i : \u03c3 \u2192\u2080 \u2115), r \u2223 coeff i \u03c6 ** rintro \u27e8\u03c6, rfl\u27e9 c ** case mp.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R c : \u03c3 \u2192\u2080 \u2115 \u22a2 r \u2223 coeff c (\u2191C r * \u03c6) ** rw [coeff_C_mul] ** case mp.intro R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R c : \u03c3 \u2192\u2080 \u2115 \u22a2 r \u2223 r * coeff c \u03c6 ** apply dvd_mul_right ** case mpr R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R \u22a2 (\u2200 (i : \u03c3 \u2192\u2080 \u2115), r \u2223 coeff i \u03c6) \u2192 \u2191C r \u2223 \u03c6 ** intro h ** case mpr R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R h : \u2200 (i : \u03c3 \u2192\u2080 \u2115), r \u2223 coeff i \u03c6 \u22a2 \u2191C r \u2223 \u03c6 ** choose C hc using h ** case mpr R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i \u22a2 \u2191MvPolynomial.C r \u2223 \u03c6 ** let c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 \u03c6.support then C i else 0 ** case mpr R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u22a2 \u2191MvPolynomial.C r \u2223 \u03c6 ** let \u03c8 : MvPolynomial \u03c3 R := \u2211 i in \u03c6.support, monomial i (c' i) ** case mpr R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) \u22a2 \u2191MvPolynomial.C r \u2223 \u03c6 ** use \u03c8 ** case h R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) \u22a2 \u03c6 = \u2191MvPolynomial.C r * \u03c8 ** apply MvPolynomial.ext ** case h.a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) \u22a2 \u2200 (m : \u03c3 \u2192\u2080 \u2115), coeff m \u03c6 = coeff m (\u2191MvPolynomial.C r * \u03c8) ** intro i ** case h.a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) i : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff i \u03c6 = coeff i (\u2191MvPolynomial.C r * \u03c8) ** simp only [coeff_C_mul, coeff_sum, coeff_monomial, Finset.sum_ite_eq'] ** case h.a R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) i : \u03c3 \u2192\u2080 \u2115 \u22a2 coeff i \u03c6 = r * if i \u2208 support \u03c6 then if i \u2208 support \u03c6 then C i else 0 else 0 ** split_ifs with hi ** case pos R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) i : \u03c3 \u2192\u2080 \u2115 hi : i \u2208 support \u03c6 \u22a2 coeff i \u03c6 = r * C i ** rw [hc] ** case neg R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) i : \u03c3 \u2192\u2080 \u2115 hi : \u00aci \u2208 support \u03c6 \u22a2 coeff i \u03c6 = r * 0 ** rw [not_mem_support_iff] at hi ** case neg R : Type u S\u2081 : Type v S\u2082 : Type w S\u2083 : Type x \u03c3 : Type u_1 a a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R inst\u271d : CommSemiring S\u2081 p q : MvPolynomial \u03c3 R r : R \u03c6 : MvPolynomial \u03c3 R C : (\u03c3 \u2192\u2080 \u2115) \u2192 R hc : \u2200 (i : \u03c3 \u2192\u2080 \u2115), coeff i \u03c6 = r * C i c' : (\u03c3 \u2192\u2080 \u2115) \u2192 R := fun i => if i \u2208 support \u03c6 then C i else 0 \u03c8 : MvPolynomial \u03c3 R := \u2211 i in support \u03c6, \u2191(monomial i) (c' i) i : \u03c3 \u2192\u2080 \u2115 hi : coeff i \u03c6 = 0 \u22a2 coeff i \u03c6 = r * 0 ** rwa [mul_zero] ** Qed", + "informal": "" + }, + { + "formal": "Real.toNNReal_sum_of_nonneg ** \u03b1 : Type u_1 s : Finset \u03b1 f : \u03b1 \u2192 \u211d hf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a \u22a2 Real.toNNReal (\u2211 a in s, f a) = \u2211 a in s, Real.toNNReal (f a) ** rw [\u2190 NNReal.coe_eq, NNReal.coe_sum, Real.coe_toNNReal _ (Finset.sum_nonneg hf)] ** \u03b1 : Type u_1 s : Finset \u03b1 f : \u03b1 \u2192 \u211d hf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a \u22a2 \u2211 i in s, f i = \u2211 a in s, \u2191(Real.toNNReal (f a)) ** exact Finset.sum_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)] ** \u03b1 : Type u_1 s : Finset \u03b1 f : \u03b1 \u2192 \u211d hf : \u2200 (a : \u03b1), a \u2208 s \u2192 0 \u2264 f a x : \u03b1 hxs : x \u2208 s \u22a2 f x = \u2191(Real.toNNReal (f x)) ** rw [Real.coe_toNNReal _ (hf x hxs)] ** Qed", + "informal": "" + }, + { + "formal": "Measurable.prod ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 \u00d7 \u03b3 hf\u2081 : Measurable fun a => (f a).1 hf\u2082 : Measurable fun a => (f a).2 \u22a2 MeasurableSpace.comap Prod.fst m\u03b2 \u2264 MeasurableSpace.map f m ** rw [MeasurableSpace.comap_le_iff_le_map, MeasurableSpace.map_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 \u00d7 \u03b3 hf\u2081 : Measurable fun a => (f a).1 hf\u2082 : Measurable fun a => (f a).2 \u22a2 m\u03b2 \u2264 MeasurableSpace.map (Prod.fst \u2218 f) m ** exact hf\u2081 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 \u00d7 \u03b3 hf\u2081 : Measurable fun a => (f a).1 hf\u2082 : Measurable fun a => (f a).2 \u22a2 MeasurableSpace.comap Prod.snd m\u03b3 \u2264 MeasurableSpace.map f m ** rw [MeasurableSpace.comap_le_iff_le_map, MeasurableSpace.map_comp] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s t u : Set \u03b1 m : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 m\u03b3 : MeasurableSpace \u03b3 f : \u03b1 \u2192 \u03b2 \u00d7 \u03b3 hf\u2081 : Measurable fun a => (f a).1 hf\u2082 : Measurable fun a => (f a).2 \u22a2 m\u03b3 \u2264 MeasurableSpace.map (Prod.snd \u2218 f) m ** exact hf\u2082 ** Qed", + "informal": "" + }, + { + "formal": "frontier_Iic' ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : DenselyOrdered \u03b1 a\u271d b : \u03b1 s : Set \u03b1 a : \u03b1 ha : Set.Nonempty (Ioi a) \u22a2 frontier (Iic a) = {a} ** simp [frontier, ha] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousAffineMap.contLinear_eq_zero_iff_exists_const ** \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 h\u2082 : \u2200 (q : Q), f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q \u22a2 contLinear f = 0 \u2194 \u2203 q, f = const R P q ** simp_rw [h\u2081, h\u2082] ** \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 h\u2082 : \u2200 (q : Q), f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q \u22a2 f.linear = 0 \u2194 \u2203 q, f.toAffineMap = AffineMap.const R P q ** exact (f : P \u2192\u1d43[R] Q).linear_eq_zero_iff_exists_const ** \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q \u22a2 contLinear f = 0 \u2194 f.linear = 0 ** refine' \u27e8fun h => _, fun h => _\u27e9 <;> ext ** case refine'_1.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h : contLinear f = 0 x\u271d : V \u22a2 \u2191f.linear x\u271d = \u21910 x\u271d ** rw [\u2190 coe_contLinear_eq_linear, h] ** case refine'_1.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h : contLinear f = 0 x\u271d : V \u22a2 \u2191\u21910 x\u271d = \u21910 x\u271d ** rfl ** case refine'_2.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h : f.linear = 0 x\u271d : V \u22a2 \u2191(contLinear f) x\u271d = \u21910 x\u271d ** rw [\u2190 coe_linear_eq_coe_contLinear, h] ** case refine'_2.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h : f.linear = 0 x\u271d : V \u22a2 \u21910 x\u271d = \u21910 x\u271d ** rfl ** \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 \u22a2 \u2200 (q : Q), f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q ** intro q ** \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 q : Q \u22a2 f = const R P q \u2194 f.toAffineMap = AffineMap.const R P q ** refine' \u27e8fun h => _, fun h => _\u27e9 <;> ext ** case refine'_1.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 q : Q h : f = const R P q p\u271d : P \u22a2 \u2191f.toAffineMap p\u271d = \u2191(AffineMap.const R P q) p\u271d ** rw [h] ** case refine'_1.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 q : Q h : f = const R P q p\u271d : P \u22a2 \u2191(const R P q).toAffineMap p\u271d = \u2191(AffineMap.const R P q) p\u271d ** rfl ** case refine'_2.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 q : Q h : f.toAffineMap = AffineMap.const R P q x\u271d : P \u22a2 \u2191f x\u271d = \u2191(const R P q) x\u271d ** rw [\u2190 coe_to_affineMap, h] ** case refine'_2.h \ud835\udd5c : Type u_1 R : Type u_2 V : Type u_3 W : Type u_4 W\u2082 : Type u_5 P : Type u_6 Q : Type u_7 Q\u2082 : Type u_8 inst\u271d\u00b9\u2076 : NormedAddCommGroup V inst\u271d\u00b9\u2075 : MetricSpace P inst\u271d\u00b9\u2074 : NormedAddTorsor V P inst\u271d\u00b9\u00b3 : NormedAddCommGroup W inst\u271d\u00b9\u00b2 : MetricSpace Q inst\u271d\u00b9\u00b9 : NormedAddTorsor W Q inst\u271d\u00b9\u2070 : NormedAddCommGroup W\u2082 inst\u271d\u2079 : MetricSpace Q\u2082 inst\u271d\u2078 : NormedAddTorsor W\u2082 Q\u2082 inst\u271d\u2077 : NormedField R inst\u271d\u2076 : NormedSpace R V inst\u271d\u2075 : NormedSpace R W inst\u271d\u2074 : NormedSpace R W\u2082 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b2 : NormedSpace \ud835\udd5c V inst\u271d\u00b9 : NormedSpace \ud835\udd5c W inst\u271d : NormedSpace \ud835\udd5c W\u2082 f : P \u2192A[R] Q h\u2081 : contLinear f = 0 \u2194 f.linear = 0 q : Q h : f.toAffineMap = AffineMap.const R P q x\u271d : P \u22a2 \u2191(AffineMap.const R P q) x\u271d = \u2191(const R P q) x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "FormalMultilinearSeries.summable_norm_apply ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \ud835\udd5c G p\u271d : FormalMultilinearSeries \ud835\udd5c E F r : \u211d\u22650 p : FormalMultilinearSeries \ud835\udd5c E F x : E hx : x \u2208 EMetric.ball 0 (radius p) \u22a2 Summable fun n => \u2016\u2191(p n) fun x_1 => x\u2016 ** rw [mem_emetric_ball_zero_iff] at hx ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \ud835\udd5c G p\u271d : FormalMultilinearSeries \ud835\udd5c E F r : \u211d\u22650 p : FormalMultilinearSeries \ud835\udd5c E F x : E hx : \u2191\u2016x\u2016\u208a < radius p \u22a2 Summable fun n => \u2016\u2191(p n) fun x_1 => x\u2016 ** refine' summable_of_nonneg_of_le\n (fun _ => norm_nonneg _) (fun n => ((p n).le_op_norm _).trans_eq _) (p.summable_norm_mul_pow hx) ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \ud835\udd5c G p\u271d : FormalMultilinearSeries \ud835\udd5c E F r : \u211d\u22650 p : FormalMultilinearSeries \ud835\udd5c E F x : E hx : \u2191\u2016x\u2016\u208a < radius p n : \u2115 \u22a2 \u2016p n\u2016 * \u220f i : Fin n, \u2016x\u2016 = \u2016p n\u2016 * \u2191\u2016x\u2016\u208a ^ n ** simp ** Qed", + "informal": "" + }, + { + "formal": "Bornology.comap_cobounded_le_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : Bornology \u03b1 s t : Set \u03b1 x : \u03b1 inst\u271d : Bornology \u03b2 f : \u03b1 \u2192 \u03b2 \u22a2 comap f (cobounded \u03b2) \u2264 cobounded \u03b1 \u2194 \u2200 \u2983s : Set \u03b1\u2984, IsBounded s \u2192 IsBounded (f '' s) ** refine'\n \u27e8fun h s hs => _, fun h t ht =>\n \u27e8(f '' t\u1d9c)\u1d9c, h <| IsCobounded.compl ht, compl_subset_comm.1 <| subset_preimage_image _ _\u27e9\u27e9 ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : Bornology \u03b1 s\u271d t : Set \u03b1 x : \u03b1 inst\u271d : Bornology \u03b2 f : \u03b1 \u2192 \u03b2 h : comap f (cobounded \u03b2) \u2264 cobounded \u03b1 s : Set \u03b1 hs : IsBounded s \u22a2 IsBounded (f '' s) ** obtain \u27e8t, ht, hts\u27e9 := h hs.compl ** case intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : Bornology \u03b1 s\u271d t\u271d : Set \u03b1 x : \u03b1 inst\u271d : Bornology \u03b2 f : \u03b1 \u2192 \u03b2 h : comap f (cobounded \u03b2) \u2264 cobounded \u03b1 s : Set \u03b1 hs : IsBounded s t : Set \u03b2 ht : t \u2208 cobounded \u03b2 hts : f \u207b\u00b9' t \u2286 s\u1d9c \u22a2 IsBounded (f '' s) ** rw [subset_compl_comm, \u2190 preimage_compl] at hts ** case intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u00b9 : Bornology \u03b1 s\u271d t\u271d : Set \u03b1 x : \u03b1 inst\u271d : Bornology \u03b2 f : \u03b1 \u2192 \u03b2 h : comap f (cobounded \u03b2) \u2264 cobounded \u03b1 s : Set \u03b1 hs : IsBounded s t : Set \u03b2 ht : t \u2208 cobounded \u03b2 hts : s \u2286 f \u207b\u00b9' t\u1d9c \u22a2 IsBounded (f '' s) ** exact (IsCobounded.compl ht).subset ((image_subset f hts).trans <| image_preimage_subset _ _) ** Qed", + "informal": "" + }, + { + "formal": "Finset.card_le_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 \u22a2 card s \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b ** obtain rfl | \u27e8x, hx\u27e9 := s.eq_empty_or_nonempty ** case inr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 x : \u03b1 hx : x \u2208 s \u22a2 card s \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b ** refine' (Nat.succ_le_of_lt (card_pos.2 \u27e8x, hx\u27e9)).le_iff_eq.trans (card_eq_one.trans \u27e8_, _\u27e9) ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 \u22a2 card \u2205 \u2264 1 \u2194 \u2200 (a : \u03b1), a \u2208 \u2205 \u2192 \u2200 (b : \u03b1), b \u2208 \u2205 \u2192 a = b ** simp ** case inr.intro.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 x : \u03b1 hx : x \u2208 s \u22a2 (\u2203 a, s = {a}) \u2192 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b ** rintro \u27e8y, rfl\u27e9 ** case inr.intro.refine'_1.intro \u03b1 : Type u_1 \u03b2 : Type u_2 t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 x y : \u03b1 hx : x \u2208 {y} \u22a2 \u2200 (a : \u03b1), a \u2208 {y} \u2192 \u2200 (b : \u03b1), b \u2208 {y} \u2192 a = b ** simp ** case inr.intro.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 s t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 x : \u03b1 hx : x \u2208 s \u22a2 (\u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 s \u2192 a = b) \u2192 \u2203 a, s = {a} ** exact fun h => \u27e8x, eq_singleton_iff_unique_mem.2 \u27e8hx, fun y hy => h _ hy _ hx\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.snorm_mono_measure ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** by_cases hp0 : p = 0 ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** by_cases hp_top : p = \u221e ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** simp_rw [snorm_eq_snorm' hp0 hp_top] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 hp_top : \u00acp = \u22a4 \u22a2 snorm' f (ENNReal.toReal p) \u03bd \u2264 snorm' f (ENNReal.toReal p) \u03bc ** exact snorm'_mono_measure f h\u03bc\u03bd ENNReal.toReal_nonneg ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : p = 0 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** simp [hp0] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192 F h\u03bc\u03bd : \u03bd \u2264 \u03bc hp0 : \u00acp = 0 hp_top : p = \u22a4 \u22a2 snorm f p \u03bd \u2264 snorm f p \u03bc ** simp [hp_top, snormEssSup_mono_measure f (Measure.absolutelyContinuous_of_le h\u03bc\u03bd)] ** Qed", + "informal": "" + }, + { + "formal": "Metric.hausdorffDist_self_closure ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 \u22a2 hausdorffDist s (closure s) = 0 ** simp [hausdorffDist] ** Qed", + "informal": "" + }, + { + "formal": "interior_eq_nhds' ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a : \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 x : \u03b1 \u22a2 x \u2208 interior s \u2194 x \u2208 {a | s \u2208 \ud835\udcdd a} ** simp only [mem_interior, mem_nhds_iff, mem_setOf_eq] ** Qed", + "informal": "" + }, + { + "formal": "Real.logb_one ** b x y : \u211d \u22a2 logb b 1 = 0 ** simp [logb] ** Qed", + "informal": "" + }, + { + "formal": "iSup_iInf_eq ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 \u22a2 \u2a06 a, \u2a05 b, f a b = \u2a05 g, \u2a06 a, f a (g a) ** refine le_antisymm iSup_iInf_le ?_ ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 \u22a2 \u2a05 g, \u2a06 a, f a (g a) \u2264 \u2a06 a, \u2a05 b, f a b ** rw [iInf_iSup_eq] ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 \u22a2 \u2a06 g, \u2a05 a, f (g a) (a (g a)) \u2264 \u2a06 a, \u2a05 b, f a b ** refine iSup_le fun g => ?_ ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 \u22a2 \u2a05 a, f (g a) (a (g a)) \u2264 \u2a06 a, \u2a05 b, f a b ** have \u27e8a, ha\u27e9 : \u2203 a, \u2200 b, \u2203 f, \u2203 h : a = g f, h \u25b8 b = f (g f) := of_not_not fun h => by\n push_neg at h\n choose h hh using h\n have := hh _ h rfl\n contradiction ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 a : \u03b9 ha : \u2200 (b : \u03ba a), \u2203 f h, h \u25b8 b = f (g f) \u22a2 \u2a05 a, f (g a) (a (g a)) \u2264 \u2a06 a, \u2a05 b, f a b ** refine le_trans ?_ (le_iSup _ a) ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 a : \u03b9 ha : \u2200 (b : \u03ba a), \u2203 f h, h \u25b8 b = f (g f) \u22a2 \u2a05 a, f (g a) (a (g a)) \u2264 \u2a05 b, f a b ** refine le_iInf fun b => ?_ ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 a : \u03b9 ha : \u2200 (b : \u03ba a), \u2203 f h, h \u25b8 b = f (g f) b : \u03ba a \u22a2 \u2a05 a, f (g a) (a (g a)) \u2264 f a b ** obtain \u27e8h, rfl, rfl\u27e9 := ha b ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 h : (a : \u03b9) \u2192 \u03ba a ha : \u2200 (b : \u03ba (g h)), \u2203 f h_1, h_1 \u25b8 b = f (g f) \u22a2 \u2a05 a, f (g a) (a (g a)) \u2264 f (g h) (h (g h)) ** refine iInf_le _ _ ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 h : \u00ac\u2203 a, \u2200 (b : \u03ba a), \u2203 f h, h \u25b8 b = f (g f) \u22a2 False ** push_neg at h ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 h : \u2200 (a : \u03b9), \u2203 b, \u2200 (f : (a : \u03b9) \u2192 \u03ba a) (h : a = g f), h \u25b8 b \u2260 f (g f) \u22a2 False ** choose h hh using h ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 h : (a : \u03b9) \u2192 \u03ba a hh : \u2200 (a : \u03b9) (f : (a : \u03b9) \u2192 \u03ba a) (h_1 : a = g f), h_1 \u25b8 h a \u2260 f (g f) \u22a2 False ** have := hh _ h rfl ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03ba : \u03b9 \u2192 Sort w' inst\u271d : CompletelyDistribLattice \u03b1 f : (a : \u03b9) \u2192 \u03ba a \u2192 \u03b1 g : ((a : \u03b9) \u2192 \u03ba a) \u2192 \u03b9 h : (a : \u03b9) \u2192 \u03ba a hh : \u2200 (a : \u03b9) (f : (a : \u03b9) \u2192 \u03ba a) (h_1 : a = g f), h_1 \u25b8 h a \u2260 f (g f) this : (_ : g h = g h) \u25b8 h (g h) \u2260 h (g h) \u22a2 False ** contradiction ** Qed", + "informal": "" + }, + { + "formal": "List.append_eq_cons_iff ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 a b c : List \u03b1 x : \u03b1 \u22a2 a ++ b = x :: c \u2194 a = [] \u2227 b = x :: c \u2228 \u2203 a', a = x :: a' \u2227 c = a' ++ b ** cases a <;>\n simp only [and_assoc, @eq_comm _ c, nil_append, cons_append, cons.injEq, true_and_iff,\n false_and_iff, exists_false, false_or_iff, or_false_iff, exists_and_left, exists_eq_left'] ** Qed", + "informal": "" + }, + { + "formal": "HasFDerivAt.mul ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u2075 : NormedAddCommGroup G' inst\u271d\u2074 : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \ud835\udd38 : Type u_6 \ud835\udd38' : Type u_7 inst\u271d\u00b3 : NormedRing \ud835\udd38 inst\u271d\u00b2 : NormedCommRing \ud835\udd38' inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38 inst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38' a b : E \u2192 \ud835\udd38 a' b' : E \u2192L[\ud835\udd5c] \ud835\udd38 c d : E \u2192 \ud835\udd38' c' d' : E \u2192L[\ud835\udd5c] \ud835\udd38' hc : HasFDerivAt c c' x hd : HasFDerivAt d d' x \u22a2 HasFDerivAt (fun y => c y * d y) (c x \u2022 d' + d x \u2022 c') x ** convert hc.mul' hd ** case h.e'_10.h.e'_6 \ud835\udd5c : Type u_1 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u2075 : NormedAddCommGroup G' inst\u271d\u2074 : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \ud835\udd38 : Type u_6 \ud835\udd38' : Type u_7 inst\u271d\u00b3 : NormedRing \ud835\udd38 inst\u271d\u00b2 : NormedCommRing \ud835\udd38' inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38 inst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38' a b : E \u2192 \ud835\udd38 a' b' : E \u2192L[\ud835\udd5c] \ud835\udd38 c d : E \u2192 \ud835\udd38' c' d' : E \u2192L[\ud835\udd5c] \ud835\udd38' hc : HasFDerivAt c c' x hd : HasFDerivAt d d' x \u22a2 d x \u2022 c' = smulRight c' (d x) ** ext z ** case h.e'_10.h.e'_6.h \ud835\udd5c : Type u_1 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u2075 : NormedAddCommGroup G' inst\u271d\u2074 : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \ud835\udd38 : Type u_6 \ud835\udd38' : Type u_7 inst\u271d\u00b3 : NormedRing \ud835\udd38 inst\u271d\u00b2 : NormedCommRing \ud835\udd38' inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c \ud835\udd38 inst\u271d : NormedAlgebra \ud835\udd5c \ud835\udd38' a b : E \u2192 \ud835\udd38 a' b' : E \u2192L[\ud835\udd5c] \ud835\udd38 c d : E \u2192 \ud835\udd38' c' d' : E \u2192L[\ud835\udd5c] \ud835\udd38' hc : HasFDerivAt c c' x hd : HasFDerivAt d d' x z : E \u22a2 \u2191(d x \u2022 c') z = \u2191(smulRight c' (d x)) z ** apply mul_comm ** Qed", + "informal": "" + }, + { + "formal": "list_reverse' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03c3 : Type u_3 inst\u271d\u00b2 : Primcodable \u03b1 inst\u271d\u00b9 : Primcodable \u03b2 inst\u271d : Primcodable \u03c3 H : Nat.Primrec fun n => encode (decode n) this : Primcodable (List \u03b2) := prim H l : List \u03b2 \u22a2 \u2200 (r : List \u03b2), List.foldl (fun s b => b :: s) r l = List.reverseAux l r ** induction l <;> simp [*, List.reverseAux] ** Qed", + "informal": "" + }, + { + "formal": "IsMinOn.of_isLocalMinOn_of_convexOn_Icc ** E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsLocalMinOn f (Icc a b) a h_conv : ConvexOn \u211d (Icc a b) f \u22a2 IsMinOn f (Icc a b) a ** rintro c hc ** E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsLocalMinOn f (Icc a b) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b \u22a2 c \u2208 {x | (fun x => f a \u2264 f x) x} ** dsimp only [mem_setOf_eq] ** E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsLocalMinOn f (Icc a b) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b \u22a2 f a \u2264 f c ** rw [IsLocalMinOn, nhdsWithin_Icc_eq_nhdsWithin_Ici a_lt_b] at h_local_min ** E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b \u22a2 f a \u2264 f c ** rcases hc.1.eq_or_lt with (rfl | a_lt_c) ** case inr E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c \u22a2 f a \u2264 f c ** have H\u2081 : \u2200\u1da0 y in \ud835\udcdd[>] a, f a \u2264 f y :=\n h_local_min.filter_mono (nhdsWithin_mono _ Ioi_subset_Ici_self) ** case inr E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y \u22a2 f a \u2264 f c ** have H\u2082 : \u2200\u1da0 y in \ud835\udcdd[>] a, y \u2208 Ioc a c := Ioc_mem_nhdsWithin_Ioi (left_mem_Ico.2 a_lt_c) ** case inr E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c \u22a2 f a \u2264 f c ** rcases (H\u2081.and H\u2082).exists with \u27e8y, hfy, hy_ac\u27e9 ** case inr.intro.intro E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c y : \u211d hfy : f a \u2264 f y hy_ac : y \u2208 Ioc a c \u22a2 f a \u2264 f c ** rcases (Convex.mem_Ioc a_lt_c).mp hy_ac with \u27e8ya, yc, ya\u2080, yc\u2080, yac, rfl\u27e9 ** case inr.intro.intro.intro.intro.intro.intro.intro E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c ya yc : \u211d ya\u2080 : 0 \u2264 ya yc\u2080 : 0 < yc yac : ya + yc = 1 hfy : f a \u2264 f (ya * a + yc * c) hy_ac : ya * a + yc * c \u2208 Ioc a c \u22a2 f a \u2264 f c ** suffices : ya \u2022 f a + yc \u2022 f a \u2264 ya \u2022 f a + yc \u2022 f c ** case inr.intro.intro.intro.intro.intro.intro.intro E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c ya yc : \u211d ya\u2080 : 0 \u2264 ya yc\u2080 : 0 < yc yac : ya + yc = 1 hfy : f a \u2264 f (ya * a + yc * c) hy_ac : ya * a + yc * c \u2208 Ioc a c this : ya \u2022 f a + yc \u2022 f a \u2264 ya \u2022 f a + yc \u2022 f c \u22a2 f a \u2264 f c case this E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c ya yc : \u211d ya\u2080 : 0 \u2264 ya yc\u2080 : 0 < yc yac : ya + yc = 1 hfy : f a \u2264 f (ya * a + yc * c) hy_ac : ya * a + yc * c \u2208 Ioc a c \u22a2 ya \u2022 f a + yc \u2022 f a \u2264 ya \u2022 f a + yc \u2022 f c ** exact (smul_le_smul_iff_of_pos yc\u2080).1 (le_of_add_le_add_left this) ** case this E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c ya yc : \u211d ya\u2080 : 0 \u2264 ya yc\u2080 : 0 < yc yac : ya + yc = 1 hfy : f a \u2264 f (ya * a + yc * c) hy_ac : ya * a + yc * c \u2208 Ioc a c \u22a2 ya \u2022 f a + yc \u2022 f a \u2264 ya \u2022 f a + yc \u2022 f c ** calc\n ya \u2022 f a + yc \u2022 f a = f a := by rw [\u2190 add_smul, yac, one_smul]\n _ \u2264 f (ya * a + yc * c) := hfy\n _ \u2264 ya \u2022 f a + yc \u2022 f c := h_conv.2 (left_mem_Icc.2 a_lt_b.le) hc ya\u2080 yc\u2080.le yac ** case inl E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f hc : a \u2208 Icc a b \u22a2 f a \u2264 f a ** exact le_rfl ** E : Type u_1 \u03b2 : Type u_2 inst\u271d\u2077 : AddCommGroup E inst\u271d\u2076 : TopologicalSpace E inst\u271d\u2075 : Module \u211d E inst\u271d\u2074 : TopologicalAddGroup E inst\u271d\u00b3 : ContinuousSMul \u211d E inst\u271d\u00b2 : OrderedAddCommGroup \u03b2 inst\u271d\u00b9 : Module \u211d \u03b2 inst\u271d : OrderedSMul \u211d \u03b2 s : Set E f : \u211d \u2192 \u03b2 a b : \u211d a_lt_b : a < b h_local_min : IsMinFilter f (\ud835\udcdd[Ici a] a) a h_conv : ConvexOn \u211d (Icc a b) f c : \u211d hc : c \u2208 Icc a b a_lt_c : a < c H\u2081 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, f a \u2264 f y H\u2082 : \u2200\u1da0 (y : \u211d) in \ud835\udcdd[Ioi a] a, y \u2208 Ioc a c ya yc : \u211d ya\u2080 : 0 \u2264 ya yc\u2080 : 0 < yc yac : ya + yc = 1 hfy : f a \u2264 f (ya * a + yc * c) hy_ac : ya * a + yc * c \u2208 Ioc a c \u22a2 ya \u2022 f a + yc \u2022 f a = f a ** rw [\u2190 add_smul, yac, one_smul] ** Qed", + "informal": "" + }, + { + "formal": "mem_pairSelfAdjointMatricesSubmodule' ** R : Type u_1 M : Type u_2 inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : AddCommMonoid M inst\u271d\u00b9\u2074 : Module R M R\u2081 : Type u_3 M\u2081 : Type u_4 inst\u271d\u00b9\u00b3 : Ring R\u2081 inst\u271d\u00b9\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9\u00b9 : Module R\u2081 M\u2081 R\u2082 : Type u_5 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : CommSemiring R\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 inst\u271d\u2078 : Module R\u2082 M\u2082 R\u2083 : Type u_7 M\u2083 : Type u_8 inst\u271d\u2077 : CommRing R\u2083 inst\u271d\u2076 : AddCommGroup M\u2083 inst\u271d\u2075 : Module R\u2083 M\u2083 V : Type u_9 K : Type u_10 inst\u271d\u2074 : Field K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V B : BilinForm R M B\u2081 : BilinForm R\u2081 M\u2081 B\u2082 : BilinForm R\u2082 M\u2082 n : Type u_11 inst\u271d\u00b9 : Fintype n b : Basis n R\u2083 M\u2083 J J\u2083 A A' : Matrix n n R\u2083 inst\u271d : DecidableEq n \u22a2 A \u2208 pairSelfAdjointMatricesSubmodule J J\u2083 \u2194 IsAdjointPair J J\u2083 A A ** simp only [mem_pairSelfAdjointMatricesSubmodule] ** Qed", + "informal": "" + }, + { + "formal": "Set.surjective_iff_surjOn_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 p : Set \u03b3 f f\u2081 f\u2082 f\u2083 : \u03b1 \u2192 \u03b2 g g\u2081 g\u2082 : \u03b2 \u2192 \u03b3 f' f\u2081' f\u2082' : \u03b2 \u2192 \u03b1 g' : \u03b3 \u2192 \u03b2 a : \u03b1 b : \u03b2 \u22a2 Surjective f \u2194 SurjOn f univ univ ** simp [Surjective, SurjOn, subset_def] ** Qed", + "informal": "" + }, + { + "formal": "Odd.nat_add_dvd_pow_add_pow ** \u03b1 : Type u x y n : \u2115 h : Odd n \u22a2 x + y \u2223 x ^ n + y ^ n ** exact_mod_cast Odd.add_dvd_pow_add_pow (x : \u2124) (\u2191y) h ** Qed", + "informal": "" + }, + { + "formal": "Submodule.prod_eq_top_iff ** R : Type u K : Type u' M : Type v V : Type v' M\u2082 : Type w V\u2082 : Type w' M\u2083 : Type y V\u2083 : Type y' M\u2084 : Type z \u03b9 : Type x M\u2085 : Type u_1 M\u2086 : Type u_2 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : AddCommMonoid M\u2082 inst\u271d\u00b9 : Module R M inst\u271d : Module R M\u2082 p : Submodule R M q : Submodule R M\u2082 p\u2081 : Submodule R M p\u2082 : Submodule R M\u2082 \u22a2 prod p\u2081 p\u2082 = \u22a4 \u2194 p\u2081 = \u22a4 \u2227 p\u2082 = \u22a4 ** simp only [eq_top_iff, le_prod_iff, \u2190 (gc_map_comap _).le_iff_le, map_top, range_fst, range_snd] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.Polynomial.isOpen_imageOfDf ** R : Type u_1 inst\u271d : CommRing R f : R[X] \u22a2 IsOpen (imageOfDf f) ** rw [imageOfDf, setOf_exists fun i (x : PrimeSpectrum R) => coeff f i \u2209 x.asIdeal] ** R : Type u_1 inst\u271d : CommRing R f : R[X] \u22a2 IsOpen (\u22c3 i, {x | \u00accoeff f i \u2208 x.asIdeal}) ** exact isOpen_iUnion fun i => isOpen_basicOpen ** Qed", + "informal": "" + }, + { + "formal": "Vector.mapAccumr_mapAccumr\u2082 ** \u03b1 : Type n : \u2115 \u03b2 : Type xs : Vector \u03b1 n ys : Vector \u03b2 n \u03b3 \u03c3\u2081 \u03b6 \u03c3\u2082 : Type s\u2082 : \u03c3\u2082 s\u2081 : \u03c3\u2081 f\u2081 : \u03b3 \u2192 \u03c3\u2081 \u2192 \u03c3\u2081 \u00d7 \u03b6 f\u2082 : \u03b1 \u2192 \u03b2 \u2192 \u03c3\u2082 \u2192 \u03c3\u2082 \u00d7 \u03b3 \u22a2 mapAccumr f\u2081 (mapAccumr\u2082 f\u2082 xs ys s\u2082).2 s\u2081 = let m := mapAccumr\u2082 (fun x y s => let r\u2082 := f\u2082 x y s.2; let r\u2081 := f\u2081 r\u2082.2 s.1; ((r\u2081.1, r\u2082.1), r\u2081.2)) xs ys (s\u2081, s\u2082); (m.1.1, m.2) ** induction xs, ys using Vector.revInductionOn\u2082 generalizing s\u2081 s\u2082 <;> simp_all ** Qed", + "informal": "" + }, + { + "formal": "mul_smul_one ** M\u271d : Type u_1 N\u271d : Type u_2 G : Type u_3 A : Type u_4 B : Type u_5 \u03b1 : Type u_6 \u03b2 : Type u_7 \u03b3 : Type u_8 \u03b4 : Type u_9 M : Type u_10 N : Type u_11 inst\u271d\u00b2 : MulOneClass N inst\u271d\u00b9 : SMul M N inst\u271d : SMulCommClass M N N x : M y : N \u22a2 y * x \u2022 1 = x \u2022 y ** rw [\u2190 smul_eq_mul, \u2190 smul_comm, smul_eq_mul, mul_one] ** Qed", + "informal": "" + }, + { + "formal": "NNReal.geom_mean_le_arith_mean3_weighted ** \u03b9 : Type u s : Finset \u03b9 w\u2081 w\u2082 w\u2083 p\u2081 p\u2082 p\u2083 : \u211d\u22650 \u22a2 w\u2081 + w\u2082 + w\u2083 = 1 \u2192 p\u2081 ^ \u2191w\u2081 * p\u2082 ^ \u2191w\u2082 * p\u2083 ^ \u2191w\u2083 \u2264 w\u2081 * p\u2081 + w\u2082 * p\u2082 + w\u2083 * p\u2083 ** simpa only [Fin.prod_univ_succ, Fin.sum_univ_succ, Finset.prod_empty, Finset.sum_empty,\n Fintype.univ_of_isEmpty, Fin.cons_succ, Fin.cons_zero, add_zero, mul_one, \u2190 add_assoc,\n mul_assoc] using geom_mean_le_arith_mean_weighted univ ![w\u2081, w\u2082, w\u2083] ![p\u2081, p\u2082, p\u2083] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.span_pair_add_mul_right ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : Semiring \u03b1 I : Ideal \u03b1 a b : \u03b1 R : Type u inst\u271d : CommRing R x y z : R \u22a2 span {x, y + x * z} = span {x, y} ** rw [span_pair_comm, span_pair_add_mul_left, span_pair_comm] ** Qed", + "informal": "" + }, + { + "formal": "mfderiv_comp ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2074 : NormedAddCommGroup E inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u00b2 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u00b9 : TopologicalSpace M inst\u271d\u00b9\u2070 : ChartedSpace H M E' : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2077 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2076 : TopologicalSpace M' inst\u271d\u2075 : ChartedSpace H' M' E'' : Type u_8 inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b9 : TopologicalSpace M'' inst\u271d : ChartedSpace H'' M'' f f\u2080 f\u2081 : M \u2192 M' x : M s t : Set M g : M' \u2192 M'' u : Set M' Is : SmoothManifoldWithCorners I M I's : SmoothManifoldWithCorners I' M' I''s : SmoothManifoldWithCorners I'' M'' f' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x) g' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x)) hg : MDifferentiableAt I' I'' g (f x) hf : MDifferentiableAt I I' f x \u22a2 mfderiv I I'' (g \u2218 f) x = ContinuousLinearMap.comp (mfderiv I' I'' g (f x)) (mfderiv I I' f x) ** apply HasMFDerivAt.mfderiv ** case h \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2074 : NormedAddCommGroup E inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b9\u00b2 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b9\u00b9 : TopologicalSpace M inst\u271d\u00b9\u2070 : ChartedSpace H M E' : Type u_5 inst\u271d\u2079 : NormedAddCommGroup E' inst\u271d\u2078 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u2077 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u2076 : TopologicalSpace M' inst\u271d\u2075 : ChartedSpace H' M' E'' : Type u_8 inst\u271d\u2074 : NormedAddCommGroup E'' inst\u271d\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b9 : TopologicalSpace M'' inst\u271d : ChartedSpace H'' M'' f f\u2080 f\u2081 : M \u2192 M' x : M s t : Set M g : M' \u2192 M'' u : Set M' Is : SmoothManifoldWithCorners I M I's : SmoothManifoldWithCorners I' M' I''s : SmoothManifoldWithCorners I'' M'' f' f\u2080' f\u2081' : TangentSpace I x \u2192L[\ud835\udd5c] TangentSpace I' (f x) g' : TangentSpace I' (f x) \u2192L[\ud835\udd5c] TangentSpace I'' (g (f x)) hg : MDifferentiableAt I' I'' g (f x) hf : MDifferentiableAt I I' f x \u22a2 HasMFDerivAt I I'' (g \u2218 f) x (ContinuousLinearMap.comp (mfderiv I' I'' g (f x)) (mfderiv I I' f x)) ** exact HasMFDerivAt.comp x hg.hasMFDerivAt hf.hasMFDerivAt ** Qed", + "informal": "" + }, + { + "formal": "Function.Surjective.nontrivial ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f \u22a2 Nontrivial \u03b1 ** rcases exists_pair_ne \u03b2 with \u27e8x, y, h\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x y : \u03b2 h : x \u2260 y \u22a2 Nontrivial \u03b1 ** rcases hf x with \u27e8x', hx'\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x y : \u03b2 h : x \u2260 y x' : \u03b1 hx' : f x' = x \u22a2 Nontrivial \u03b1 ** rcases hf y with \u27e8y', hy'\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x y : \u03b2 h : x \u2260 y x' : \u03b1 hx' : f x' = x y' : \u03b1 hy' : f y' = y \u22a2 Nontrivial \u03b1 ** have : x' \u2260 y' := by\n refine fun H \u21a6 h ?_\n rw [\u2190 hx', \u2190 hy', H] ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x y : \u03b2 h : x \u2260 y x' : \u03b1 hx' : f x' = x y' : \u03b1 hy' : f y' = y this : x' \u2260 y' \u22a2 Nontrivial \u03b1 ** exact \u27e8\u27e8x', y', this\u27e9\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x y : \u03b2 h : x \u2260 y x' : \u03b1 hx' : f x' = x y' : \u03b1 hy' : f y' = y \u22a2 x' \u2260 y' ** refine fun H \u21a6 h ?_ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : Nontrivial \u03b2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x y : \u03b2 h : x \u2260 y x' : \u03b1 hx' : f x' = x y' : \u03b1 hy' : f y' = y H : x' = y' \u22a2 x = y ** rw [\u2190 hx', \u2190 hy', H] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.mirror_smul ** R : Type u_1 inst\u271d\u00b9 : Ring R p q : R[X] inst\u271d : NoZeroDivisors R a : R \u22a2 mirror (a \u2022 p) = a \u2022 mirror p ** rw [\u2190 C_mul', \u2190 C_mul', mirror_mul_of_domain, mirror_C] ** Qed", + "informal": "" + }, + { + "formal": "HasFDerivAtFilter.sum ** \ud835\udd5c : Type u_1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \u03b9 : Type u_6 u : Finset \u03b9 A : \u03b9 \u2192 E \u2192 F A' : \u03b9 \u2192 E \u2192L[\ud835\udd5c] F h : \u2200 (i : \u03b9), i \u2208 u \u2192 HasFDerivAtFilter (A i) (A' i) x L \u22a2 HasFDerivAtFilter (fun y => \u2211 i in u, A i y) (\u2211 i in u, A' i) x L ** dsimp [HasFDerivAtFilter] at * ** \ud835\udd5c : Type u_1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \u03b9 : Type u_6 u : Finset \u03b9 A : \u03b9 \u2192 E \u2192 F A' : \u03b9 \u2192 E \u2192L[\ud835\udd5c] F h : \u2200 (i : \u03b9), i \u2208 u \u2192 (fun x' => A i x' - A i x - \u2191(A' i) (x' - x)) =o[L] fun x' => x' - x \u22a2 (fun x' => \u2211 i in u, A i x' - \u2211 i in u, A i x - \u2191(\u2211 i in u, A' i) (x' - x)) =o[L] fun x' => x' - x ** convert IsLittleO.sum h ** case h.e'_7.h \ud835\udd5c : Type u_1 inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' f f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \u03b9 : Type u_6 u : Finset \u03b9 A : \u03b9 \u2192 E \u2192 F A' : \u03b9 \u2192 E \u2192L[\ud835\udd5c] F h : \u2200 (i : \u03b9), i \u2208 u \u2192 (fun x' => A i x' - A i x - \u2191(A' i) (x' - x)) =o[L] fun x' => x' - x x\u271d : E \u22a2 \u2211 i in u, A i x\u271d - \u2211 i in u, A i x - \u2191(\u2211 i in u, A' i) (x\u271d - x) = \u2211 i in u, (A i x\u271d - A i x - \u2191(A' i) (x\u271d - x)) ** simp [ContinuousLinearMap.sum_apply] ** Qed", + "informal": "" + }, + { + "formal": "Equiv.swap_apply_eq_iff ** \u03b1 : Sort u_1 inst\u271d : DecidableEq \u03b1 x y z w : \u03b1 \u22a2 \u2191(swap x y) z = w \u2194 z = \u2191(swap x y) w ** rw [apply_eq_iff_eq_symm_apply, symm_swap] ** Qed", + "informal": "" + }, + { + "formal": "lowerSemicontinuousAt_iSup ** \u03b1 : Type u_1 inst\u271d\u00b3 : TopologicalSpace \u03b1 \u03b2 : Type u_2 inst\u271d\u00b2 : Preorder \u03b2 f\u271d g : \u03b1 \u2192 \u03b2 x : \u03b1 s t : Set \u03b1 y z : \u03b2 \u03b9 : Sort u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 inst\u271d\u00b9 : CompleteLinearOrder \u03b4 inst\u271d : ConditionallyCompleteLinearOrder \u03b4' f : \u03b9 \u2192 \u03b1 \u2192 \u03b4 h : \u2200 (i : \u03b9), LowerSemicontinuousAt (f i) x \u22a2 \u2200\u1da0 (y : \u03b1) in \ud835\udcdd x, BddAbove (range fun i => f i y) ** simp ** Qed", + "informal": "" + }, + { + "formal": "List.get?_pmap ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 h : \u2200 (a : \u03b1), a \u2208 l \u2192 p a n : \u2115 \u22a2 get? (pmap f l h) n = Option.pmap f (get? l n) (_ : \u2200 (x : \u03b1), x \u2208 get? l n \u2192 p x) ** induction' l with hd tl hl generalizing n ** case nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 h\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a n\u271d : \u2115 h : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a n : \u2115 \u22a2 get? (pmap f [] h) n = Option.pmap f (get? [] n) (_ : \u2200 (x : \u03b1), x \u2208 get? [] n \u2192 p x) ** simp ** case cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 h\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a n\u271d : \u2115 hd : \u03b1 tl : List \u03b1 hl : \u2200 (h : \u2200 (a : \u03b1), a \u2208 tl \u2192 p a) (n : \u2115), get? (pmap f tl h) n = Option.pmap f (get? tl n) (_ : \u2200 (x : \u03b1), x \u2208 get? tl n \u2192 p x) h : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 p a n : \u2115 \u22a2 get? (pmap f (hd :: tl) h) n = Option.pmap f (get? (hd :: tl) n) (_ : \u2200 (x : \u03b1), x \u2208 get? (hd :: tl) n \u2192 p x) ** cases' n with n ** case cons.zero \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 h\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a n : \u2115 hd : \u03b1 tl : List \u03b1 hl : \u2200 (h : \u2200 (a : \u03b1), a \u2208 tl \u2192 p a) (n : \u2115), get? (pmap f tl h) n = Option.pmap f (get? tl n) (_ : \u2200 (x : \u03b1), x \u2208 get? tl n \u2192 p x) h : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 p a \u22a2 get? (pmap f (hd :: tl) h) zero = Option.pmap f (get? (hd :: tl) zero) (_ : \u2200 (x : \u03b1), x \u2208 get? (hd :: tl) zero \u2192 p x) ** simp ** case cons.succ \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 h\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a n\u271d : \u2115 hd : \u03b1 tl : List \u03b1 hl : \u2200 (h : \u2200 (a : \u03b1), a \u2208 tl \u2192 p a) (n : \u2115), get? (pmap f tl h) n = Option.pmap f (get? tl n) (_ : \u2200 (x : \u03b1), x \u2208 get? tl n \u2192 p x) h : \u2200 (a : \u03b1), a \u2208 hd :: tl \u2192 p a n : \u2115 \u22a2 get? (pmap f (hd :: tl) h) (succ n) = Option.pmap f (get? (hd :: tl) (succ n)) (_ : \u2200 (x : \u03b1), x \u2208 get? (hd :: tl) (succ n) \u2192 p x) ** simp [hl] ** Qed", + "informal": "" + }, + { + "formal": "IsTorsionFree.quotient_torsion ** G : Type u_1 H : Type u_2 inst\u271d : CommGroup G g : G \u29f8 torsion G hne : g \u2260 1 hfin : IsOfFinOrder g \u22a2 g = 1 ** induction' g using QuotientGroup.induction_on' with g ** case H G : Type u_1 H : Type u_2 inst\u271d : CommGroup G g\u271d : G \u29f8 torsion G hne\u271d : g\u271d \u2260 1 hfin\u271d : IsOfFinOrder g\u271d g : G hne : \u2191g \u2260 1 hfin : IsOfFinOrder \u2191g \u22a2 \u2191g = 1 ** obtain \u27e8m, mpos, hm\u27e9 := (isOfFinOrder_iff_pow_eq_one _).mp hfin ** case H.intro.intro G : Type u_1 H : Type u_2 inst\u271d : CommGroup G g\u271d : G \u29f8 torsion G hne\u271d : g\u271d \u2260 1 hfin\u271d : IsOfFinOrder g\u271d g : G hne : \u2191g \u2260 1 hfin : IsOfFinOrder \u2191g m : \u2115 mpos : 0 < m hm : \u2191g ^ m = 1 \u22a2 \u2191g = 1 ** obtain \u27e8n, npos, hn\u27e9 := (isOfFinOrder_iff_pow_eq_one _).mp ((QuotientGroup.eq_one_iff _).mp hm) ** case H.intro.intro.intro.intro G : Type u_1 H : Type u_2 inst\u271d : CommGroup G g\u271d : G \u29f8 torsion G hne\u271d : g\u271d \u2260 1 hfin\u271d : IsOfFinOrder g\u271d g : G hne : \u2191g \u2260 1 hfin : IsOfFinOrder \u2191g m : \u2115 mpos : 0 < m hm : \u2191g ^ m = 1 n : \u2115 npos : 0 < n hn : (fun x => x ^ m) g ^ n = 1 \u22a2 \u2191g = 1 ** exact\n (QuotientGroup.eq_one_iff g).mpr\n ((isOfFinOrder_iff_pow_eq_one _).mpr \u27e8m * n, mul_pos mpos npos, (pow_mul g m n).symm \u25b8 hn\u27e9) ** Qed", + "informal": "" + }, + { + "formal": "Nat.choose_pos ** x\u271d : \u2115 hk : x\u271d \u2264 0 \u22a2 0 < choose 0 x\u271d ** rw [Nat.eq_zero_of_le_zero hk] ** x\u271d : \u2115 hk : x\u271d \u2264 0 \u22a2 0 < choose 0 0 ** decide ** n : \u2115 x\u271d : 0 \u2264 n + 1 \u22a2 0 < choose (n + 1) 0 ** simp ** n k : \u2115 hk : k + 1 \u2264 n + 1 \u22a2 0 < choose (n + 1) (k + 1) ** rw [choose_succ_succ] ** n k : \u2115 hk : k + 1 \u2264 n + 1 \u22a2 0 < choose n k + choose n (succ k) ** exact add_pos_of_pos_of_nonneg (choose_pos (le_of_succ_le_succ hk)) (Nat.zero_le _) ** Qed", + "informal": "" + }, + { + "formal": "Set.mem_of_mem_coe ** \u03b1 \u03b2\u271d : Type u s : Set \u03b1 f : \u03b1 \u2192 Set \u03b2\u271d g : Set (\u03b1 \u2192 \u03b2\u271d) \u03b2 : Set \u03b1 \u03b3 : Set \u2191\u03b2 a : \u03b1 ha : a \u2208 Lean.Internal.coeM \u03b3 \u22a2 { val := a, property := (_ : a \u2208 \u03b2) } \u2208 \u03b3 ** rcases ha with \u27e8_, \u27e8_, rfl\u27e9, _, \u27e8ha, rfl\u27e9, _\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1 \u03b2\u271d : Type u s : Set \u03b1 f : \u03b1 \u2192 Set \u03b2\u271d g : Set (\u03b1 \u2192 \u03b2\u271d) \u03b2 : Set \u03b1 \u03b3 : Set \u2191\u03b2 a : \u03b1 w\u271d : \u2191\u03b2 ha : w\u271d \u2208 \u03b3 right\u271d : a \u2208 (fun h => (fun a => pure (CoeT.coe a)) w\u271d) ha \u22a2 { val := a, property := (_ : a \u2208 \u03b2) } \u2208 \u03b3 ** convert ha ** Qed", + "informal": "" + }, + { + "formal": "Orientation.inner_rightAngleRotation_swap' ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E \u22a2 inner (\u2191(rightAngleRotation o) x) y = -inner x (\u2191(rightAngleRotation o) y) ** simp [o.inner_rightAngleRotation_swap x y] ** Qed", + "informal": "" + }, + { + "formal": "Ultrafilter.eq_pure_of_finite_mem ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 f g : Ultrafilter \u03b1 s t : Set \u03b1 p q : \u03b1 \u2192 Prop h : Set.Finite s h' : s \u2208 f \u22a2 \u2203 x, x \u2208 s \u2227 f = pure x ** rw [\u2190 biUnion_of_singleton s] at h' ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 f g : Ultrafilter \u03b1 s t : Set \u03b1 p q : \u03b1 \u2192 Prop h : Set.Finite s h' : \u22c3 x \u2208 s, {x} \u2208 f \u22a2 \u2203 x, x \u2208 s \u2227 f = pure x ** rcases (Ultrafilter.finite_biUnion_mem_iff h).mp h' with \u27e8a, has, haf\u27e9 ** case intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 f g : Ultrafilter \u03b1 s t : Set \u03b1 p q : \u03b1 \u2192 Prop h : Set.Finite s h' : \u22c3 x \u2208 s, {x} \u2208 f a : \u03b1 has : a \u2208 s haf : {a} \u2208 f \u22a2 \u2203 x, x \u2208 s \u2227 f = pure x ** exact \u27e8a, has, eq_of_le (Filter.le_pure_iff.2 haf)\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Complex.rat_cast_re ** q : \u211a \u22a2 (\u2191q).re = \u2191q ** show (Rat.castRec q : \u2102).re = _ ** q : \u211a \u22a2 (Rat.castRec q).re = \u2191q ** cases q ** case mk' num\u271d : \u2124 den\u271d : \u2115 den_nz\u271d : den\u271d \u2260 0 reduced\u271d : Nat.Coprime (Int.natAbs num\u271d) den\u271d \u22a2 (Rat.castRec (Rat.mk' num\u271d den\u271d)).re = \u2191(Rat.mk' num\u271d den\u271d) ** simp [Rat.castRec, normSq, Rat.mk_eq_divInt, Rat.mkRat_eq_div, div_eq_mul_inv, *] ** Qed", + "informal": "" + }, + { + "formal": "GeneralizedContinuedFraction.squashSeq_succ_n_tail_eq_squashSeq_tail_n ** K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n ** cases' s_succ_succ_nth_eq : s.get? (n + 2) with gp_succ_succ_n ** case none K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = none \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n case some K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n ** case none =>\n cases s_succ_nth_eq : s.get? (n + 1) <;>\n simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq] ** K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = none \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n ** cases s_succ_nth_eq : s.get? (n + 1) <;>\n simp only [squashSeq, Stream'.Seq.get?_tail, s_succ_nth_eq, s_succ_succ_nth_eq] ** K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n ** obtain \u27e8gp_succ_n, s_succ_nth_eq\u27e9 : \u2203 gp_succ_n, s.get? (n + 1) = some gp_succ_n ** K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n \u22a2 \u2203 gp_succ_n, Stream'.Seq.get? s (n + 1) = some gp_succ_n case intro K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n ** exact s.ge_stable (n + 1).le_succ s_succ_succ_nth_eq ** case intro K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n \u22a2 Stream'.Seq.tail (squashSeq s (n + 1)) = squashSeq (Stream'.Seq.tail s) n ** ext1 m ** case intro.h K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** cases' Decidable.em (m = n) with m_eq_n m_ne_n ** case intro.h.inl K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_eq_n : m = n \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** simp [*, squashSeq] ** case intro.h.inr K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** have : s.tail.get? m = s.get? (m + 1) := s.get?_tail m ** case intro.h.inr K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1) \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** cases s_succ_mth_eq : s.get? (m + 1) ** case intro.h.inr.none K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1) s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m case intro.h.inr.some K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1) val\u271d : Pair K s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val\u271d \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** all_goals have _ := this.trans s_succ_mth_eq ** case intro.h.inr.some K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1) val\u271d : Pair K s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val\u271d \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** have _ := this.trans s_succ_mth_eq ** case intro.h.inr.none K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1) s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = none x\u271d : Stream'.Seq.get? (Stream'.Seq.tail s) m = none \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** simp only [*, squashSeq, Stream'.Seq.get?_tail, Stream'.Seq.get?_zipWith,\n Option.map\u2082_none_right] ** case intro.h.inr.some K : Type u_1 n : \u2115 g : GeneralizedContinuedFraction K s : Stream'.Seq (Pair K) inst\u271d : DivisionRing K gp_succ_succ_n : Pair K s_succ_succ_nth_eq : Stream'.Seq.get? s (n + 2) = some gp_succ_succ_n gp_succ_n : Pair K s_succ_nth_eq : Stream'.Seq.get? s (n + 1) = some gp_succ_n m : \u2115 m_ne_n : \u00acm = n this : Stream'.Seq.get? (Stream'.Seq.tail s) m = Stream'.Seq.get? s (m + 1) val\u271d : Pair K s_succ_mth_eq : Stream'.Seq.get? s (m + 1) = some val\u271d x\u271d : Stream'.Seq.get? (Stream'.Seq.tail s) m = some val\u271d \u22a2 Stream'.Seq.get? (Stream'.Seq.tail (squashSeq s (n + 1))) m = Stream'.Seq.get? (squashSeq (Stream'.Seq.tail s) n) m ** simp [*, squashSeq] ** Qed", + "informal": "" + }, + { + "formal": "exists_zpow_eq_one ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Group G x\u271d y : G n : \u2115 inst\u271d : Finite G x : G \u22a2 \u2203 i x_1, x ^ i = 1 ** rcases exists_pow_eq_one x with \u27e8w, hw1, hw2\u27e9 ** case intro.intro G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Group G x\u271d y : G n : \u2115 inst\u271d : Finite G x : G w : \u2115 hw1 : w > 0 hw2 : IsPeriodicPt ((fun x x_1 => x * x_1) x) w 1 \u22a2 \u2203 i x_1, x ^ i = 1 ** refine' \u27e8w, Int.coe_nat_ne_zero.mpr (_root_.ne_of_gt hw1), _\u27e9 ** case intro.intro G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Group G x\u271d y : G n : \u2115 inst\u271d : Finite G x : G w : \u2115 hw1 : w > 0 hw2 : IsPeriodicPt ((fun x x_1 => x * x_1) x) w 1 \u22a2 x ^ \u2191w = 1 ** rw [zpow_ofNat] ** case intro.intro G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Group G x\u271d y : G n : \u2115 inst\u271d : Finite G x : G w : \u2115 hw1 : w > 0 hw2 : IsPeriodicPt ((fun x x_1 => x * x_1) x) w 1 \u22a2 x ^ w = 1 ** exact (isPeriodicPt_mul_iff_pow_eq_one _).mp hw2 ** Qed", + "informal": "" + }, + { + "formal": "IsAlgClosed.exists_eq_mul_self ** k : Type u inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k x : k \u22a2 \u2203 z, x = z * z ** rcases exists_pow_nat_eq x zero_lt_two with \u27e8z, rfl\u27e9 ** case intro k : Type u inst\u271d\u00b9 : Field k inst\u271d : IsAlgClosed k z : k \u22a2 \u2203 z_1, z ^ 2 = z_1 * z_1 ** exact \u27e8z, sq z\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Nat.choose_mul ** n k s : \u2115 hkn : k \u2264 n hsk : s \u2264 k \u22a2 (n - k)! * (k - s)! * s ! \u2260 0 ** apply_rules [factorial_ne_zero, mul_ne_zero] ** n k s : \u2115 hkn : k \u2264 n hsk : s \u2264 k h : (n - k)! * (k - s)! * s ! \u2260 0 \u22a2 choose n k * choose k s * ((n - k)! * (k - s)! * s !) = choose n k * (choose k s * s ! * (k - s)!) * (n - k)! ** rw [mul_assoc, mul_assoc, mul_assoc, mul_assoc _ s !, mul_assoc, mul_comm (n - k)!,\nmul_comm s !] ** n k s : \u2115 hkn : k \u2264 n hsk : s \u2264 k h : (n - k)! * (k - s)! * s ! \u2260 0 \u22a2 choose n k * (choose k s * s ! * (k - s)!) * (n - k)! = n ! ** rw [choose_mul_factorial_mul_factorial hsk, choose_mul_factorial_mul_factorial hkn] ** n k s : \u2115 hkn : k \u2264 n hsk : s \u2264 k h : (n - k)! * (k - s)! * s ! \u2260 0 \u22a2 n ! = choose n s * s ! * (choose (n - s) (k - s) * (k - s)! * (n - s - (k - s))!) ** rw [choose_mul_factorial_mul_factorial (tsub_le_tsub_right hkn _),\nchoose_mul_factorial_mul_factorial (hsk.trans hkn)] ** n k s : \u2115 hkn : k \u2264 n hsk : s \u2264 k h : (n - k)! * (k - s)! * s ! \u2260 0 \u22a2 choose n s * s ! * (choose (n - s) (k - s) * (k - s)! * (n - s - (k - s))!) = choose n s * choose (n - s) (k - s) * ((n - k)! * (k - s)! * s !) ** rw [tsub_tsub_tsub_cancel_right hsk, mul_assoc, mul_left_comm s !, mul_assoc,\nmul_comm (k - s)!, mul_comm s !, mul_right_comm, \u2190 mul_assoc] ** Qed", + "informal": "" + }, + { + "formal": "IsHilbertSum.mk ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u2075 : IsROrC \ud835\udd5c E : Type u_3 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E cplt : CompleteSpace E G : \u03b9 \u2192 Type u_4 inst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i) inst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i) V : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E F : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i) hVortho : OrthogonalFamily \ud835\udd5c G V hVtotal : \u22a4 \u2264 topologicalClosure (\u2a06 i, LinearMap.range (V i).toLinearMap) \u22a2 Function.Surjective \u2191(OrthogonalFamily.linearIsometry hVortho) ** rw [\u2190 LinearIsometry.coe_toLinearMap] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u2075 : IsROrC \ud835\udd5c E : Type u_3 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E cplt : CompleteSpace E G : \u03b9 \u2192 Type u_4 inst\u271d\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (G i) inst\u271d\u00b9 : (i : \u03b9) \u2192 InnerProductSpace \ud835\udd5c (G i) V : (i : \u03b9) \u2192 G i \u2192\u2097\u1d62[\ud835\udd5c] E F : \u03b9 \u2192 Submodule \ud835\udd5c E inst\u271d : \u2200 (i : \u03b9), CompleteSpace (G i) hVortho : OrthogonalFamily \ud835\udd5c G V hVtotal : \u22a4 \u2264 topologicalClosure (\u2a06 i, LinearMap.range (V i).toLinearMap) \u22a2 Function.Surjective \u2191(OrthogonalFamily.linearIsometry hVortho).toLinearMap ** exact LinearMap.range_eq_top.mp\n (eq_top_iff.mpr <| hVtotal.trans_eq hVortho.range_linearIsometry.symm) ** Qed", + "informal": "" + }, + { + "formal": "IsPrincipalIdealRing.of_finite_primes ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R h : Set.Finite {I | IsPrime I} I : Ideal R \u22a2 Submodule.IsPrincipal I ** obtain rfl | hI := eq_or_ne I \u22a5 ** case inr R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R h : Set.Finite {I | IsPrime I} I : Ideal R hI : I \u2260 \u22a5 \u22a2 Submodule.IsPrincipal I ** apply Ideal.IsPrincipal.of_finite_maximals_of_isUnit ** case inl R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R h : Set.Finite {I | IsPrime I} \u22a2 Submodule.IsPrincipal \u22a5 ** exact bot_isPrincipal ** case inr.hf R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R h : Set.Finite {I | IsPrime I} I : Ideal R hI : I \u2260 \u22a5 \u22a2 Set.Finite {I | IsMaximal I} ** apply h.subset ** case inr.hf R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R h : Set.Finite {I | IsPrime I} I : Ideal R hI : I \u2260 \u22a5 \u22a2 {I | IsMaximal I} \u2286 {I | IsPrime I} ** exact @Ideal.IsMaximal.isPrime _ _ ** case inr.hI R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : IsDedekindDomain R h : Set.Finite {I | IsPrime I} I : Ideal R hI : I \u2260 \u22a5 \u22a2 IsUnit \u2191I ** exact isUnit_of_mul_eq_one _ _ (FractionalIdeal.coe_ideal_mul_inv I hI) ** Qed", + "informal": "" + }, + { + "formal": "HahnSeries.C_injective ** \u0393 : Type u_1 R : Type u_2 inst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393 inst\u271d : NonAssocSemiring R \u22a2 Injective \u2191C ** intro r s rs ** \u0393 : Type u_1 R : Type u_2 inst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393 inst\u271d : NonAssocSemiring R r s : R rs : \u2191C r = \u2191C s \u22a2 r = s ** rw [HahnSeries.ext_iff, Function.funext_iff] at rs ** \u0393 : Type u_1 R : Type u_2 inst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393 inst\u271d : NonAssocSemiring R r s : R rs : \u2200 (a : \u0393), coeff (\u2191C r) a = coeff (\u2191C s) a \u22a2 r = s ** have h := rs 0 ** \u0393 : Type u_1 R : Type u_2 inst\u271d\u00b9 : OrderedCancelAddCommMonoid \u0393 inst\u271d : NonAssocSemiring R r s : R rs : \u2200 (a : \u0393), coeff (\u2191C r) a = coeff (\u2191C s) a h : coeff (\u2191C r) 0 = coeff (\u2191C s) 0 \u22a2 r = s ** rwa [C_apply, single_coeff_same, C_apply, single_coeff_same] at h ** Qed", + "informal": "" + }, + { + "formal": "inter_maximals_preimage_inter_eq_of_rel_iff_rel_on ** \u03b2 : Type u_2 \u03b1 : Type u_1 r\u271d r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d t : Set \u03b1 a b : \u03b1 inst\u271d : PartialOrder \u03b1 f : \u03b1 \u2192 \u03b2 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop x : Set \u03b1 hf : \u2200 \u2983a a' : \u03b1\u2984, a \u2208 x \u2192 a' \u2208 x \u2192 (r a a' \u2194 s (f a) (f a')) y : Set \u03b2 \u22a2 x \u2229 f \u207b\u00b9' maximals s (f '' x \u2229 y) = maximals r (x \u2229 f \u207b\u00b9' y) ** apply inter_minimals_preimage_inter_eq_of_rel_iff_rel_on ** case hf \u03b2 : Type u_2 \u03b1 : Type u_1 r\u271d r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop s\u271d t : Set \u03b1 a b : \u03b1 inst\u271d : PartialOrder \u03b1 f : \u03b1 \u2192 \u03b2 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop x : Set \u03b1 hf : \u2200 \u2983a a' : \u03b1\u2984, a \u2208 x \u2192 a' \u2208 x \u2192 (r a a' \u2194 s (f a) (f a')) y : Set \u03b2 \u22a2 \u2200 \u2983a a' : \u03b1\u2984, a \u2208 x \u2192 a' \u2208 x \u2192 (r a' a \u2194 s (f a') (f a)) ** exact fun _ _ a b \u21a6 hf b a ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.exists_finset_of_splits ** F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f \u22a2 \u2203 s, map i f = \u2191C (\u2191i (leadingCoeff f)) * \u220f a in s, (X - \u2191C a) ** obtain \u27e8s, h\u27e9 := (splits_iff_exists_multiset _).1 sp ** case intro F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 \u2203 s, map i f = \u2191C (\u2191i (leadingCoeff f)) * \u220f a in s, (X - \u2191C a) ** use s.toFinset ** case h F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 map i f = \u2191C (\u2191i (leadingCoeff f)) * \u220f a in Multiset.toFinset s, (X - \u2191C a) ** rw [h, Finset.prod_eq_multiset_prod, \u2190 Multiset.toFinset_eq] ** case h F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 Multiset.Nodup s ** apply nodup_of_separable_prod ** case h.hs F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 Separable (Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)) ** apply Separable.of_mul_right ** case h.hs.h F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 Separable (?h.hs.f * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s)) case h.hs.f F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 K[X] ** rw [\u2190 h] ** case h.hs.h F : Type u inst\u271d\u00b9 : Field F K : Type v inst\u271d : Field K i\u271d i : F \u2192+* K f : F[X] sep : Separable f sp : Splits i f s : Multiset K h : map i f = \u2191C (\u2191i (leadingCoeff f)) * Multiset.prod (Multiset.map (fun a => X - \u2191C a) s) \u22a2 Separable (map i f) ** exact sep.map ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.dist_div_tan_oangle_left_of_oangle_eq_pi_div_two ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2) \u22a2 dist p\u2083 p\u2082 / Real.Angle.tan (\u2221 p\u2083 p\u2081 p\u2082) = dist p\u2081 p\u2082 ** have hs : (\u2221 p\u2083 p\u2081 p\u2082).sign = 1 := by rw [\u2190 oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2) hs : Real.Angle.sign (\u2221 p\u2083 p\u2081 p\u2082) = 1 \u22a2 dist p\u2083 p\u2082 / Real.Angle.tan (\u2221 p\u2083 p\u2081 p\u2082) = dist p\u2081 p\u2082 ** rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.tan_coe,\n dist_div_tan_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)\n (Or.inl (right_ne_of_oangle_eq_pi_div_two h))] ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) p\u2081 p\u2082 p\u2083 : P h : \u2221 p\u2081 p\u2082 p\u2083 = \u2191(\u03c0 / 2) \u22a2 Real.Angle.sign (\u2221 p\u2083 p\u2081 p\u2082) = 1 ** rw [\u2190 oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] ** Qed", + "informal": "" + }, + { + "formal": "separatedNhds_iff_disjoint ** \u03b1 : Type u \u03b2 : Type v inst\u271d : TopologicalSpace \u03b1 s t : Set \u03b1 \u22a2 SeparatedNhds s t \u2194 Disjoint (\ud835\udcdd\u02e2 s) (\ud835\udcdd\u02e2 t) ** simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, \u2190\n exists_and_left, and_assoc, and_comm, and_left_comm] ** Qed", + "informal": "" + }, + { + "formal": "jacobiSym.quadratic_reciprocity_one_mod_four ** a b : \u2115 ha : a % 4 = 1 hb : Odd b \u22a2 J(\u2191a | b) = J(\u2191b | a) ** rw [quadratic_reciprocity (odd_iff.mpr (odd_of_mod_four_eq_one ha)) hb, pow_mul,\n neg_one_pow_div_two_of_one_mod_four ha, one_pow, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "mulSalemSpencer_mul_right_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \ud835\udd5c : Type u_4 E : Type u_5 inst\u271d : CancelCommMonoid \u03b1 s : Set \u03b1 a : \u03b1 hs : MulSalemSpencer ((fun x => x * a) '' s) b c d : \u03b1 hb : b \u2208 s hc : c \u2208 s hd : d \u2208 s h : b * c = d * d \u22a2 b * a * (c * a) = d * a * (d * a) ** rw [mul_mul_mul_comm, h, mul_mul_mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "List.splitWrtComposition_join ** n : \u2115 \u03b1 : Type u_1 L : List (List \u03b1) c : Composition (length (join L)) h : map length L = c.blocks \u22a2 splitWrtComposition (join L) c = L ** simp only [eq_self_iff_true, and_self_iff, eq_iff_join_eq, join_splitWrtComposition,\n map_length_splitWrtComposition, h] ** Qed", + "informal": "" + }, + { + "formal": "List.take_all_of_le ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 n : \u2115 a : \u03b1 l : List \u03b1 h : length (a :: l) \u2264 n + 1 \u22a2 take (n + 1) (a :: l) = a :: l ** change a :: take n l = a :: l ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 n : \u2115 a : \u03b1 l : List \u03b1 h : length (a :: l) \u2264 n + 1 \u22a2 a :: take n l = a :: l ** rw [take_all_of_le (le_of_succ_le_succ h)] ** Qed", + "informal": "" + }, + { + "formal": "ModelWithCorners.extChartAt_transDiffeomorph_target ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E E' : Type u_3 inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E' F : Type u_4 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F H : Type u_5 inst\u271d\u00b9\u00b9 : TopologicalSpace H H' : Type u_6 inst\u271d\u00b9\u2070 : TopologicalSpace H' G : Type u_7 inst\u271d\u2079 : TopologicalSpace G G' : Type u_8 inst\u271d\u2078 : TopologicalSpace G' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' J : ModelWithCorners \ud835\udd5c F G J' : ModelWithCorners \ud835\udd5c F G' M : Type u_9 inst\u271d\u2077 : TopologicalSpace M inst\u271d\u2076 : ChartedSpace H M M' : Type u_10 inst\u271d\u2075 : TopologicalSpace M' inst\u271d\u2074 : ChartedSpace H' M' N : Type u_11 inst\u271d\u00b3 : TopologicalSpace N inst\u271d\u00b2 : ChartedSpace G N N' : Type u_12 inst\u271d\u00b9 : TopologicalSpace N' inst\u271d : ChartedSpace G' N' n : \u2115\u221e e : E \u2243\u2098\u27ee\ud835\udcd8(\ud835\udd5c, E), \ud835\udcd8(\ud835\udd5c, E')\u27ef E' x : M \u22a2 (extChartAt (transDiffeomorph I e) x).target = \u2191e.symm \u207b\u00b9' (extChartAt I x).target ** simp only [e.range_comp, preimage_preimage, mfld_simps] ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2078 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u2077 : NormedAddCommGroup E inst\u271d\u00b9\u2076 : NormedSpace \ud835\udd5c E E' : Type u_3 inst\u271d\u00b9\u2075 : NormedAddCommGroup E' inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c E' F : Type u_4 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F H : Type u_5 inst\u271d\u00b9\u00b9 : TopologicalSpace H H' : Type u_6 inst\u271d\u00b9\u2070 : TopologicalSpace H' G : Type u_7 inst\u271d\u2079 : TopologicalSpace G G' : Type u_8 inst\u271d\u2078 : TopologicalSpace G' I : ModelWithCorners \ud835\udd5c E H I' : ModelWithCorners \ud835\udd5c E' H' J : ModelWithCorners \ud835\udd5c F G J' : ModelWithCorners \ud835\udd5c F G' M : Type u_9 inst\u271d\u2077 : TopologicalSpace M inst\u271d\u2076 : ChartedSpace H M M' : Type u_10 inst\u271d\u2075 : TopologicalSpace M' inst\u271d\u2074 : ChartedSpace H' M' N : Type u_11 inst\u271d\u00b3 : TopologicalSpace N inst\u271d\u00b2 : ChartedSpace G N N' : Type u_12 inst\u271d\u00b9 : TopologicalSpace N' inst\u271d : ChartedSpace G' N' n : \u2115\u221e e : E \u2243\u2098\u27ee\ud835\udcd8(\ud835\udd5c, E), \ud835\udcd8(\ud835\udd5c, E')\u27ef E' x : M \u22a2 \u2191e.symm \u207b\u00b9' range \u2191I \u2229 \u2191(ModelWithCorners.symm I) \u2218 \u2191e.symm \u207b\u00b9' (chartAt H x).toLocalEquiv.target = \u2191e.symm \u207b\u00b9' range \u2191I \u2229 (fun x => \u2191(ModelWithCorners.symm I) (\u2191e.symm x)) \u207b\u00b9' (chartAt H x).toLocalEquiv.target ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Ideal.Quotient.subsingleton_iff ** R : Type u inst\u271d : CommRing R I\u271d : Ideal R a b : R S : Type v x y : R I : Ideal R \u22a2 Subsingleton (R \u29f8 I) \u2194 I = \u22a4 ** rw [eq_top_iff_one, \u2190 subsingleton_iff_zero_eq_one, eq_comm, \u2190 (mk I).map_one,\n Quotient.eq_zero_iff_mem] ** Qed", + "informal": "" + }, + { + "formal": "Bitraversable.tsnd_eq_snd_id ** t : Type u \u2192 Type u \u2192 Type u inst\u271d\u2075 : Bitraversable t \u03b2\u271d : Type u F G : Type u \u2192 Type u inst\u271d\u2074 : Applicative F inst\u271d\u00b3 : Applicative G inst\u271d\u00b2 : LawfulBitraversable t inst\u271d\u00b9 : LawfulApplicative F inst\u271d : LawfulApplicative G \u03b1 \u03b2 \u03b2' : Type u f : \u03b2 \u2192 \u03b2' x : t \u03b1 \u03b2 \u22a2 tsnd (pure \u2218 f) x = pure (snd f x) ** apply bitraverse_eq_bimap_id ** Qed", + "informal": "" + }, + { + "formal": "NormedRing.inverse_add_norm ** R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 \u22a2 (fun t => inverse (\u2191x + t)) =O[\ud835\udcdd 0] fun _t => 1 ** refine EventuallyEq.trans_isBigO (inverse_add x) (one_mul (1 : \u211d) \u25b8 ?_) ** R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 \u22a2 (fun x_1 => inverse (1 + \u2191x\u207b\u00b9 * x_1) * \u2191x\u207b\u00b9) =O[\ud835\udcdd 0] fun _t => 1 * 1 ** simp only [\u2190 sub_neg_eq_add, \u2190 neg_mul] ** R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R\u02e3 hzero : Tendsto (fun x_1 => -\u2191x\u207b\u00b9 * x_1) (\ud835\udcdd 0) (\ud835\udcdd 0) \u22a2 (fun x_1 => inverse (1 - -\u2191x\u207b\u00b9 * x_1) * \u2191x\u207b\u00b9) =O[\ud835\udcdd 0] fun _t => 1 * 1 ** exact (inverse_one_sub_norm.comp_tendsto hzero).mul (isBigO_const_const _ one_ne_zero _) ** Qed", + "informal": "" + }, + { + "formal": "Order.not_isSuccLimit_iff ** \u03b1 : Type u_1 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : SuccOrder \u03b1 a b : \u03b1 C : \u03b1 \u2192 Sort u_2 \u22a2 \u00acIsSuccLimit a \u2194 \u2203 b, \u00acIsMax b \u2227 succ b = a ** rw [not_isSuccLimit_iff_exists_covby] ** \u03b1 : Type u_1 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : SuccOrder \u03b1 a b : \u03b1 C : \u03b1 \u2192 Sort u_2 \u22a2 (\u2203 b, b \u22d6 a) \u2194 \u2203 b, \u00acIsMax b \u2227 succ b = a ** refine' exists_congr fun b => \u27e8fun hba => \u27e8hba.lt.not_isMax, (Covby.succ_eq hba)\u27e9, _\u27e9 ** \u03b1 : Type u_1 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : SuccOrder \u03b1 a b\u271d : \u03b1 C : \u03b1 \u2192 Sort u_2 b : \u03b1 \u22a2 \u00acIsMax b \u2227 succ b = a \u2192 b \u22d6 a ** rintro \u27e8h, rfl\u27e9 ** case intro \u03b1 : Type u_1 inst\u271d\u00b9 : PartialOrder \u03b1 inst\u271d : SuccOrder \u03b1 b\u271d : \u03b1 C : \u03b1 \u2192 Sort u_2 b : \u03b1 h : \u00acIsMax b \u22a2 b \u22d6 succ b ** exact covby_succ_of_not_isMax h ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.finrank_range_of_inj ** K : Type u V : Type v inst\u271d\u2074 : Ring K inst\u271d\u00b3 : AddCommGroup V inst\u271d\u00b2 : Module K V V\u2082 : Type v' inst\u271d\u00b9 : AddCommGroup V\u2082 inst\u271d : Module K V\u2082 f : V \u2192\u2097[K] V\u2082 hf : Injective \u2191f \u22a2 finrank K { x // x \u2208 range f } = finrank K V ** rw [(LinearEquiv.ofInjective f hf).finrank_eq] ** Qed", + "informal": "" + }, + { + "formal": "MeasurableSpace.generateFrom_singleton_empty ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u_6 s t u : Set \u03b1 \u22a2 \u2200 (t : Set \u03b1), t \u2208 {\u2205} \u2192 MeasurableSet t ** simp [@MeasurableSet.empty \u03b1 \u22a5] ** Qed", + "informal": "" + }, + { + "formal": "Real.cos_arctan ** x : \u211d \u22a2 cos (arctan x) = \u21911 / sqrt (\u21911 + x ^ 2) ** rw_mod_cast [one_div, \u2190 inv_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan] ** Qed", + "informal": "" + }, + { + "formal": "Submonoid.LocalizationMap.eq_of_eq ** M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x y : M h : \u2191(toMap f) x = \u2191(toMap f) y \u22a2 \u2191g x = \u2191g y ** obtain \u27e8c, hc\u27e9 := f.eq_iff_exists.1 h ** case intro M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x y : M h : \u2191(toMap f) x = \u2191(toMap f) y c : { x // x \u2208 S } hc : \u2191c * x = \u2191c * y \u22a2 \u2191g x = \u2191g y ** rw [\u2190 one_mul (g x), \u2190 IsUnit.liftRight_inv_mul (g.restrict S) hg c] ** case intro M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x y : M h : \u2191(toMap f) x = \u2191(toMap f) y c : { x // x \u2208 S } hc : \u2191c * x = \u2191c * y \u22a2 \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)\u207b\u00b9 * \u2191(MonoidHom.restrict g S) c * \u2191g x = \u2191g y ** show _ * g c * _ = _ ** case intro M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N g : M \u2192* P hg : \u2200 (y : { x // x \u2208 S }), IsUnit (\u2191g \u2191y) x y : M h : \u2191(toMap f) x = \u2191(toMap f) y c : { x // x \u2208 S } hc : \u2191c * x = \u2191c * y \u22a2 \u2191(\u2191(IsUnit.liftRight (MonoidHom.restrict g S) hg) c)\u207b\u00b9 * \u2191g \u2191c * \u2191g x = \u2191g y ** rw [mul_assoc, \u2190 g.map_mul, hc, mul_comm, mul_inv_left hg, g.map_mul] ** Qed", + "informal": "" + }, + { + "formal": "List.formPerm_reverse ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l \u22a2 formPerm (reverse l) = (formPerm l)\u207b\u00b9 ** rw [eq_comm, inv_eq_iff_mul_eq_one] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l \u22a2 formPerm l * formPerm (reverse l) = 1 ** ext x ** case H \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l : List \u03b1 h : Nodup l x : \u03b1 \u22a2 \u2191(formPerm l * formPerm (reverse l)) x = \u21911 x ** rw [mul_apply, one_apply] ** case H \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l : List \u03b1 h : Nodup l x : \u03b1 \u22a2 \u2191(formPerm l) (\u2191(formPerm (reverse l)) x) = x ** cases' Classical.em (x \u2208 l) with hx hx ** case H.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l : List \u03b1 h : Nodup l x : \u03b1 hx : x \u2208 l \u22a2 \u2191(formPerm l) (\u2191(formPerm (reverse l)) x) = x ** obtain \u27e8k, hk, rfl\u27e9 := nthLe_of_mem (mem_reverse.mpr hx) ** case H.inl.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l \u22a2 \u2191(formPerm l) (\u2191(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk ** have h1 : l.length - 1 - k < l.length := by\n rw [Nat.sub_sub, add_comm]\n exact Nat.sub_lt_self (Nat.succ_pos _) (Nat.succ_le_of_lt (by simpa using hk)) ** case H.inl.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 \u2191(formPerm l) (\u2191(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk ** have h2 : length l - 1 - (k + 1) % length (reverse l) < length l := by\n rw [Nat.sub_sub, length_reverse];\n exact Nat.sub_lt_self (by rw [add_comm]; exact Nat.succ_pos _)\n (by rw [add_comm]; exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx))) ** case H.inl.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length (reverse l) < length l \u22a2 \u2191(formPerm l) (\u2191(formPerm (reverse l)) (nthLe (reverse l) k hk)) = nthLe (reverse l) k hk ** rw [formPerm_apply_nthLe l.reverse (nodup_reverse.mpr h), nthLe_reverse' _ _ _ h1,\n nthLe_reverse' _ _ _ h2, formPerm_apply_nthLe _ h] ** case H.inl.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length (reverse l) < length l \u22a2 nthLe l ((length l - 1 - (k + 1) % length (reverse l) + 1) % length l) (_ : (length l - 1 - (k + 1) % length (reverse l) + 1) % length l < length l) = nthLe l (length l - 1 - k) h1 ** congr ** case H.inl.intro.intro.e_n \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length (reverse l) < length l \u22a2 (length l - 1 - (k + 1) % length (reverse l) + 1) % length l = length l - 1 - k ** rw [length_reverse] at * ** case H.inl.intro.intro.e_n \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk\u271d : k < length (reverse l) hk : k < length l hx : nthLe (reverse l) k hk\u271d \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length l < length l \u22a2 (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k ** cases' lt_or_eq_of_le (Nat.succ_le_of_lt hk) with h h ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l \u22a2 length l - 1 - k < length l ** rw [Nat.sub_sub, add_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l \u22a2 length l - (k + 1) < length l ** exact Nat.sub_lt_self (Nat.succ_pos _) (Nat.succ_le_of_lt (by simpa using hk)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l \u22a2 k < length l ** simpa using hk ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 length l - 1 - (k + 1) % length (reverse l) < length l ** rw [Nat.sub_sub, length_reverse] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 length l - (1 + (k + 1) % length l) < length l ** exact Nat.sub_lt_self (by rw [add_comm]; exact Nat.succ_pos _)\n (by rw [add_comm]; exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx))) ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 0 < 1 + (k + 1) % length l ** rw [add_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 0 < (k + 1) % length l + 1 ** exact Nat.succ_pos _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 1 + (k + 1) % length l \u2264 length l ** rw [add_comm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h : Nodup l k : \u2115 hk : k < length (reverse l) hx : nthLe (reverse l) k hk \u2208 l h1 : length l - 1 - k < length l \u22a2 (k + 1) % length l + 1 \u2264 length l ** exact Nat.succ_le_of_lt (Nat.mod_lt _ (length_pos_of_mem hx)) ** case H.inl.intro.intro.e_n.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h\u271d : Nodup l k : \u2115 hk\u271d : k < length (reverse l) hk : k < length l hx : nthLe (reverse l) k hk\u271d \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length l < length l h : Nat.succ k < length l \u22a2 (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k ** rw [Nat.mod_eq_of_lt h, \u2190 Nat.sub_add_comm, Nat.succ_sub_succ_eq_sub,\n Nat.mod_eq_of_lt h1] ** case H.inl.intro.intro.e_n.inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h\u271d : Nodup l k : \u2115 hk\u271d : k < length (reverse l) hk : k < length l hx : nthLe (reverse l) k hk\u271d \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length l < length l h : Nat.succ k < length l \u22a2 Nat.succ k \u2264 length l - 1 ** exact (Nat.le_sub_iff_add_le (length_pos_of_mem hx)).2 (Nat.succ_le_of_lt h) ** case H.inl.intro.intro.e_n.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h\u271d : Nodup l k : \u2115 hk\u271d : k < length (reverse l) hk : k < length l hx : nthLe (reverse l) k hk\u271d \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length l < length l h : Nat.succ k = length l \u22a2 (length l - 1 - (k + 1) % length l + 1) % length l = length l - 1 - k ** rw [\u2190 h] ** case H.inl.intro.intro.e_n.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x : \u03b1 l : List \u03b1 h\u271d : Nodup l k : \u2115 hk\u271d : k < length (reverse l) hk : k < length l hx : nthLe (reverse l) k hk\u271d \u2208 l h1 : length l - 1 - k < length l h2 : length l - 1 - (k + 1) % length l < length l h : Nat.succ k = length l \u22a2 (Nat.succ k - 1 - (k + 1) % Nat.succ k + 1) % Nat.succ k = Nat.succ k - 1 - k ** simp ** case H.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l : List \u03b1 h : Nodup l x : \u03b1 hx : \u00acx \u2208 l \u22a2 \u2191(formPerm l) (\u2191(formPerm (reverse l)) x) = x ** rw [formPerm_apply_of_not_mem x l.reverse, formPerm_apply_of_not_mem _ _ hx] ** case H.inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d : \u03b1 l : List \u03b1 h : Nodup l x : \u03b1 hx : \u00acx \u2208 l \u22a2 \u00acx \u2208 reverse l ** simpa using hx ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.lift_bit1 ** \u03b1 \u03b2 : Type u a : Cardinal.{u_1} \u22a2 lift.{v, u_1} (bit1 a) = bit1 (lift.{v, u_1} a) ** simp [bit1] ** Qed", + "informal": "" + }, + { + "formal": "Finset.max_singleton ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 a : \u03b1 \u22a2 Finset.max {a} = \u2191a ** rw [\u2190 insert_emptyc_eq] ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d : LinearOrder \u03b1 a : \u03b1 \u22a2 Finset.max (insert a \u2205) = \u2191a ** exact max_insert ** Qed", + "informal": "" + }, + { + "formal": "Int.two_pow_sub_pow' ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x \u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n ** have hx_odd : Odd x := by rwa [Int.odd_iff_not_even, even_iff_two_dvd] ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x \u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n ** have hxy_even : Even (x - y) := even_iff_two_dvd.mpr (dvd_trans (by norm_num) hxy) ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) \u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n ** have hy_odd : Odd y := by simpa using hx_odd.sub_even hxy_even ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y \u22a2 multiplicity 2 (x ^ n - y ^ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191n ** cases' n with n ** case succ R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 \u22a2 multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n) ** have h : (multiplicity 2 n.succ).Dom := multiplicity.finite_nat_iff.mpr \u27e8by norm_num, n.succ_pos\u27e9 ** case succ R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom \u22a2 multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n) ** rcases multiplicity.eq_coe_iff.mp (PartENat.natCast_get h).symm with \u27e8\u27e8k, hk\u27e9, hpn\u27e9 ** case succ.intro.intro R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 multiplicity 2 (x ^ Nat.succ n - y ^ Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n) ** rw [hk, pow_mul, pow_mul, multiplicity.pow_sub_pow_of_prime,\n Int.two_pow_two_pow_sub_pow_two_pow _ hxy hx, \u2190 hk, PartENat.natCast_get] ** case succ.intro.intro.hn R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 \u00ac2 \u2223 \u2191k ** erw [Int.coe_nat_dvd] ** case succ.intro.intro.hn R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 \u00ac2 \u2223 k ** contrapose! hpn ** case succ.intro.intro.hn R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k hpn : 2 \u2223 k \u22a2 2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n ** rw [pow_succ'] ** case succ.intro.intro.hn R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k hpn : 2 \u2223 k \u22a2 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * 2 \u2223 Nat.succ n ** conv_rhs => rw [hk] ** case succ.intro.intro.hn R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k hpn : 2 \u2223 k \u22a2 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * 2 \u2223 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k ** exact mul_dvd_mul_left _ hpn ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x \u22a2 Odd x ** rwa [Int.odd_iff_not_even, even_iff_two_dvd] ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x \u22a2 2 \u2223 4 ** norm_num ** R : Type u_1 n\u271d : \u2115 x y : \u2124 n : \u2115 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) \u22a2 Odd y ** simpa using hx_odd.sub_even hxy_even ** case zero R : Type u_1 n : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y \u22a2 multiplicity 2 (x ^ Nat.zero - y ^ Nat.zero) = multiplicity 2 (x - y) + multiplicity 2 \u2191Nat.zero ** simp only [pow_zero, sub_self, multiplicity.zero, Int.ofNat_zero, Nat.zero_eq, add_top] ** R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 \u22a2 2 \u2260 1 ** norm_num ** case succ.intro.intro R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 multiplicity 2 (x - y) + multiplicity 2 (Nat.succ n) = multiplicity 2 (x - y) + multiplicity 2 \u2191(Nat.succ n) ** norm_cast ** case succ.intro.intro.hp R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 Prime 2 ** exact Int.prime_two ** case succ.intro.intro.hxy R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 2 \u2223 x ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h - y ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h ** simpa only [even_iff_two_dvd] using hx_odd.pow.sub_odd hy_odd.pow ** case succ.intro.intro.hx R : Type u_1 n\u271d : \u2115 x y : \u2124 hxy : 4 \u2223 x - y hx : \u00ac2 \u2223 x hx_odd : Odd x hxy_even : Even (x - y) hy_odd : Odd y n : \u2115 h : (multiplicity 2 (Nat.succ n)).Dom hpn : \u00ac2 ^ (Part.get (multiplicity 2 (Nat.succ n)) h + 1) \u2223 Nat.succ n k : \u2115 hk : Nat.succ n = 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h * k \u22a2 \u00ac2 \u2223 x ^ 2 ^ Part.get (multiplicity 2 (Nat.succ n)) h ** simpa only [even_iff_two_dvd, Int.odd_iff_not_even] using hx_odd.pow ** Qed", + "informal": "" + }, + { + "formal": "Ideal.isCoprime_iff_add ** R : Type u \u03b9 : Type u_1 inst\u271d : CommSemiring R I J K L : Ideal R \u22a2 IsCoprime I J \u2194 I + J = 1 ** rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] ** Qed", + "informal": "" + }, + { + "formal": "Real.sin_eq_zero_iff_of_lt_of_lt ** x : \u211d hx\u2081 : -\u03c0 < x hx\u2082 : x < \u03c0 h : sin x = 0 \u22a2 x = 0 ** contrapose! h ** x : \u211d hx\u2081 : -\u03c0 < x hx\u2082 : x < \u03c0 h : x \u2260 0 \u22a2 sin x \u2260 0 ** cases h.lt_or_lt with\n| inl h0 => exact (sin_neg_of_neg_of_neg_pi_lt h0 hx\u2081).ne\n| inr h0 => exact (sin_pos_of_pos_of_lt_pi h0 hx\u2082).ne' ** case inl x : \u211d hx\u2081 : -\u03c0 < x hx\u2082 : x < \u03c0 h : x \u2260 0 h0 : x < 0 \u22a2 sin x \u2260 0 ** exact (sin_neg_of_neg_of_neg_pi_lt h0 hx\u2081).ne ** case inr x : \u211d hx\u2081 : -\u03c0 < x hx\u2082 : x < \u03c0 h : x \u2260 0 h0 : 0 < x \u22a2 sin x \u2260 0 ** exact (sin_pos_of_pos_of_lt_pi h0 hx\u2082).ne' ** x : \u211d hx\u2081 : -\u03c0 < x hx\u2082 : x < \u03c0 h : x = 0 \u22a2 sin x = 0 ** simp [h] ** Qed", + "informal": "" + }, + { + "formal": "AlgHom.snd_prod ** R : Type u_1 A : Type u_2 B : Type u_3 C : Type u_4 inst\u271d\u2076 : CommSemiring R inst\u271d\u2075 : Semiring A inst\u271d\u2074 : Algebra R A inst\u271d\u00b3 : Semiring B inst\u271d\u00b2 : Algebra R B inst\u271d\u00b9 : Semiring C inst\u271d : Algebra R C f : A \u2192\u2090[R] B g : A \u2192\u2090[R] C \u22a2 comp (snd R B C) (prod f g) = g ** ext ** case H R : Type u_1 A : Type u_2 B : Type u_3 C : Type u_4 inst\u271d\u2076 : CommSemiring R inst\u271d\u2075 : Semiring A inst\u271d\u2074 : Algebra R A inst\u271d\u00b3 : Semiring B inst\u271d\u00b2 : Algebra R B inst\u271d\u00b9 : Semiring C inst\u271d : Algebra R C f : A \u2192\u2090[R] B g : A \u2192\u2090[R] C x\u271d : A \u22a2 \u2191(comp (snd R B C) (prod f g)) x\u271d = \u2191g x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Set.div_mem_centralizer ** M : Type u_1 S T : Set M a b : M inst\u271d : Group M ha : a \u2208 centralizer S hb : b \u2208 centralizer S \u22a2 a / b \u2208 centralizer S ** rw [div_eq_mul_inv] ** M : Type u_1 S T : Set M a b : M inst\u271d : Group M ha : a \u2208 centralizer S hb : b \u2208 centralizer S \u22a2 a * b\u207b\u00b9 \u2208 centralizer S ** exact mul_mem_centralizer ha (inv_mem_centralizer hb) ** Qed", + "informal": "" + }, + { + "formal": "PosNum.cmp_to_nat_lemma ** \u03b1 : Type u_1 m n : PosNum \u22a2 \u2191m < \u2191n \u2192 \u2191m + \u2191m + 1 + 1 \u2264 \u2191n + \u2191n ** intro h ** \u03b1 : Type u_1 m n : PosNum h : \u2191m < \u2191n \u22a2 \u2191m + \u2191m + 1 + 1 \u2264 \u2191n + \u2191n ** rw [Nat.add_right_comm m m 1, add_assoc] ** \u03b1 : Type u_1 m n : PosNum h : \u2191m < \u2191n \u22a2 \u2191m + 1 + (\u2191m + 1) \u2264 \u2191n + \u2191n ** exact add_le_add h h ** Qed", + "informal": "" + }, + { + "formal": "MonoidHom.transfer_eq_pow_aux ** G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G key : \u2200 (k : \u2115) (g\u2080 : G), g\u2080\u207b\u00b9 * g ^ k * g\u2080 \u2208 H \u2192 g\u2080\u207b\u00b9 * g ^ k * g\u2080 = g ^ k \u22a2 g ^ index H \u2208 H ** by_cases hH : H.index = 0 ** case neg G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G key : \u2200 (k : \u2115) (g\u2080 : G), g\u2080\u207b\u00b9 * g ^ k * g\u2080 \u2208 H \u2192 g\u2080\u207b\u00b9 * g ^ k * g\u2080 = g ^ k hH : \u00acindex H = 0 \u22a2 g ^ index H \u2208 H ** letI := fintypeOfIndexNeZero hH ** case pos G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G key : \u2200 (k : \u2115) (g\u2080 : G), g\u2080\u207b\u00b9 * g ^ k * g\u2080 \u2208 H \u2192 g\u2080\u207b\u00b9 * g ^ k * g\u2080 = g ^ k hH : index H = 0 \u22a2 g ^ index H \u2208 H ** rw [hH, pow_zero] ** case pos G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G key : \u2200 (k : \u2115) (g\u2080 : G), g\u2080\u207b\u00b9 * g ^ k * g\u2080 \u2208 H \u2192 g\u2080\u207b\u00b9 * g ^ k * g\u2080 = g ^ k hH : index H = 0 \u22a2 1 \u2208 H ** exact H.one_mem ** case neg G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G hH : \u00acindex H = 0 this : Fintype (G \u29f8 H) := fintypeOfIndexNeZero hH key : \u2200 (q : G \u29f8 H), g ^ Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) g) q \u2208 H f : Quotient (orbitRel { x // x \u2208 zpowers g } (G \u29f8 H)) \u2192 { x // x \u2208 zpowers g } := fun q => { val := g, property := (_ : g \u2208 zpowers g) } ^ Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) \u22a2 g ^ index H \u2208 H ** have hf : \u2200 q, f q \u2208 H.subgroupOf (zpowers g) := fun q => key q.out' ** case neg G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G hH : \u00acindex H = 0 this : Fintype (G \u29f8 H) := fintypeOfIndexNeZero hH key : \u2200 (q : G \u29f8 H), g ^ Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) g) q \u2208 H f : Quotient (orbitRel { x // x \u2208 zpowers g } (G \u29f8 H)) \u2192 { x // x \u2208 zpowers g } := fun q => { val := g, property := (_ : g \u2208 zpowers g) } ^ Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) hf : \u2200 (q : Quotient (orbitRel { x // x \u2208 zpowers g } (G \u29f8 H))), f q \u2208 subgroupOf H (zpowers g) \u22a2 g ^ index H \u2208 H ** replace key :=\n Subgroup.prod_mem (H.subgroupOf (zpowers g)) fun q (_ : q \u2208 Finset.univ) => hf q ** case neg G : Type u_1 inst\u271d\u00b9 : Group G H : Subgroup G A : Type u_2 inst\u271d : CommGroup A \u03d5 : { x // x \u2208 H } \u2192* A T : \u2191(leftTransversals \u2191H) g : G hH : \u00acindex H = 0 this : Fintype (G \u29f8 H) := fintypeOfIndexNeZero hH f : Quotient (orbitRel { x // x \u2208 zpowers g } (G \u29f8 H)) \u2192 { x // x \u2208 zpowers g } := fun q => { val := g, property := (_ : g \u2208 zpowers g) } ^ Function.minimalPeriod ((fun x x_1 => x \u2022 x_1) g) (Quotient.out' q) hf : \u2200 (q : Quotient (orbitRel { x // x \u2208 zpowers g } (G \u29f8 H))), f q \u2208 subgroupOf H (zpowers g) key : \u220f c : Quotient (orbitRel { x // x \u2208 zpowers g } (G \u29f8 H)), f c \u2208 subgroupOf H (zpowers g) \u22a2 g ^ index H \u2208 H ** simpa only [minimalPeriod_eq_card, Finset.prod_pow_eq_pow_sum, Fintype.card_sigma,\n Fintype.card_congr (selfEquivSigmaOrbits (zpowers g) (G \u29f8 H)), index_eq_card] using key ** Qed", + "informal": "" + }, + { + "formal": "Std.RBNode.Path.ins_eq_fill ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat x\u271d : RBNode \u03b1 v\u271d : \u03b1 parent\u271d : Path \u03b1 ha : Balanced x\u271d black n H : Path.Balanced c\u2080 n\u2080 parent\u271d red n hb : Balanced t black n \u22a2 ins (right red x\u271d v\u271d parent\u271d) t = setBlack (fill (right red x\u271d v\u271d parent\u271d) t) ** unfold ins ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat x\u271d : RBNode \u03b1 v\u271d : \u03b1 parent\u271d : Path \u03b1 ha : Balanced x\u271d black n H : Path.Balanced c\u2080 n\u2080 parent\u271d red n hb : Balanced t black n \u22a2 ins parent\u271d (node red x\u271d v\u271d t) = setBlack (fill (right red x\u271d v\u271d parent\u271d) t) ** exact ins_eq_fill H (.red ha hb) ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c\u271d : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat c : RBColor y\u271d : RBNode \u03b1 c\u2082\u271d : RBColor parent\u271d : Path \u03b1 v\u271d : \u03b1 hb : Balanced y\u271d c\u2082\u271d n H : Path.Balanced c\u2080 n\u2080 parent\u271d black (n + 1) ha : Balanced t c n \u22a2 ins (left black parent\u271d v\u271d y\u271d) t = setBlack (fill (left black parent\u271d v\u271d y\u271d) t) ** rw [ins, fill, \u2190 ins_eq_fill H (.black ha hb), balance1_eq ha] ** \u03b1 : Type u_1 c\u2080 : RBColor n\u2080 : Nat c\u271d : RBColor n\u271d : Nat path : Path \u03b1 t : RBNode \u03b1 n : Nat c : RBColor x\u271d : RBNode \u03b1 c\u2081\u271d : RBColor v\u271d : \u03b1 parent\u271d : Path \u03b1 ha : Balanced x\u271d c\u2081\u271d n H : Path.Balanced c\u2080 n\u2080 parent\u271d black (n + 1) hb : Balanced t c n \u22a2 ins (right black x\u271d v\u271d parent\u271d) t = setBlack (fill (right black x\u271d v\u271d parent\u271d) t) ** rw [ins, fill, \u2190 ins_eq_fill H (.black ha hb), balance2_eq hb] ** Qed", + "informal": "" + }, + { + "formal": "Rat.mkRat_mul_mkRat ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** if z\u2081 : d\u2081 = 0 then simp [z\u2081] else if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\nrw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : d\u2081 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** simp [z\u2081] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : \u00acd\u2081 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** if z\u2082 : d\u2082 = 0 then simp [z\u2082] else\nrw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : \u00acd\u2081 = 0 z\u2082 : d\u2082 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** simp [z\u2082] ** n\u2081 n\u2082 : Int d\u2081 d\u2082 : Nat z\u2081 : \u00acd\u2081 = 0 z\u2082 : \u00acd\u2082 = 0 \u22a2 mkRat n\u2081 d\u2081 * mkRat n\u2082 d\u2082 = mkRat (n\u2081 * n\u2082) (d\u2081 * d\u2082) ** rw [\u2190 normalize_eq_mkRat z\u2081, \u2190 normalize_eq_mkRat z\u2082, normalize_mul_normalize, normalize_eq_mkRat] ** Qed", + "informal": "" + }, + { + "formal": "List.Func.length_pointwise ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : Inhabited \u03b1 inst\u271d : Inhabited \u03b2 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 head\u271d : \u03b2 bs : List \u03b2 \u22a2 length (pointwise f [] (head\u271d :: bs)) = max (length []) (length (head\u271d :: bs)) ** simp only [pointwise, length, length_map, max_eq_right (Nat.zero_le (length bs + 1))] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : Inhabited \u03b1 inst\u271d : Inhabited \u03b2 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 head\u271d : \u03b1 as : List \u03b1 \u22a2 length (pointwise f (head\u271d :: as) []) = max (length (head\u271d :: as)) (length []) ** simp only [pointwise, length, length_map, max_eq_left (Nat.zero_le (length as + 1))] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b9 : Inhabited \u03b1 inst\u271d : Inhabited \u03b2 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 head\u271d\u00b9 : \u03b1 as : List \u03b1 head\u271d : \u03b2 bs : List \u03b2 \u22a2 length (pointwise f (head\u271d\u00b9 :: as) (head\u271d :: bs)) = max (length (head\u271d\u00b9 :: as)) (length (head\u271d :: bs)) ** simp only [pointwise, length, Nat.max_succ_succ, @length_pointwise _ as bs] ** Qed", + "informal": "" + }, + { + "formal": "List.nil_zipWith ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 l : List \u03b2 \u22a2 zipWith f [] l = [] ** cases l <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.trace_one ** R : Type u_1 inst\u271d\u00b9\u00b2 : CommRing R M : Type u_2 inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M N : Type u_3 P : Type u_4 inst\u271d\u2079 : AddCommGroup N inst\u271d\u2078 : Module R N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : Module R P \u03b9 : Type u_5 inst\u271d\u2075 : Module.Free R M inst\u271d\u2074 : Module.Finite R M inst\u271d\u00b3 : Module.Free R N inst\u271d\u00b2 : Module.Finite R N inst\u271d\u00b9 : Module.Free R P inst\u271d : Module.Finite R P \u22a2 \u2191(trace R M) 1 = \u2191(finrank R M) ** cases subsingleton_or_nontrivial R ** case inr R : Type u_1 inst\u271d\u00b9\u00b2 : CommRing R M : Type u_2 inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M N : Type u_3 P : Type u_4 inst\u271d\u2079 : AddCommGroup N inst\u271d\u2078 : Module R N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : Module R P \u03b9 : Type u_5 inst\u271d\u2075 : Module.Free R M inst\u271d\u2074 : Module.Finite R M inst\u271d\u00b3 : Module.Free R N inst\u271d\u00b2 : Module.Finite R N inst\u271d\u00b9 : Module.Free R P inst\u271d : Module.Finite R P h\u271d : Nontrivial R \u22a2 \u2191(trace R M) 1 = \u2191(finrank R M) ** have b := Module.Free.chooseBasis R M ** case inr R : Type u_1 inst\u271d\u00b9\u00b2 : CommRing R M : Type u_2 inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M N : Type u_3 P : Type u_4 inst\u271d\u2079 : AddCommGroup N inst\u271d\u2078 : Module R N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : Module R P \u03b9 : Type u_5 inst\u271d\u2075 : Module.Free R M inst\u271d\u2074 : Module.Finite R M inst\u271d\u00b3 : Module.Free R N inst\u271d\u00b2 : Module.Finite R N inst\u271d\u00b9 : Module.Free R P inst\u271d : Module.Finite R P h\u271d : Nontrivial R b : Basis (Module.Free.ChooseBasisIndex R M) R M \u22a2 \u2191(trace R M) 1 = \u2191(finrank R M) ** rw [trace_eq_matrix_trace R b, toMatrix_one, finrank_eq_card_chooseBasisIndex] ** case inr R : Type u_1 inst\u271d\u00b9\u00b2 : CommRing R M : Type u_2 inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M N : Type u_3 P : Type u_4 inst\u271d\u2079 : AddCommGroup N inst\u271d\u2078 : Module R N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : Module R P \u03b9 : Type u_5 inst\u271d\u2075 : Module.Free R M inst\u271d\u2074 : Module.Finite R M inst\u271d\u00b3 : Module.Free R N inst\u271d\u00b2 : Module.Finite R N inst\u271d\u00b9 : Module.Free R P inst\u271d : Module.Finite R P h\u271d : Nontrivial R b : Basis (Module.Free.ChooseBasisIndex R M) R M \u22a2 Matrix.trace 1 = \u2191(Fintype.card (Module.Free.ChooseBasisIndex R M)) ** simp ** case inl R : Type u_1 inst\u271d\u00b9\u00b2 : CommRing R M : Type u_2 inst\u271d\u00b9\u00b9 : AddCommGroup M inst\u271d\u00b9\u2070 : Module R M N : Type u_3 P : Type u_4 inst\u271d\u2079 : AddCommGroup N inst\u271d\u2078 : Module R N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : Module R P \u03b9 : Type u_5 inst\u271d\u2075 : Module.Free R M inst\u271d\u2074 : Module.Finite R M inst\u271d\u00b3 : Module.Free R N inst\u271d\u00b2 : Module.Finite R N inst\u271d\u00b9 : Module.Free R P inst\u271d : Module.Finite R P h\u271d : Subsingleton R \u22a2 \u2191(trace R M) 1 = \u2191(finrank R M) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.mapDomain_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : AddCommMonoid M v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M f : \u03b1 \u2192 \u03b2 hf : Injective f x : \u03b1 \u2192\u2080 M a : \u03b1 \u22a2 \u2191(mapDomain f x) (f a) = \u2191x a ** rw [mapDomain, sum_apply, sum_eq_single a, single_eq_same] ** case h\u2080 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : AddCommMonoid M v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M f : \u03b1 \u2192 \u03b2 hf : Injective f x : \u03b1 \u2192\u2080 M a : \u03b1 \u22a2 \u2200 (b : \u03b1), \u2191x b \u2260 0 \u2192 b \u2260 a \u2192 (\u2191fun\u2080 | f b => \u2191x b) (f a) = 0 ** intro b _ hba ** case h\u2080 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : AddCommMonoid M v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M f : \u03b1 \u2192 \u03b2 hf : Injective f x : \u03b1 \u2192\u2080 M a b : \u03b1 a\u271d : \u2191x b \u2260 0 hba : b \u2260 a \u22a2 (\u2191fun\u2080 | f b => \u2191x b) (f a) = 0 ** exact single_eq_of_ne (hf.ne hba) ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : AddCommMonoid M v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M f : \u03b1 \u2192 \u03b2 hf : Injective f x : \u03b1 \u2192\u2080 M a : \u03b1 \u22a2 \u2191x a = 0 \u2192 (\u2191fun\u2080 | f a => 0) (f a) = 0 ** intro _ ** case h\u2081 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : AddCommMonoid M v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M f : \u03b1 \u2192 \u03b2 hf : Injective f x : \u03b1 \u2192\u2080 M a : \u03b1 a\u271d : \u2191x a = 0 \u22a2 (\u2191fun\u2080 | f a => 0) (f a) = 0 ** rw [single_zero, coe_zero, Pi.zero_apply] ** Qed", + "informal": "" + }, + { + "formal": "Finset.Icc_subset_Icc ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c x : \u03b1 ha : a\u2082 \u2264 a\u2081 hb : b\u2081 \u2264 b\u2082 \u22a2 Icc a\u2081 b\u2081 \u2286 Icc a\u2082 b\u2082 ** simpa [\u2190 coe_subset] using Set.Icc_subset_Icc ha hb ** Qed", + "informal": "" + }, + { + "formal": "LieSubmodule.sInf_coe_toSubmodule' ** R : Type u L : Type v M : Type w inst\u271d\u2076 : CommRing R inst\u271d\u2075 : LieRing L inst\u271d\u2074 : LieAlgebra R L inst\u271d\u00b3 : AddCommGroup M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : LieRingModule L M inst\u271d : LieModule R L M N N' : LieSubmodule R L M I J : LieIdeal R L S : Set (LieSubmodule R L M) \u22a2 \u2191(sInf S) = \u2a05 N \u2208 S, \u2191N ** rw [sInf_coe_toSubmodule, \u2190 Set.image, sInf_image] ** Qed", + "informal": "" + }, + { + "formal": "exp_sum_of_commute ** \ud835\udd42 : Type u_1 \ud835\udd38 : Type u_2 \ud835\udd39 : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd42 inst\u271d\u2074 : NormedRing \ud835\udd38 inst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38 inst\u271d\u00b2 : NormedRing \ud835\udd39 inst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39 inst\u271d : CompleteSpace \ud835\udd38 \u03b9 : Type u_4 s : Finset \u03b9 f : \u03b9 \u2192 \ud835\udd38 h : Set.Pairwise \u2191s fun i j => Commute (f i) (f j) \u22a2 exp \ud835\udd42 (\u2211 i in s, f i) = Finset.noncommProd s (fun i => exp \ud835\udd42 (f i)) (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j))) ** induction' s using Finset.induction_on with a s ha ih ** case insert \ud835\udd42 : Type u_1 \ud835\udd38 : Type u_2 \ud835\udd39 : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd42 inst\u271d\u2074 : NormedRing \ud835\udd38 inst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38 inst\u271d\u00b2 : NormedRing \ud835\udd39 inst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39 inst\u271d : CompleteSpace \ud835\udd38 \u03b9 : Type u_4 s\u271d : Finset \u03b9 f : \u03b9 \u2192 \ud835\udd38 h\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (f i) (f j) a : \u03b9 s : Finset \u03b9 ha : \u00aca \u2208 s ih : \u2200 (h : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)), exp \ud835\udd42 (\u2211 i in s, f i) = Finset.noncommProd s (fun i => exp \ud835\udd42 (f i)) (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j))) h : Set.Pairwise \u2191(insert a s) fun i j => Commute (f i) (f j) \u22a2 exp \ud835\udd42 (\u2211 i in insert a s, f i) = Finset.noncommProd (insert a s) (fun i => exp \ud835\udd42 (f i)) (_ : \u2200 (i : \u03b9), i \u2208 \u2191(insert a s) \u2192 \u2200 (j : \u03b9), j \u2208 \u2191(insert a s) \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j))) ** rw [Finset.noncommProd_insert_of_not_mem _ _ _ _ ha, Finset.sum_insert ha, exp_add_of_commute,\n ih (h.mono <| Finset.subset_insert _ _)] ** case insert \ud835\udd42 : Type u_1 \ud835\udd38 : Type u_2 \ud835\udd39 : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd42 inst\u271d\u2074 : NormedRing \ud835\udd38 inst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38 inst\u271d\u00b2 : NormedRing \ud835\udd39 inst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39 inst\u271d : CompleteSpace \ud835\udd38 \u03b9 : Type u_4 s\u271d : Finset \u03b9 f : \u03b9 \u2192 \ud835\udd38 h\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (f i) (f j) a : \u03b9 s : Finset \u03b9 ha : \u00aca \u2208 s ih : \u2200 (h : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)), exp \ud835\udd42 (\u2211 i in s, f i) = Finset.noncommProd s (fun i => exp \ud835\udd42 (f i)) (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j))) h : Set.Pairwise \u2191(insert a s) fun i j => Commute (f i) (f j) \u22a2 Commute (f a) (\u2211 x in s, f x) ** refine' Commute.sum_right _ _ _ fun i hi => _ ** case insert \ud835\udd42 : Type u_1 \ud835\udd38 : Type u_2 \ud835\udd39 : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd42 inst\u271d\u2074 : NormedRing \ud835\udd38 inst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38 inst\u271d\u00b2 : NormedRing \ud835\udd39 inst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39 inst\u271d : CompleteSpace \ud835\udd38 \u03b9 : Type u_4 s\u271d : Finset \u03b9 f : \u03b9 \u2192 \ud835\udd38 h\u271d : Set.Pairwise \u2191s\u271d fun i j => Commute (f i) (f j) a : \u03b9 s : Finset \u03b9 ha : \u00aca \u2208 s ih : \u2200 (h : Set.Pairwise \u2191s fun i j => Commute (f i) (f j)), exp \ud835\udd42 (\u2211 i in s, f i) = Finset.noncommProd s (fun i => exp \ud835\udd42 (f i)) (_ : \u2200 (i : \u03b9), i \u2208 \u2191s \u2192 \u2200 (j : \u03b9), j \u2208 \u2191s \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j))) h : Set.Pairwise \u2191(insert a s) fun i j => Commute (f i) (f j) i : \u03b9 hi : i \u2208 s \u22a2 Commute (f a) (f i) ** exact h.of_refl (Finset.mem_insert_self _ _) (Finset.mem_insert_of_mem hi) ** case empty \ud835\udd42 : Type u_1 \ud835\udd38 : Type u_2 \ud835\udd39 : Type u_3 inst\u271d\u2075 : IsROrC \ud835\udd42 inst\u271d\u2074 : NormedRing \ud835\udd38 inst\u271d\u00b3 : NormedAlgebra \ud835\udd42 \ud835\udd38 inst\u271d\u00b2 : NormedRing \ud835\udd39 inst\u271d\u00b9 : NormedAlgebra \ud835\udd42 \ud835\udd39 inst\u271d : CompleteSpace \ud835\udd38 \u03b9 : Type u_4 s : Finset \u03b9 f : \u03b9 \u2192 \ud835\udd38 h\u271d : Set.Pairwise \u2191s fun i j => Commute (f i) (f j) h : Set.Pairwise \u2191\u2205 fun i j => Commute (f i) (f j) \u22a2 exp \ud835\udd42 (\u2211 i in \u2205, f i) = Finset.noncommProd \u2205 (fun i => exp \ud835\udd42 (f i)) (_ : \u2200 (i : \u03b9), i \u2208 \u2191\u2205 \u2192 \u2200 (j : \u03b9), j \u2208 \u2191\u2205 \u2192 i \u2260 j \u2192 Commute (exp \ud835\udd42 (f i)) (exp \ud835\udd42 (f j))) ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.tendsto_measure_Iic_atTop ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ)) ** cases isEmpty_or_nonempty \u03b1 ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ)) ** have h_mono : Monotone fun x => \u03bc (Iic x) := fun i j hij => measure_mono (Iic_subset_Iic.mpr hij) ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ)) ** convert tendsto_atTop_iSup h_mono ** case h.e'_5.h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) \u22a2 \u2191\u2191\u03bc univ = \u2a06 i, \u2191\u2191\u03bc (Iic i) ** obtain \u27e8xs, hxs_mono, hxs_tendsto\u27e9 := exists_seq_monotone_tendsto_atTop_atTop \u03b1 ** case h.e'_5.h.e'_3.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop \u22a2 \u2191\u2191\u03bc univ = \u2a06 i, \u2191\u2191\u03bc (Iic i) ** have h_univ : (univ : Set \u03b1) = \u22c3 n, Iic (xs n) := by\n ext1 x\n simp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff]\n obtain \u27e8n, hn\u27e9 := tendsto_atTop_atTop.mp hxs_tendsto x\n exact \u27e8n, hn n le_rfl\u27e9 ** case h.e'_5.h.e'_3.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop h_univ : univ = \u22c3 n, Iic (xs n) \u22a2 \u2191\u2191\u03bc univ = \u2a06 i, \u2191\u2191\u03bc (Iic i) ** rw [h_univ, measure_iUnion_eq_iSup, iSup_eq_iSup_subseq_of_monotone h_mono hxs_tendsto] ** case h.e'_5.h.e'_3.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop h_univ : univ = \u22c3 n, Iic (xs n) \u22a2 Directed (fun x x_1 => x \u2286 x_1) fun n => Iic (xs n) ** exact Monotone.directed_le fun i j hij => Iic_subset_Iic.mpr (hxs_mono hij) ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : IsEmpty \u03b1 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ)) ** have h1 : \u2200 x : \u03b1, Iic x = \u2205 := fun x => Subsingleton.elim _ _ ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : IsEmpty \u03b1 h1 : \u2200 (x : \u03b1), Iic x = \u2205 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ)) ** have h2 : (univ : Set \u03b1) = \u2205 := Subsingleton.elim _ _ ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : IsEmpty \u03b1 h1 : \u2200 (x : \u03b1), Iic x = \u2205 h2 : univ = \u2205 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc (Iic x)) atTop (\ud835\udcdd (\u2191\u2191\u03bc univ)) ** simp_rw [h1, h2] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : IsEmpty \u03b1 h1 : \u2200 (x : \u03b1), Iic x = \u2205 h2 : univ = \u2205 \u22a2 Tendsto (fun x => \u2191\u2191\u03bc \u2205) atTop (\ud835\udcdd (\u2191\u2191\u03bc \u2205)) ** exact tendsto_const_nhds ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop \u22a2 univ = \u22c3 n, Iic (xs n) ** ext1 x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop x : \u03b1 \u22a2 x \u2208 univ \u2194 x \u2208 \u22c3 n, Iic (xs n) ** simp only [mem_univ, mem_iUnion, mem_Iic, true_iff_iff] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop x : \u03b1 \u22a2 \u2203 i, x \u2264 xs i ** obtain \u27e8n, hn\u27e9 := tendsto_atTop_atTop.mp hxs_tendsto x ** case h.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m0 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 inst\u271d\u00b2 : MeasurableSpace \u03b3 \u03bc\u271d \u03bc\u2081 \u03bc\u2082 \u03bc\u2083 \u03bd \u03bd' \u03bd\u2081 \u03bd\u2082 : Measure \u03b1 s s' t : Set \u03b1 inst\u271d\u00b9 : SemilatticeSup \u03b1 inst\u271d : IsCountablyGenerated atTop \u03bc : Measure \u03b1 h\u271d : Nonempty \u03b1 h_mono : Monotone fun x => \u2191\u2191\u03bc (Iic x) xs : \u2115 \u2192 \u03b1 hxs_mono : Monotone xs hxs_tendsto : Tendsto xs atTop atTop x : \u03b1 n : \u2115 hn : \u2200 (a : \u2115), n \u2264 a \u2192 x \u2264 xs a \u22a2 \u2203 i, x \u2264 xs i ** exact \u27e8n, hn n le_rfl\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "List.length_pos_of_prod_ne_one ** \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : Monoid N inst\u271d : Monoid P l l\u2081 l\u2082 : List M a : M L : List M h : prod L \u2260 1 \u22a2 0 < length L ** cases L ** case nil \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : Monoid N inst\u271d : Monoid P l l\u2081 l\u2082 : List M a : M h : prod [] \u2260 1 \u22a2 0 < length [] ** contrapose h ** case nil \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : Monoid N inst\u271d : Monoid P l l\u2081 l\u2082 : List M a : M h : \u00ac0 < length [] \u22a2 \u00acprod [] \u2260 1 ** simp ** case cons \u03b9 : Type u_1 \u03b1 : Type u_2 M : Type u_3 N : Type u_4 P : Type u_5 M\u2080 : Type u_6 G : Type u_7 R : Type u_8 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : Monoid N inst\u271d : Monoid P l l\u2081 l\u2082 : List M a head\u271d : M tail\u271d : List M h : prod (head\u271d :: tail\u271d) \u2260 1 \u22a2 0 < length (head\u271d :: tail\u271d) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Nat.Partrec.Code.rec_prim' ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf \u22a2 let PR := fun a cf cg hf hg => pr a (cf, cg, hf, hg); let CO := fun a cf cg hf hg => co a (cf, cg, hf, hg); let PC := fun a cf cg hf hg => pc a (cf, cg, hf, hg); let RF := fun a cf hf => rf a (cf, hf); let F := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR a) (CO a) (PC a) (RF a); Primrec fun a => F a (c a) ** intros _ _ _ _ F ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) \u22a2 Primrec fun a => F a (c a) ** let G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p =>\n let a := p.1.1\n let IH := p.1.2\n let n := p.2.1\n let m := p.2.2\n (IH.get? m).bind fun s =>\n (IH.get? m.unpair.1).bind fun s\u2081 =>\n (IH.get? m.unpair.2).map fun s\u2082 =>\n cond n.bodd\n (cond n.div2.bodd (rf a (ofNat Code m, s))\n (pc a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082)))\n (cond n.div2.bodd (co a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))\n (pr a (ofNat Code m.unpair.1, ofNat Code m.unpair.2, s\u2081, s\u2082))) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec fun a => F a (c a) ** have : Primrec G\u2081 := by\n refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _\n unfold Primrec\u2082\n refine'\n option_bind\n ((list_get?.comp (snd.comp fst)\n (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) _\n unfold Primrec\u2082\n refine'\n option_map\n ((list_get?.comp (snd.comp fst)\n (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _\n have a : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n fst.comp (fst.comp <| fst.comp <| fst.comp fst)\n have n : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n fst.comp (snd.comp <| fst.comp <| fst.comp fst)\n have m : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n snd.comp (snd.comp <| fst.comp <| fst.comp fst)\n have m\u2081 := fst.comp (Primrec.unpair.comp m)\n have m\u2082 := snd.comp (Primrec.unpair.comp m)\n have s : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n snd.comp (fst.comp fst)\n have s\u2081 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n snd.comp fst\n have s\u2082 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n snd\n unfold Primrec\u2082\n exact\n (nat_bodd.comp n).cond\n ((nat_bodd.comp <| nat_div2.comp n).cond (hrf.comp a (((Primrec.ofNat Code).comp m).pair s))\n (hpc.comp a\n (((Primrec.ofNat Code).comp m\u2081).pair <|\n ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n (Primrec.cond (nat_bodd.comp <| nat_div2.comp n)\n (hco.comp a\n (((Primrec.ofNat Code).comp m\u2081).pair <|\n ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n (hpr.comp a\n (((Primrec.ofNat Code).comp m\u2081).pair <|\n ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 \u22a2 Primrec fun a => F a (c a) ** let G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH =>\n IH.length.casesOn (some (z a)) fun n =>\n n.casesOn (some (s a)) fun n =>\n n.casesOn (some (l a)) fun n =>\n n.casesOn (some (r a)) fun n => G\u2081 ((a, IH), n, n.div2.div2) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec fun a => F a (c a) ** have : Primrec\u2082 G := by\n unfold Primrec\u2082\n refine nat_casesOn\n (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_\n unfold Primrec\u2082\n refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_\n unfold Primrec\u2082\n refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_\n unfold Primrec\u2082\n refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_\n unfold Primrec\u2082\n exact this.comp <|\n ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n snd.pair <| nat_div2.comp <| nat_div2.comp snd ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G \u22a2 Primrec fun a => F a (c a) ** refine'\n ((nat_strong_rec (fun a n => F a (ofNat Code n)) this.to\u2082 fun a n => _).comp\n _root_.Primrec.id <| encode_iff.2 hc).of_eq fun a => by simp ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 \u22a2 G (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).1 (a, List.map ((fun a n => F a (ofNat Code n)) a) (List.range n)).2 = some ((fun a n => F a (ofNat Code n)) a n) ** simp (config := { zeta := false }) ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 \u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) = some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** simp only [] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 \u22a2 Nat.rec (some (z a)) (fun n_1 n_ih => Nat.rec (some (s a)) (fun n_2 n_ih => Nat.rec (some (l a)) (fun n_3 n_ih => Nat.rec (some (r a)) (fun n_4 n_ih => Option.bind (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (div2 (div2 n_4))) fun s_1 => Option.bind (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (unpair (div2 (div2 n_4))).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n_4 then bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1) else pc a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082) else bif bodd (div2 n_4) then co a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082)) (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (unpair (div2 (div2 n_4))).2)) n_3) n_2) n_1) (List.length (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n))))))) = some (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** rw [List.length_map, List.length_range] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 \u22a2 Nat.rec (some (z a)) (fun n_1 n_ih => Nat.rec (some (s a)) (fun n_2 n_ih => Nat.rec (some (l a)) (fun n_3 n_ih => Nat.rec (some (r a)) (fun n_4 n_ih => Option.bind (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (div2 (div2 n_4))) fun s_1 => Option.bind (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (unpair (div2 (div2 n_4))).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n_4 then bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1) else pc a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082) else bif bodd (div2 n_4) then co a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082)) (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (unpair (div2 (div2 n_4))).2)) n_3) n_2) n_1) (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) = some (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** let m := n.div2.div2 ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 Nat.rec (some (z a)) (fun n_1 n_ih => Nat.rec (some (s a)) (fun n_2 n_ih => Nat.rec (some (l a)) (fun n_3 n_ih => Nat.rec (some (r a)) (fun n_4 n_ih => Option.bind (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (div2 (div2 n_4))) fun s_1 => Option.bind (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (unpair (div2 (div2 n_4))).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n_4 then bif bodd (div2 n_4) then rf a (ofNat Code (div2 (div2 n_4)), s_1) else pc a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082) else bif bodd (div2 n_4) then co a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair (div2 (div2 n_4))).1, ofNat Code (unpair (div2 (div2 n_4))).2, s\u2081, s\u2082)) (List.get? (List.map (fun n => rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) (unpair (div2 (div2 n_4))).2)) n_3) n_2) n_1) (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))) = some (rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (Nat.succ (Nat.succ (Nat.succ (Nat.succ n)))))) ** show\n G\u2081 ((a, (List.range (n + 4)).map fun n => F a (ofNat Code n)), n, m) =\n some (F a (ofNat Code (n + 4))) ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** have hm : m < n + 4 := by\n simp only [div2_val]\n exact\n lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n (Nat.succ_le_succ (Nat.le_add_right _ _)) ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** have m1 : m.unpair.1 < n + 4 := lt_of_le_of_lt m.unpair_left_le hm ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** have m2 : m.unpair.2 < n + 4 := lt_of_le_of_lt m.unpair_right_le hm ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 G\u2081 ((a, List.map (fun n => F a (ofNat Code n)) (List.range (n + 4))), n, m) = some (F a (ofNat Code (n + 4))) ** simp [List.get?_map, List.get?_range, hm, m1, m2] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 (bif bodd n then bif bodd (div2 n) then rf a (ofNat Code (div2 (div2 n)), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n)))) else pc a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2)) else bif bodd (div2 n) then co a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2)) else pr a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) = rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (n + 4)) ** rw [show ofNat Code (n + 4) = ofNatCode (n + 4) from rfl] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 (bif bodd n then bif bodd (div2 n) then rf a (ofNat Code (div2 (div2 n)), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n)))) else pc a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2)) else bif bodd (div2 n) then co a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2)) else pr a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) = rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNatCode (n + 4)) ** simp [ofNatCode] ** case succ.succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) hm : m < n + 4 m1 : (unpair m).1 < n + 4 m2 : (unpair m).2 < n + 4 \u22a2 (bif bodd n then bif bodd (div2 n) then rf a (ofNat Code (div2 (div2 n)), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (div2 (div2 n)))) else pc a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2)) else bif bodd (div2 n) then co a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2)) else pr a (ofNat Code (unpair (div2 (div2 n))).1, ofNat Code (unpair (div2 (div2 n))).2, rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).1), rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (ofNat Code (unpair (div2 (div2 n))).2))) = rec (z a) (s a) (l a) (r a) (fun cf cg hf hg => pr a (cf, cg, hf, hg)) (fun cf cg hf hg => co a (cf, cg, hf, hg)) (fun cf cg hf hg => pc a (cf, cg, hf, hg)) (fun cf hf => rf a (cf, hf)) (match bodd n, bodd (div2 n) with | false, false => pair (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2) | false, true => comp (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2) | true, false => prec (ofNatCode (unpair (div2 (div2 n))).1) (ofNatCode (unpair (div2 (div2 n))).2) | true, true => rfind' (ofNatCode (div2 (div2 n)))) ** cases n.bodd <;> cases n.div2.bodd <;> rfl ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec G\u2081 ** refine' option_bind (list_get?.comp (snd.comp fst) (snd.comp snd)) _ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec\u2082 fun p s => Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd p.2.1 then bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s) else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082) else bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082) else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)) (List.get? p.1.2 (unpair p.2.2).2) ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec fun p => (fun p s => Option.bind (List.get? p.1.2 (unpair p.2.2).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd p.2.1 then bif bodd (div2 p.2.1) then rf p.1.1 (ofNat Code p.2.2, s) else pc p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082) else bif bodd (div2 p.2.1) then co p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082) else pr p.1.1 (ofNat Code (unpair p.2.2).1, ofNat Code (unpair p.2.2).2, s\u2081, s\u2082)) (List.get? p.1.2 (unpair p.2.2).2)) p.1 p.2 ** refine'\n option_bind\n ((list_get?.comp (snd.comp fst)\n (fst.comp <| Primrec.unpair.comp (snd.comp snd))).comp fst) _ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec\u2082 fun p s\u2081 => Option.map (fun s\u2082 => bif bodd p.1.2.1 then bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2) else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082) else bif bodd (div2 p.1.2.1) then co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082) else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)) (List.get? p.1.1.2 (unpair p.1.2.2).2) ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec fun p => (fun p s\u2081 => Option.map (fun s\u2082 => bif bodd p.1.2.1 then bif bodd (div2 p.1.2.1) then rf p.1.1.1 (ofNat Code p.1.2.2, p.2) else pc p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082) else bif bodd (div2 p.1.2.1) then co p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082) else pr p.1.1.1 (ofNat Code (unpair p.1.2.2).1, ofNat Code (unpair p.1.2.2).2, s\u2081, s\u2082)) (List.get? p.1.1.2 (unpair p.1.2.2).2)) p.1 p.2 ** refine'\n option_map\n ((list_get?.comp (snd.comp fst)\n (snd.comp <| Primrec.unpair.comp (snd.comp snd))).comp <| fst.comp fst) _ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have a : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.1.1) :=\n fst.comp (fst.comp <| fst.comp <| fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have n : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.1) :=\n fst.comp (snd.comp <| fst.comp <| fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have m : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.1.2.2) :=\n snd.comp (snd.comp <| fst.comp <| fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have m\u2081 := fst.comp (Primrec.unpair.comp m) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 m\u2081 : Primrec fun a => (unpair a.1.1.1.2.2).1 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have m\u2082 := snd.comp (Primrec.unpair.comp m) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 m\u2081 : Primrec fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Primrec fun a => (unpair a.1.1.1.2.2).2 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have s : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.1.2) :=\n snd.comp (fst.comp fst) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s\u271d : \u03b1 \u2192 \u03c3 hs : Primrec s\u271d l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 m\u2081 : Primrec fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Primrec fun a => (unpair a.1.1.1.2.2).2 s : Primrec fun p => p.1.1.2 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have s\u2081 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.1.2) :=\n snd.comp fst ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s\u271d : \u03b1 \u2192 \u03c3 hs : Primrec s\u271d l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 m\u2081 : Primrec fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Primrec fun a => (unpair a.1.1.1.2.2).2 s : Primrec fun p => p.1.1.2 s\u2081 : Primrec fun p => p.1.2 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** have s\u2082 : Primrec (fun p : ((((\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115) \u00d7 \u03c3) \u00d7 \u03c3) \u00d7 \u03c3 => p.2) :=\n snd ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s\u271d : \u03b1 \u2192 \u03c3 hs : Primrec s\u271d l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 m\u2081 : Primrec fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Primrec fun a => (unpair a.1.1.1.2.2).2 s : Primrec fun p => p.1.1.2 s\u2081 : Primrec fun p => p.1.2 s\u2082 : Primrec fun p => p.2 \u22a2 Primrec\u2082 fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s\u271d : \u03b1 \u2192 \u03c3 hs : Primrec s\u271d l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s\u271d a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) a : Primrec fun p => p.1.1.1.1.1 n : Primrec fun p => p.1.1.1.2.1 m : Primrec fun p => p.1.1.1.2.2 m\u2081 : Primrec fun a => (unpair a.1.1.1.2.2).1 m\u2082 : Primrec fun a => (unpair a.1.1.1.2.2).2 s : Primrec fun p => p.1.1.2 s\u2081 : Primrec fun p => p.1.2 s\u2082 : Primrec fun p => p.2 \u22a2 Primrec fun p => (fun p s\u2082 => bif bodd p.1.1.2.1 then bif bodd (div2 p.1.1.2.1) then rf p.1.1.1.1 (ofNat Code p.1.1.2.2, p.1.2) else pc p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else bif bodd (div2 p.1.1.2.1) then co p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082) else pr p.1.1.1.1 (ofNat Code (unpair p.1.1.2.2).1, ofNat Code (unpair p.1.1.2.2).2, p.2, s\u2082)) p.1 p.2 ** exact\n (nat_bodd.comp n).cond\n ((nat_bodd.comp <| nat_div2.comp n).cond (hrf.comp a (((Primrec.ofNat Code).comp m).pair s))\n (hpc.comp a\n (((Primrec.ofNat Code).comp m\u2081).pair <|\n ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082)))\n (Primrec.cond (nat_bodd.comp <| nat_div2.comp n)\n (hco.comp a\n (((Primrec.ofNat Code).comp m\u2081).pair <|\n ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))\n (hpr.comp a\n (((Primrec.ofNat Code).comp m\u2081).pair <|\n ((Primrec.ofNat Code).comp m\u2082).pair <| s\u2081.pair s\u2082))) ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec\u2082 G ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec fun p => G p.1 p.2 ** refine nat_casesOn\n (list_length.comp snd) (option_some_iff.2 (hz.comp fst)) ?_ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec\u2082 fun p n => (fun n => Nat.casesOn n (some (s p.1)) fun n => Nat.casesOn n (some (l p.1)) fun n => Nat.casesOn n (some (r p.1)) fun n => G\u2081 ((p.1, p.2), n, div2 (div2 n))) n ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec fun p => (fun p n => (fun n => Nat.casesOn n (some (s p.1)) fun n => Nat.casesOn n (some (l p.1)) fun n => Nat.casesOn n (some (r p.1)) fun n => G\u2081 ((p.1, p.2), n, div2 (div2 n))) n) p.1 p.2 ** refine nat_casesOn snd (option_some_iff.2 (hs.comp (fst.comp fst))) ?_ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec\u2082 fun p n => (fun n => Nat.casesOn n (some (l p.1.1)) fun n => Nat.casesOn n (some (r p.1.1)) fun n => G\u2081 ((p.1.1, p.1.2), n, div2 (div2 n))) n ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec fun p => (fun p n => (fun n => Nat.casesOn n (some (l p.1.1)) fun n => Nat.casesOn n (some (r p.1.1)) fun n => G\u2081 ((p.1.1, p.1.2), n, div2 (div2 n))) n) p.1 p.2 ** refine nat_casesOn snd (option_some_iff.2 (hl.comp (fst.comp <| fst.comp fst))) ?_ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec\u2082 fun p n => (fun n => Nat.casesOn n (some (r p.1.1.1)) fun n => G\u2081 ((p.1.1.1, p.1.1.2), n, div2 (div2 n))) n ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec fun p => (fun p n => (fun n => Nat.casesOn n (some (r p.1.1.1)) fun n => G\u2081 ((p.1.1.1, p.1.1.2), n, div2 (div2 n))) n) p.1 p.2 ** refine nat_casesOn snd (option_some_iff.2 (hr.comp (fst.comp <| fst.comp <| fst.comp fst))) ?_ ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec\u2082 fun p n => (fun n => G\u2081 ((p.1.1.1.1, p.1.1.1.2), n, div2 (div2 n))) n ** unfold Primrec\u2082 ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) \u22a2 Primrec fun p => (fun p n => (fun n => G\u2081 ((p.1.1.1.1, p.1.1.1.2), n, div2 (div2 n))) n) p.1 p.2 ** exact this.comp <|\n ((fst.pair snd).comp <| fst.comp <| fst.comp <| fst.comp <| fst).pair <|\n snd.pair <| nat_div2.comp <| nat_div2.comp snd ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 \u22a2 F (id a) (ofNat Code (encode (c a))) = F a (c a) ** simp ** case succ.succ.succ \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 \u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ n))))) = some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ n))))) ** cases' n with n ** case succ.succ.succ.zero \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 \u22a2 G a (List.map (fun n => F a (ofNat Code n)) (List.range (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) = some (F a (ofNat Code (Nat.succ (Nat.succ (Nat.succ Nat.zero))))) ** simp (config := { zeta := false }) [ofNatCode_eq, ofNatCode] ** case succ.succ.succ.zero \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 \u22a2 G a (List.map (fun n => F a (ofNatCode n)) (List.range (Nat.succ (Nat.succ (Nat.succ 0))))) = some (F a right) ** rfl ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 m < n + 4 ** simp only [div2_val] ** \u03b1 : Type u_1 \u03c3 : Type u_2 inst\u271d\u00b9 : Primcodable \u03b1 inst\u271d : Primcodable \u03c3 c : \u03b1 \u2192 Code hc : Primrec c z : \u03b1 \u2192 \u03c3 hz : Primrec z s : \u03b1 \u2192 \u03c3 hs : Primrec s l : \u03b1 \u2192 \u03c3 hl : Primrec l r : \u03b1 \u2192 \u03c3 hr : Primrec r pr : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpr : Primrec\u2082 pr co : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hco : Primrec\u2082 co pc : \u03b1 \u2192 Code \u00d7 Code \u00d7 \u03c3 \u00d7 \u03c3 \u2192 \u03c3 hpc : Primrec\u2082 pc rf : \u03b1 \u2192 Code \u00d7 \u03c3 \u2192 \u03c3 hrf : Primrec\u2082 rf PR\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pr a (cf, cg, hf, hg) CO\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => co a (cf, cg, hf, hg) PC\u271d : \u03b1 \u2192 Code \u2192 Code \u2192 \u03c3 \u2192 \u03c3 \u2192 \u03c3 := fun a cf cg hf hg => pc a (cf, cg, hf, hg) RF\u271d : \u03b1 \u2192 Code \u2192 \u03c3 \u2192 \u03c3 := fun a cf hf => rf a (cf, hf) F : \u03b1 \u2192 Code \u2192 \u03c3 := fun a c => Code.recOn c (z a) (s a) (l a) (r a) (PR\u271d a) (CO\u271d a) (PC\u271d a) (RF\u271d a) G\u2081 : (\u03b1 \u00d7 List \u03c3) \u00d7 \u2115 \u00d7 \u2115 \u2192 Option \u03c3 := fun p => let a := p.1.1; let IH := p.1.2; let n := p.2.1; let m := p.2.2; Option.bind (List.get? IH m) fun s => Option.bind (List.get? IH (unpair m).1) fun s\u2081 => Option.map (fun s\u2082 => bif bodd n then bif bodd (div2 n) then rf a (ofNat Code m, s) else pc a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else bif bodd (div2 n) then co a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082) else pr a (ofNat Code (unpair m).1, ofNat Code (unpair m).2, s\u2081, s\u2082)) (List.get? IH (unpair m).2) this\u271d : Primrec G\u2081 G : \u03b1 \u2192 List \u03c3 \u2192 Option \u03c3 := fun a IH => Nat.casesOn (List.length IH) (some (z a)) fun n => Nat.casesOn n (some (s a)) fun n => Nat.casesOn n (some (l a)) fun n => Nat.casesOn n (some (r a)) fun n => G\u2081 ((a, IH), n, div2 (div2 n)) this : Primrec\u2082 G a : \u03b1 n : \u2115 m : \u2115 := div2 (div2 n) \u22a2 n / 2 / 2 < n + 4 ** exact\n lt_of_le_of_lt (le_trans (Nat.div_le_self _ _) (Nat.div_le_self _ _))\n (Nat.succ_le_succ (Nat.le_add_right _ _)) ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.isRegular_succ ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u22a2 c < Ordinal.cof (ord (succ c)) ** cases' Quotient.exists_rep (@succ Cardinal _ _ c) with \u03b1 \u03b1e ** case intro \u03b1\u271d : Type u_1 r : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : Quotient.mk isEquivalent \u03b1 = succ c \u22a2 c < Ordinal.cof (ord (succ c)) ** simp at \u03b1e ** case intro \u03b1\u271d : Type u_1 r : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c \u22a2 c < Ordinal.cof (ord (succ c)) ** rcases ord_eq \u03b1 with \u27e8r, wo, re\u27e9 ** case intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r \u22a2 c < Ordinal.cof (ord (succ c)) ** have := ord_isLimit (h.trans (le_succ _)) ** case intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (ord (succ c)) \u22a2 c < Ordinal.cof (ord (succ c)) ** rw [\u2190 \u03b1e, re] at this \u22a2 ** case intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) \u22a2 c < Ordinal.cof (type r) ** rcases cof_eq' r this with \u27e8S, H, Se\u27e9 ** case intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) \u22a2 c < Ordinal.cof (type r) ** rw [\u2190 Se] ** case intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) \u22a2 c < #\u2191S ** apply lt_imp_lt_of_le_imp_le fun h => mul_le_mul_right' h c ** case intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) \u22a2 c * c < #\u2191S * c ** rw [mul_eq_self h, \u2190 succ_le_iff, \u2190 \u03b1e, \u2190 sum_const'] ** case intro.intro.intro.intro.intro \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) \u22a2 #\u03b1 \u2264 sum fun x => c ** refine' le_trans _ (sum_le_sum (fun (x : S) => card (typein r (x : \u03b1))) _ fun i => _) ** case intro.intro.intro.intro.intro.refine'_1 \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) \u22a2 #\u03b1 \u2264 sum fun x => card (typein r \u2191x) ** simp only [\u2190 card_typein, \u2190 mk_sigma] ** case intro.intro.intro.intro.intro.refine'_1 \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) \u22a2 #\u03b1 \u2264 #((i : \u2191S) \u00d7 { y // r y \u2191i }) ** exact\n \u27e8Embedding.ofSurjective (fun x => x.2.1) fun a =>\n let \u27e8b, h, ab\u27e9 := H a\n \u27e8\u27e8\u27e8_, h\u27e9, _, ab\u27e9, rfl\u27e9\u27e9 ** case intro.intro.intro.intro.intro.refine'_2 \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) i : \u2191S \u22a2 (fun x => card (typein r \u2191x)) i \u2264 c ** rw [\u2190 lt_succ_iff, \u2190 lt_ord, \u2190 \u03b1e, re] ** case intro.intro.intro.intro.intro.refine'_2 \u03b1\u271d : Type u_1 r\u271d : \u03b1\u271d \u2192 \u03b1\u271d \u2192 Prop c : Cardinal.{u} h : \u2135\u2080 \u2264 c \u03b1 : Type u \u03b1e : #\u03b1 = succ c r : \u03b1 \u2192 \u03b1 \u2192 Prop wo : IsWellOrder \u03b1 r re : ord #\u03b1 = type r this : Ordinal.IsLimit (type r) S : Set \u03b1 H : \u2200 (a : \u03b1), \u2203 b, b \u2208 S \u2227 r a b Se : #\u2191S = Ordinal.cof (type r) i : \u2191S \u22a2 typein r \u2191i < type r ** apply typein_lt_type ** Qed", + "informal": "" + }, + { + "formal": "Vector.ext ** n : \u2115 \u03b1 : Type u_1 v : List \u03b1 hv : List.length v = n w : List \u03b1 hw : List.length w = n h : \u2200 (m : Fin n), get { val := v, property := hv } m = get { val := w, property := hw } m \u22a2 List.length \u2191{ val := v, property := hv } = List.length \u2191{ val := w, property := hw } ** rw [hv, hw] ** Qed", + "informal": "" + }, + { + "formal": "Quiver.Path.eq_of_length_zero ** V : Type u inst\u271d : Quiver V a b c d : V p : Path a b hzero : length p = 0 \u22a2 a = b ** cases p ** case nil V : Type u inst\u271d : Quiver V a c d : V hzero : length nil = 0 \u22a2 a = a ** rfl ** case cons V : Type u inst\u271d : Quiver V a b c d b\u271d : V a\u271d\u00b9 : Path a b\u271d a\u271d : b\u271d \u27f6 b hzero : length (cons a\u271d\u00b9 a\u271d) = 0 \u22a2 a = b ** cases Nat.succ_ne_zero _ hzero ** Qed", + "informal": "" + }, + { + "formal": "ContDiffBump.neg ** E : Type u_1 X : Type u_2 inst\u271d\u2074 : NormedAddCommGroup E inst\u271d\u00b3 : NormedSpace \u211d E inst\u271d\u00b2 : NormedAddCommGroup X inst\u271d\u00b9 : NormedSpace \u211d X inst\u271d : HasContDiffBump E c : E f\u271d : ContDiffBump c x\u271d : E n : \u2115\u221e f : ContDiffBump 0 x : E \u22a2 \u2191f (-x) = \u2191f x ** simp_rw [\u2190 zero_sub, f.sub, zero_add] ** Qed", + "informal": "" + }, + { + "formal": "Nat.psp_from_prime_psp ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) \u22a2 FermatPsp (Nat.psp_from_prime b p) b ** unfold psp_from_prime ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) \u22a2 FermatPsp ((b ^ p - 1) / (b - 1) * ((b ^ p + 1) / (b + 1))) b ** set A := (b ^ p - 1) / (b - 1) ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) \u22a2 FermatPsp (A * ((b ^ p + 1) / (b + 1))) b ** set B := (b ^ p + 1) / (b + 1) ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) \u22a2 FermatPsp (A * B) b ** have hi_A : 1 < A := a_id_helper (Nat.succ_le_iff.mp b_ge_two) (Nat.Prime.one_lt p_prime) ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A \u22a2 FermatPsp (A * B) b ** have hi_B : 1 < B := b_id_helper (Nat.succ_le_iff.mp b_ge_two) p_gt_two ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B \u22a2 FermatPsp (A * B) b ** have hi_AB : 1 < A * B := one_lt_mul'' hi_A hi_B ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B \u22a2 FermatPsp (A * B) b ** have hi_b : 0 < b := by linarith ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b \u22a2 FermatPsp (A * B) b ** have hi_p : 1 \u2264 p := Nat.one_le_of_lt p_gt_two ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p \u22a2 FermatPsp (A * B) b ** have hi_bsquared : 0 < b ^ 2 - 1 := by\n have h0 := mul_le_mul b_ge_two b_ge_two zero_le_two hi_b.le\n have h1 : 1 < 2 * 2 := by linarith\n have := tsub_pos_of_lt (h1.trans_le h0)\n rwa [pow_two] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 \u22a2 FermatPsp (A * B) b ** have hi_bpowtwop : 1 \u2264 b ^ (2 * p) := Nat.one_le_pow (2 * p) b hi_b ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) \u22a2 FermatPsp (A * B) b ** have hi_bpowpsubone : 1 \u2264 b ^ (p - 1) := Nat.one_le_pow (p - 1) b hi_b ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) \u22a2 FermatPsp (A * B) b ** have p_odd : Odd p := p_prime.odd_of_ne_two p_gt_two.ne.symm ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p \u22a2 FermatPsp (A * B) b ** have AB_not_prime : \u00acNat.Prime (A * B) := Nat.not_prime_mul hi_A hi_B ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) \u22a2 FermatPsp (A * B) b ** have AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) := AB_id_helper _ _ b_ge_two p_odd ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) \u22a2 FermatPsp (A * B) b ** have hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 := by\n simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 \u22a2 FermatPsp (A * B) b ** refine' \u27e8_, AB_not_prime, hi_AB\u27e9 ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 \u22a2 ProbablePrime (A * B) b ** have ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) := by\n apply_fun fun x => x * (b ^ 2 - 1) at AB_id\n rw [Nat.div_mul_cancel hd] at AB_id\n apply_fun fun x => x - (b ^ 2 - 1) at AB_id\n nth_rw 2 [\u2190 one_mul (b ^ 2 - 1)] at AB_id\n rw [\u2190 Nat.mul_sub_right_distrib, mul_comm] at AB_id\n rw [AB_id]\n exact bp_helper hi_b hi_p ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) \u22a2 ProbablePrime (A * B) b ** have ha\u2082 : 2 \u2223 b ^ p + b := by\n rw [\u2190 even_iff_two_dvd, Nat.even_add, Nat.even_pow' p_prime.ne_zero] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b \u22a2 ProbablePrime (A * B) b ** have ha\u2083 : p \u2223 b ^ (p - 1) - 1 := by\n have : \u00acp \u2223 b := mt (fun h : p \u2223 b => dvd_mul_of_dvd_left h _) not_dvd\n have : p.Coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) this\n have : IsCoprime (b : \u2124) \u2191p := this.symm.isCoprime\n have : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] := Int.ModEq.pow_card_sub_one_eq_one p_prime this\n have : \u2191p \u2223 \u2191b ^ (p - 1) - \u21911 := by exact_mod_cast Int.ModEq.dvd (Int.ModEq.symm this)\n exact_mod_cast this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 \u22a2 ProbablePrime (A * B) b ** have ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 := by\n cases' p_odd with k hk\n have : 2 \u2223 p - 1 := \u27e8k, by simp [hk]\u27e9\n cases' this with c hc\n have : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 := by\n simpa only [one_pow] using nat_sub_dvd_pow_sub_pow _ 1 c\n have : b ^ 2 - 1 \u2223 b ^ (2 * c) - 1 := by rwa [\u2190 pow_mul] at this\n rwa [\u2190 hc] at this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) \u22a2 ProbablePrime (A * B) b ** have ha\u2086 : 2 * p \u2223 A * B - 1 := by\n rw [mul_comm] at ha\u2085\n exact Nat.dvd_of_mul_dvd_mul_left hi_bsquared ha\u2085 ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2086 : 2 * p \u2223 A * B - 1 \u22a2 ProbablePrime (A * B) b ** have ha\u2087 : A * B \u2223 b ^ (2 * p) - 1 := by\n use b ^ 2 - 1\n have : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1) :=\n congr_arg (fun x : \u2115 => x * (b ^ 2 - 1)) AB_id\n simpa only [add_comm, Nat.div_mul_cancel hd, Nat.sub_add_cancel hi_bpowtwop] using this.symm ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2086 : 2 * p \u2223 A * B - 1 ha\u2087 : A * B \u2223 b ^ (2 * p) - 1 \u22a2 ProbablePrime (A * B) b ** cases' ha\u2086 with q hq ** case intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2087 : A * B \u2223 b ^ (2 * p) - 1 q : \u2115 hq : A * B - 1 = 2 * p * q \u22a2 ProbablePrime (A * B) b ** have ha\u2088 : b ^ (2 * p) - 1 \u2223 b ^ (A * B - 1) - 1 := by\n simpa only [one_pow, pow_mul, hq] using nat_sub_dvd_pow_sub_pow _ 1 q ** case intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2087 : A * B \u2223 b ^ (2 * p) - 1 q : \u2115 hq : A * B - 1 = 2 * p * q ha\u2088 : b ^ (2 * p) - 1 \u2223 b ^ (A * B - 1) - 1 \u22a2 ProbablePrime (A * B) b ** exact dvd_trans ha\u2087 ha\u2088 ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B \u22a2 0 < b ** linarith ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p \u22a2 0 < b ^ 2 - 1 ** have h0 := mul_le_mul b_ge_two b_ge_two zero_le_two hi_b.le ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p h0 : 2 * 2 \u2264 b * b \u22a2 0 < b ^ 2 - 1 ** have h1 : 1 < 2 * 2 := by linarith ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p h0 : 2 * 2 \u2264 b * b h1 : 1 < 2 * 2 \u22a2 0 < b ^ 2 - 1 ** have := tsub_pos_of_lt (h1.trans_le h0) ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p h0 : 2 * 2 \u2264 b * b h1 : 1 < 2 * 2 this : 0 < b * b - 1 \u22a2 0 < b ^ 2 - 1 ** rwa [pow_two] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p h0 : 2 * 2 \u2264 b * b \u22a2 1 < 2 * 2 ** linarith ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) \u22a2 b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ** simpa only [one_pow, pow_mul] using nat_sub_dvd_pow_sub_pow _ 1 p ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 \u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** apply_fun fun x => x * (b ^ 2 - 1) at AB_id ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 AB_id : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1) \u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** rw [Nat.div_mul_cancel hd] at AB_id ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 AB_id : A * B * (b ^ 2 - 1) = b ^ (2 * p) - 1 \u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** apply_fun fun x => x - (b ^ 2 - 1) at AB_id ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 AB_id : A * B * (b ^ 2 - 1) - (b ^ 2 - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1) \u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** nth_rw 2 [\u2190 one_mul (b ^ 2 - 1)] at AB_id ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 AB_id : A * B * (b ^ 2 - 1) - 1 * (b ^ 2 - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1) \u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** rw [\u2190 Nat.mul_sub_right_distrib, mul_comm] at AB_id ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 AB_id : (b ^ 2 - 1) * (A * B - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1) \u22a2 (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** rw [AB_id] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 AB_id : (b ^ 2 - 1) * (A * B - 1) = b ^ (2 * p) - 1 - (b ^ 2 - 1) \u22a2 b ^ (2 * p) - 1 - (b ^ 2 - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ** exact bp_helper hi_b hi_p ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) \u22a2 2 \u2223 b ^ p + b ** rw [\u2190 even_iff_two_dvd, Nat.even_add, Nat.even_pow' p_prime.ne_zero] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b \u22a2 p \u2223 b ^ (p - 1) - 1 ** have : \u00acp \u2223 b := mt (fun h : p \u2223 b => dvd_mul_of_dvd_left h _) not_dvd ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b this : \u00acp \u2223 b \u22a2 p \u2223 b ^ (p - 1) - 1 ** have : p.Coprime b := Or.resolve_right (Nat.coprime_or_dvd_of_prime p_prime b) this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b this\u271d : \u00acp \u2223 b this : Coprime p b \u22a2 p \u2223 b ^ (p - 1) - 1 ** have : IsCoprime (b : \u2124) \u2191p := this.symm.isCoprime ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b this\u271d\u00b9 : \u00acp \u2223 b this\u271d : Coprime p b this : IsCoprime \u2191b \u2191p \u22a2 p \u2223 b ^ (p - 1) - 1 ** have : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] := Int.ModEq.pow_card_sub_one_eq_one p_prime this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b this\u271d\u00b2 : \u00acp \u2223 b this\u271d\u00b9 : Coprime p b this\u271d : IsCoprime \u2191b \u2191p this : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] \u22a2 p \u2223 b ^ (p - 1) - 1 ** have : \u2191p \u2223 \u2191b ^ (p - 1) - \u21911 := by exact_mod_cast Int.ModEq.dvd (Int.ModEq.symm this) ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b this\u271d\u00b3 : \u00acp \u2223 b this\u271d\u00b2 : Coprime p b this\u271d\u00b9 : IsCoprime \u2191b \u2191p this\u271d : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] this : p \u2223 b ^ (p - 1) - 1 \u22a2 p \u2223 b ^ (p - 1) - 1 ** exact_mod_cast this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b this\u271d\u00b2 : \u00acp \u2223 b this\u271d\u00b9 : Coprime p b this\u271d : IsCoprime \u2191b \u2191p this : \u2191b ^ (p - 1) \u2261 1 [ZMOD \u2191p] \u22a2 p \u2223 b ^ (p - 1) - 1 ** exact_mod_cast Int.ModEq.dvd (Int.ModEq.symm this) ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 \u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ** cases' p_odd with k hk ** case intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 \u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ** have : 2 \u2223 p - 1 := \u27e8k, by simp [hk]\u27e9 ** case intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 this : 2 \u2223 p - 1 \u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ** cases' this with c hc ** case intro.intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 c : \u2115 hc : p - 1 = 2 * c \u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ** have : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 := by\n simpa only [one_pow] using nat_sub_dvd_pow_sub_pow _ 1 c ** case intro.intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 c : \u2115 hc : p - 1 = 2 * c this : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 \u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ** have : b ^ 2 - 1 \u2223 b ^ (2 * c) - 1 := by rwa [\u2190 pow_mul] at this ** case intro.intro b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 c : \u2115 hc : p - 1 = 2 * c this\u271d : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 this : b ^ 2 - 1 \u2223 b ^ (2 * c) - 1 \u22a2 b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ** rwa [\u2190 hc] at this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 \u22a2 p - 1 = 2 * k ** simp [hk] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 c : \u2115 hc : p - 1 = 2 * c \u22a2 b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 ** simpa only [one_pow] using nat_sub_dvd_pow_sub_pow _ 1 c ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 k : \u2115 hk : p = 2 * k + 1 c : \u2115 hc : p - 1 = 2 * c this : b ^ 2 - 1 \u2223 (b ^ 2) ^ c - 1 \u22a2 b ^ 2 - 1 \u2223 b ^ (2 * c) - 1 ** rwa [\u2190 pow_mul] at this ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 \u22a2 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ** suffices q : 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) ** case q b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 \u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) ** have q\u2081 : Nat.Coprime p (b ^ 2 - 1) :=\n haveI q\u2082 : \u00acp \u2223 b ^ 2 - 1 := by\n rw [mul_comm] at not_dvd\n exact mt (fun h : p \u2223 b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd\n (Nat.Prime.coprime_iff_not_dvd p_prime).mpr q\u2082 ** case q b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 q\u2081 : Coprime p (b ^ 2 - 1) \u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) ** have q\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1 := Nat.Coprime.mul_dvd_of_dvd_of_dvd q\u2081 ha\u2083 ha\u2084 ** case q b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 q\u2081 : Coprime p (b ^ 2 - 1) q\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1 \u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) ** have q\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b) := mul_dvd_mul q\u2082 ha\u2082 ** case q b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 q\u2081 : Coprime p (b ^ 2 - 1) q\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1 q\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b) \u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) ** have q\u2084 : p * (b ^ 2 - 1) * 2 \u2223 b * ((b ^ (p - 1) - 1) * (b ^ p + b)) :=\n dvd_mul_of_dvd_right q\u2083 _ ** case q b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 q\u2081 : Coprime p (b ^ 2 - 1) q\u2082 : p * (b ^ 2 - 1) \u2223 b ^ (p - 1) - 1 q\u2083 : p * (b ^ 2 - 1) * 2 \u2223 (b ^ (p - 1) - 1) * (b ^ p + b) q\u2084 : p * (b ^ 2 - 1) * 2 \u2223 b * ((b ^ (p - 1) - 1) * (b ^ p + b)) \u22a2 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) ** rwa [mul_assoc, mul_comm, mul_assoc b] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 q : 2 * p * (b ^ 2 - 1) \u2223 b * (b ^ (p - 1) - 1) * (b ^ p + b) \u22a2 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ** rwa [ha\u2081] ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 \u22a2 \u00acp \u2223 b ^ 2 - 1 ** rw [mul_comm] at not_dvd ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 (b ^ 2 - 1) * b A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 \u22a2 \u00acp \u2223 b ^ 2 - 1 ** exact mt (fun h : p \u2223 b ^ 2 - 1 => dvd_mul_of_dvd_left h _) not_dvd ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) \u22a2 2 * p \u2223 A * B - 1 ** rw [mul_comm] at ha\u2085 ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : (b ^ 2 - 1) * (2 * p) \u2223 (b ^ 2 - 1) * (A * B - 1) \u22a2 2 * p \u2223 A * B - 1 ** exact Nat.dvd_of_mul_dvd_mul_left hi_bsquared ha\u2085 ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2086 : 2 * p \u2223 A * B - 1 \u22a2 A * B \u2223 b ^ (2 * p) - 1 ** use b ^ 2 - 1 ** case h b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2086 : 2 * p \u2223 A * B - 1 \u22a2 b ^ (2 * p) - 1 = A * B * (b ^ 2 - 1) ** have : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1) :=\n congr_arg (fun x : \u2115 => x * (b ^ 2 - 1)) AB_id ** case h b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2086 : 2 * p \u2223 A * B - 1 this : A * B * (b ^ 2 - 1) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) * (b ^ 2 - 1) \u22a2 b ^ (2 * p) - 1 = A * B * (b ^ 2 - 1) ** simpa only [add_comm, Nat.div_mul_cancel hd, Nat.sub_add_cancel hi_bpowtwop] using this.symm ** b : \u2115 b_ge_two : 2 \u2264 b p : \u2115 p_prime : Prime p p_gt_two : 2 < p not_dvd : \u00acp \u2223 b * (b ^ 2 - 1) A : \u2115 := (b ^ p - 1) / (b - 1) B : \u2115 := (b ^ p + 1) / (b + 1) hi_A : 1 < A hi_B : 1 < B hi_AB : 1 < A * B hi_b : 0 < b hi_p : 1 \u2264 p hi_bsquared : 0 < b ^ 2 - 1 hi_bpowtwop : 1 \u2264 b ^ (2 * p) hi_bpowpsubone : 1 \u2264 b ^ (p - 1) p_odd : Odd p AB_not_prime : \u00acPrime (A * B) AB_id : A * B = (b ^ (2 * p) - 1) / (b ^ 2 - 1) hd : b ^ 2 - 1 \u2223 b ^ (2 * p) - 1 ha\u2081 : (b ^ 2 - 1) * (A * B - 1) = b * (b ^ (p - 1) - 1) * (b ^ p + b) ha\u2082 : 2 \u2223 b ^ p + b ha\u2083 : p \u2223 b ^ (p - 1) - 1 ha\u2084 : b ^ 2 - 1 \u2223 b ^ (p - 1) - 1 ha\u2085 : 2 * p * (b ^ 2 - 1) \u2223 (b ^ 2 - 1) * (A * B - 1) ha\u2087 : A * B \u2223 b ^ (2 * p) - 1 q : \u2115 hq : A * B - 1 = 2 * p * q \u22a2 b ^ (2 * p) - 1 \u2223 b ^ (A * B - 1) - 1 ** simpa only [one_pow, pow_mul, hq] using nat_sub_dvd_pow_sub_pow _ 1 q ** Qed", + "informal": "" + }, + { + "formal": "Seminorm.uniformContinuous_of_continuousAt_zero ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : SeminormedRing \ud835\udd5d inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \ud835\udd5c E inst\u271d\u00b2 : Module \ud835\udd5d E inst\u271d\u00b9 : UniformSpace E inst\u271d : UniformAddGroup E p : Seminorm \ud835\udd5d E hp : ContinuousAt (\u2191p) 0 \u22a2 UniformContinuous \u2191p ** have hp : Filter.Tendsto p (\ud835\udcdd 0) (\ud835\udcdd 0) := map_zero p \u25b8 hp ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : SeminormedRing \ud835\udd5d inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \ud835\udd5c E inst\u271d\u00b2 : Module \ud835\udd5d E inst\u271d\u00b9 : UniformSpace E inst\u271d : UniformAddGroup E p : Seminorm \ud835\udd5d E hp\u271d : ContinuousAt (\u2191p) 0 hp : Tendsto (\u2191p) (\ud835\udcdd 0) (\ud835\udcdd 0) \u22a2 UniformContinuous \u2191p ** rw [UniformContinuous, uniformity_eq_comap_nhds_zero_swapped,\n Metric.uniformity_eq_comap_nhds_zero, Filter.tendsto_comap_iff] ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2075 : SeminormedRing \ud835\udd5d inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : Module \ud835\udd5c E inst\u271d\u00b2 : Module \ud835\udd5d E inst\u271d\u00b9 : UniformSpace E inst\u271d : UniformAddGroup E p : Seminorm \ud835\udd5d E hp\u271d : ContinuousAt (\u2191p) 0 hp : Tendsto (\u2191p) (\ud835\udcdd 0) (\ud835\udcdd 0) \u22a2 Tendsto ((fun p => dist p.1 p.2) \u2218 fun x => (\u2191p x.1, \u2191p x.2)) (comap (fun x => x.1 - x.2) (\ud835\udcdd 0)) (\ud835\udcdd 0) ** exact\n tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (hp.comp Filter.tendsto_comap)\n (fun xy => dist_nonneg) fun xy => p.norm_sub_map_le_sub _ _ ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_monomial_mul ** R : Type u S : Type v a b : R n\u271d m : \u2115 inst\u271d : Semiring R p\u271d q r\u271d p : R[X] n d : \u2115 r : R \u22a2 coeff (\u2191(monomial n) r * p) (d + n) = r * coeff p d ** rw [\u2190 C_mul_X_pow_eq_monomial, mul_assoc, coeff_C_mul, X_pow_mul, coeff_mul_X_pow] ** Qed", + "informal": "" + }, + { + "formal": "Int.natAbs_eq_iff_mul_self_eq ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 \u22a2 natAbs a = natAbs b \u2194 a * a = b * b ** rw [\u2190 abs_eq_iff_mul_self_eq, abs_eq_natAbs, abs_eq_natAbs] ** a\u271d b\u271d : \u2124 n : \u2115 a b : \u2124 \u22a2 natAbs a = natAbs b \u2194 \u2191(natAbs a) = \u2191(natAbs b) ** exact Int.coe_nat_inj'.symm ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.measure_compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 h\u2081 : MeasurableSet s h_fin : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc s\u1d9c = \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s ** rw [compl_eq_univ_diff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b9 : Type u_5 R : Type u_6 R' : Type u_7 m : MeasurableSpace \u03b1 \u03bc \u03bc\u2081 \u03bc\u2082 : Measure \u03b1 s s\u2081 s\u2082 t : Set \u03b1 h\u2081 : MeasurableSet s h_fin : \u2191\u2191\u03bc s \u2260 \u22a4 \u22a2 \u2191\u2191\u03bc (univ \\ s) = \u2191\u2191\u03bc univ - \u2191\u2191\u03bc s ** exact measure_diff (subset_univ s) h\u2081 h_fin ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_lt_one ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : OrderedCancelCommMonoid M f g : \u03b9 \u2192 M s t : Finset \u03b9 h : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < 1 hs : Finset.Nonempty s \u22a2 \u220f i in s, 1 \u2264 1 ** rw [prod_const_one] ** Qed", + "informal": "" + }, + { + "formal": "Mathlib.Meta.NormNum.LegendreSym.to_jacobiSym ** p : \u2115 pp : Fact (Nat.Prime p) a r : \u2124 hr : IsInt (jacobiSym a p) r \u22a2 IsInt (legendreSym p a) r ** rwa [@jacobiSym.legendreSym.to_jacobiSym p pp a] ** Qed", + "informal": "" + }, + { + "formal": "Real.surjOn_log' ** x\u271d\u00b9 y x : \u211d x\u271d : x \u2208 univ \u22a2 log (-rexp x) = x ** rw [log_neg_eq_log, log_exp] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.dvd_iff_modByMonic_eq_zero ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : p %\u2098 q = 0 \u22a2 q \u2223 p ** rw [\u2190 modByMonic_add_div p hq, h, zero_add] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : p %\u2098 q = 0 \u22a2 q \u2223 q * (p /\u2098 q) ** exact dvd_mul_right _ _ ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u22a2 p %\u2098 q = 0 ** nontriviality R ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R \u22a2 p %\u2098 q = 0 ** obtain \u27e8r, hr\u27e9 := exists_eq_mul_right_of_dvd h ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r \u22a2 p %\u2098 q = 0 ** by_contra hpq0 ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 \u22a2 False ** have hmod : p %\u2098 q = q * (r - p /\u2098 q) := by rw [modByMonic_eq_sub_mul_div _ hq, mul_sub, \u2190 hr] ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) \u22a2 False ** have : degree (q * (r - p /\u2098 q)) < degree q := hmod \u25b8 degree_modByMonic_lt _ hq ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) this : degree (q * (r - p /\u2098 q)) < degree q \u22a2 False ** have hrpq0 : leadingCoeff (r - p /\u2098 q) \u2260 0 := fun h =>\n hpq0 <|\n leadingCoeff_eq_zero.1\n (by rw [hmod, leadingCoeff_eq_zero.1 h, mul_zero, leadingCoeff_zero]) ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) this : degree (q * (r - p /\u2098 q)) < degree q hrpq0 : leadingCoeff (r - p /\u2098 q) \u2260 0 \u22a2 False ** have hlc : leadingCoeff q * leadingCoeff (r - p /\u2098 q) \u2260 0 := by rwa [Monic.def.1 hq, one_mul] ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) this : degree (q * (r - p /\u2098 q)) < degree q hrpq0 : leadingCoeff (r - p /\u2098 q) \u2260 0 hlc : leadingCoeff q * leadingCoeff (r - p /\u2098 q) \u2260 0 \u22a2 False ** rw [degree_mul' hlc, degree_eq_natDegree hq.ne_zero,\n degree_eq_natDegree (mt leadingCoeff_eq_zero.2 hrpq0)] at this ** case intro R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) this : \u2191(natDegree q) + \u2191(natDegree (r - p /\u2098 q)) < \u2191(natDegree q) hrpq0 : leadingCoeff (r - p /\u2098 q) \u2260 0 hlc : leadingCoeff q * leadingCoeff (r - p /\u2098 q) \u2260 0 \u22a2 False ** exact not_lt_of_ge (Nat.le_add_right _ _) (WithBot.some_lt_some.1 this) ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 \u22a2 p %\u2098 q = q * (r - p /\u2098 q) ** rw [modByMonic_eq_sub_mul_div _ hq, mul_sub, \u2190 hr] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h\u271d : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) this : degree (q * (r - p /\u2098 q)) < degree q h : leadingCoeff (r - p /\u2098 q) = 0 \u22a2 leadingCoeff (p %\u2098 q) = 0 ** rw [hmod, leadingCoeff_eq_zero.1 h, mul_zero, leadingCoeff_zero] ** R : Type u S : Type v T : Type w A : Type z a b : R n : \u2115 inst\u271d : CommRing R p q : R[X] hq : Monic q h : q \u2223 p \u271d : Nontrivial R r : R[X] hr : p = q * r hpq0 : \u00acp %\u2098 q = 0 hmod : p %\u2098 q = q * (r - p /\u2098 q) this : degree (q * (r - p /\u2098 q)) < degree q hrpq0 : leadingCoeff (r - p /\u2098 q) \u2260 0 \u22a2 leadingCoeff q * leadingCoeff (r - p /\u2098 q) \u2260 0 ** rwa [Monic.def.1 hq, one_mul] ** Qed", + "informal": "" + }, + { + "formal": "Int.units_inv_eq_self ** u : \u2124\u02e3 \u22a2 u\u207b\u00b9 = u ** rw [inv_eq_iff_mul_eq_one, units_mul_self] ** Qed", + "informal": "" + }, + { + "formal": "self_diff_frontier ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w a : \u03b1 s\u271d s\u2081 s\u2082 t : Set \u03b1 p p\u2081 p\u2082 : \u03b1 \u2192 Prop inst\u271d : TopologicalSpace \u03b1 s : Set \u03b1 \u22a2 s \\ frontier s = interior s ** rw [frontier, diff_diff_right, diff_eq_empty.2 subset_closure,\n inter_eq_self_of_subset_right interior_subset, empty_union] ** Qed", + "informal": "" + }, + { + "formal": "StructureGroupoid.LocalInvariantProp.right_invariance ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source \u22a2 P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) \u2194 P f s x ** refine' \u27e8fun h \u21a6 _, hG.right_invariance' he hxe\u27e9 ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) \u22a2 P f s x ** have := hG.right_invariance' (G.symm he) (e.mapsTo hxe) h ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e))) (\u2191(LocalHomeomorph.symm (LocalHomeomorph.symm e)) \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) (\u2191(LocalHomeomorph.symm e) (\u2191e x)) \u22a2 P f s x ** rw [e.symm_symm, e.left_inv hxe] at this ** H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) x \u22a2 P f s x ** refine' hG.congr _ ((hG.congr_set _).mp this) ** case refine'_1 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) x \u22a2 (f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e =\u1da0[\ud835\udcdd x] f ** refine' eventually_of_mem (e.open_source.mem_nhds hxe) fun x' hx' \u21a6 _ ** case refine'_1 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) x x' : H hx' : x' \u2208 e.source \u22a2 ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) x' = f x' ** simp_rw [Function.comp_apply, e.left_inv hx'] ** case refine'_2 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) x \u22a2 \u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) =\u1da0[\ud835\udcdd x] s ** rw [eventuallyEq_set] ** case refine'_2 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) x \u22a2 \u2200\u1da0 (x : H) in \ud835\udcdd x, x \u2208 \u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2194 x \u2208 s ** refine' eventually_of_mem (e.open_source.mem_nhds hxe) fun x' hx' \u21a6 _ ** case refine'_2 H : Type u_1 M : Type u_2 H' : Type u_3 M' : Type u_4 X : Type u_5 inst\u271d\u2076 : TopologicalSpace H inst\u271d\u2075 : TopologicalSpace M inst\u271d\u2074 : ChartedSpace H M inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' inst\u271d\u00b9 : ChartedSpace H' M' inst\u271d : TopologicalSpace X G : StructureGroupoid H G' : StructureGroupoid H' P : (H \u2192 H') \u2192 Set H \u2192 H \u2192 Prop s\u271d t u : Set H x\u271d : H hG : LocalInvariantProp G G' P s : Set H x : H f : H \u2192 H' e : LocalHomeomorph H H he : e \u2208 G hxe : x \u2208 e.source h : P (f \u2218 \u2191(LocalHomeomorph.symm e)) (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) (\u2191e x) this : P ((f \u2218 \u2191(LocalHomeomorph.symm e)) \u2218 \u2191e) (\u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s)) x x' : H hx' : x' \u2208 e.source \u22a2 x' \u2208 \u2191e \u207b\u00b9' (\u2191(LocalHomeomorph.symm e) \u207b\u00b9' s) \u2194 x' \u2208 s ** simp_rw [mem_preimage, e.left_inv hx'] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.eq_zero_of_basicOpen_eq_bot ** X\u271d : Scheme X : Scheme hX : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 \u22a2 s = 0 ** apply TopCat.Presheaf.section_ext X.sheaf U ** case h X\u271d : Scheme X : Scheme hX : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 \u22a2 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) 0 ** conv => intro x; rw [RingHom.map_zero] ** case h X\u271d : Scheme X : Scheme hX : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 \u22a2 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** refine' (@reduce_to_affine_global (fun X U =>\n \u2200 [IsReduced X] (s : X.presheaf.obj (op U)),\n X.basicOpen s = \u22a5 \u2192 \u2200 x, (X.sheaf.presheaf.germ x) s = 0) _ _ _) X U s hs ** case h.refine'_1 X\u271d : Scheme X : Scheme hX : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 \u22a2 \u2200 (X : Scheme) (U : Opens \u2191\u2191X.toPresheafedSpace), (\u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V) \u2192 (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X U ** intro X U hx hX s hs x ** case h.refine'_1 X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** obtain \u27e8V, hx, i, H\u27e9 := hx x ** case h.refine'_1.intro.intro.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx\u271d : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } V : Opens \u2191\u2191X.toPresheafedSpace hx : \u2191x \u2208 V i : V \u27f6 U H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op V))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** specialize H (X.presheaf.map i.op s) ** case h.refine'_1.intro.intro.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx\u271d : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } V : Opens \u2191\u2191X.toPresheafedSpace hx : \u2191x \u2208 V i : V \u27f6 U H : Scheme.basicOpen X (\u2191(X.presheaf.map i.op) s) = \u22a5 \u2192 \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) (\u2191(X.presheaf.map i.op) s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** erw [Scheme.basicOpen_res] at H ** case h.refine'_1.intro.intro.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx\u271d : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } V : Opens \u2191\u2191X.toPresheafedSpace hx : \u2191x \u2208 V i : V \u27f6 U H : V \u2293 Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) (\u2191(X.presheaf.map i.op) s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** rw [hs] at H ** case h.refine'_1.intro.intro.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx\u271d : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } V : Opens \u2191\u2191X.toPresheafedSpace hx : \u2191x \u2208 V i : V \u27f6 U H : V \u2293 \u22a5 = \u22a5 \u2192 \u2200 (x : { x // x \u2208 V }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) (\u2191(X.presheaf.map i.op) s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** specialize H inf_bot_eq \u27e8x, hx\u27e9 ** case h.refine'_1.intro.intro.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx\u271d : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } V : Opens \u2191\u2191X.toPresheafedSpace hx : \u2191x \u2208 V i : V \u27f6 U H : \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) { val := \u2191x, property := hx }) (\u2191(X.presheaf.map i.op) s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** erw [TopCat.Presheaf.germ_res_apply] at H ** case h.refine'_1.intro.intro.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U\u271d : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U\u271d)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X : Scheme U : Opens \u2191\u2191X.toPresheafedSpace hx\u271d : \u2200 (x : { x // x \u2208 U }), \u2203 V x x, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X V hX : IsReduced X s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 x : { x // x \u2208 U } V : Opens \u2191\u2191X.toPresheafedSpace hx : \u2191x \u2208 V i : V \u27f6 U H : \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) ((fun x => { val := \u2191x, property := (_ : \u2191x \u2208 \u2191U) }) { val := \u2191x, property := hx })) s = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 ** exact H ** case h.refine'_2 X\u271d : Scheme X : Scheme hX : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 \u22a2 \u2200 {X Y : Scheme} (f : X \u27f6 Y) [hf : IsOpenImmersion f], \u2203 U V hU hV, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) Y { carrier := V, is_open' := (_ : IsOpen V) } ** rintro X Y f hf ** case h.refine'_2 X\u271d\u00b9 : Scheme X\u271d : Scheme hX : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s : \u2191(X\u271d.presheaf.obj (op U)) hs : Scheme.basicOpen X\u271d s = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f \u22a2 \u2203 U V hU hV, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) Y { carrier := V, is_open' := (_ : IsOpen V) } ** have e : f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ := by\n rw [\u2190 Set.image_univ, Set.preimage_image_eq _ hf.base_open.inj] ** case h.refine'_2 X\u271d\u00b9 : Scheme X\u271d : Scheme hX : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s : \u2191(X\u271d.presheaf.obj (op U)) hs : Scheme.basicOpen X\u271d s = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ \u22a2 \u2203 U V hU hV, (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X { carrier := U, is_open' := (_ : IsOpen U) } \u2192 (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) Y { carrier := V, is_open' := (_ : IsOpen V) } ** refine' \u27e8_, _, e, rfl, _\u27e9 ** case h.refine'_2 X\u271d\u00b9 : Scheme X\u271d : Scheme hX : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s : \u2191(X\u271d.presheaf.obj (op U)) hs : Scheme.basicOpen X\u271d s = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ \u22a2 (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) X { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } \u2192 (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) Y { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) } ** rintro H hX s hs \u27e8_, x, rfl\u27e9 ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y)) { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) }) s = 0 ** haveI := isReducedOfOpenImmersion f ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y)) { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) }) s = 0 ** specialize H (f.1.c.app _ s) _ \u27e8x, by rw [Opens.mem_mk, e]; trivial\u27e9 ** X\u271d\u00b9 : Scheme X\u271d : Scheme hX : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s : \u2191(X\u271d.presheaf.obj (op U)) hs : Scheme.basicOpen X\u271d s = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f \u22a2 \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ ** rw [\u2190 Set.image_univ, Set.preimage_image_eq _ hf.base_open.inj] ** X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X \u22a2 x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } ** rw [Opens.mem_mk, e] ** X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X \u22a2 x \u2208 Set.univ ** trivial ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X \u22a2 Scheme.basicOpen X (\u2191(f.val.c.app (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) s) = \u22a5 ** rw [\u2190 Scheme.preimage_basicOpen, hs] ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X \u22a2 f\u207b\u00b9\u1d41 \u22a5 = \u22a5 ** ext1 ** case h.refine'_2.mk.intro.h X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ H : \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) } }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0 hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X \u22a2 \u2191(f\u207b\u00b9\u1d41 \u22a5) = \u2191\u22a5 ** simp [Opens.map] ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X H : \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) { val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) }) (\u2191(f.val.c.app (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y)) { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) }) s = 0 ** erw [\u2190 PresheafedSpace.stalkMap_germ_apply f.1 \u27e8_, _\u27e9 \u27e8x, _\u27e9] at H ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X H : \u2191(PresheafedSpace.stalkMap f.val \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) }) (\u2191(Presheaf.germ Y.presheaf { val := \u2191f.val.base \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) }, property := (_ : \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) } \u2208 f\u207b\u00b9\u1d41 { carrier := fun x => \u2203 y, \u2191f.val.base y = x, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }) }) s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y)) { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) }) s = 0 ** apply_fun inv <| PresheafedSpace.stalkMap f.val x at H ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X H : \u2191(inv (PresheafedSpace.stalkMap f.val x)) (\u2191(PresheafedSpace.stalkMap f.val \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) }) (\u2191(Presheaf.germ Y.presheaf { val := \u2191f.val.base \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) }, property := (_ : \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) } \u2208 f\u207b\u00b9\u1d41 { carrier := fun x => \u2203 y, \u2191f.val.base y = x, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }) }) s)) = \u2191(inv (PresheafedSpace.stalkMap f.val x)) 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y)) { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) }) s = 0 ** erw [CategoryTheory.IsIso.hom_inv_id_apply, map_zero] at H ** case h.refine'_2.mk.intro X\u271d\u00b9 : Scheme X\u271d : Scheme hX\u271d : IsReduced X\u271d U : Opens \u2191\u2191X\u271d.toPresheafedSpace s\u271d : \u2191(X\u271d.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X\u271d s\u271d = \u22a5 X Y : Scheme f : X \u27f6 Y hf : IsOpenImmersion f e : \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base = Set.univ hX : IsReduced Y s : \u2191(Y.presheaf.obj (op { carrier := Set.range \u2191f.val.base, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) })) hs : Scheme.basicOpen Y s = \u22a5 x : (forget TopCat).obj \u2191X.toPresheafedSpace this : IsReduced X H : \u2191(Presheaf.germ Y.presheaf { val := \u2191f.val.base \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) }, property := (_ : \u2191{ val := x, property := (_ : x \u2208 { carrier := \u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base, is_open' := (_ : IsOpen (\u2191f.val.base \u207b\u00b9' Set.range \u2191f.val.base)) }) } \u2208 f\u207b\u00b9\u1d41 { carrier := fun x => \u2203 y, \u2191f.val.base y = x, is_open' := (_ : IsOpen (Set.range \u2191f.val.base)) }) }) s = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf Y)) { val := \u2191f.val.base x, property := (_ : \u2203 y, \u2191f.val.base y = \u2191f.val.base x) }) s = 0 ** exact H ** case h.refine'_3 X\u271d : Scheme X : Scheme hX : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s : \u2191(X.presheaf.obj (op U)) hs : Scheme.basicOpen X s = \u22a5 \u22a2 \u2200 (R : CommRingCat), (fun X U => \u2200 [inst : IsReduced X] (s : \u2191(X.presheaf.obj (op U))), Scheme.basicOpen X s = \u22a5 \u2192 \u2200 (x : { x // x \u2208 U }), \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf X)) x) s = 0) (Scheme.Spec.obj (op R)) \u22a4 ** intro R hX s hs x ** case h.refine'_3 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) hs : Scheme.basicOpen (Scheme.Spec.obj (op R)) s = \u22a5 x : { x // x \u2208 \u22a4 } \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0 ** erw [basicOpen_eq_of_affine', PrimeSpectrum.basicOpen_eq_bot_iff] at hs ** case h.refine'_3 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) hs : IsNilpotent (\u2191(Spec\u0393Identity.app R).hom s) x : { x // x \u2208 \u22a4 } \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0 ** replace hs := hs.map (Spec\u0393Identity.app R).inv ** case h.refine'_3 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) x : { x // x \u2208 \u22a4 } hs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s)) \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0 ** replace hs := @IsNilpotent.eq_zero _ _ _ _ (show _ from ?_) hs ** case h.refine'_3.refine_2 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) x : { x // x \u2208 \u22a4 } hs : \u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s) = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0 case h.refine'_3.refine_1 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) x : { x // x \u2208 \u22a4 } hs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s)) \u22a2 _root_.IsReduced ((fun x => \u2191((Spec.toLocallyRingedSpace.rightOp \u22d9 LocallyRingedSpace.\u0393).obj R)) (\u2191(Spec\u0393Identity.app R).hom s)) ** rw [Iso.hom_inv_id_apply] at hs ** case h.refine'_3.refine_2 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) x : { x // x \u2208 \u22a4 } hs : s = 0 \u22a2 \u2191(Presheaf.germ (Sheaf.presheaf (Scheme.sheaf (Scheme.Spec.obj (op R)))) x) s = 0 case h.refine'_3.refine_1 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) x : { x // x \u2208 \u22a4 } hs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s)) \u22a2 _root_.IsReduced ((fun x => \u2191((Spec.toLocallyRingedSpace.rightOp \u22d9 LocallyRingedSpace.\u0393).obj R)) (\u2191(Spec\u0393Identity.app R).hom s)) ** rw [hs, map_zero] ** case h.refine'_3.refine_1 X\u271d : Scheme X : Scheme hX\u271d : IsReduced X U : Opens \u2191\u2191X.toPresheafedSpace s\u271d : \u2191(X.presheaf.obj (op U)) hs\u271d : Scheme.basicOpen X s\u271d = \u22a5 R : CommRingCat hX : IsReduced (Scheme.Spec.obj (op R)) s : \u2191((Scheme.Spec.obj (op R)).presheaf.obj (op \u22a4)) x : { x // x \u2208 \u22a4 } hs : IsNilpotent (\u2191(Spec\u0393Identity.app R).inv (\u2191(Spec\u0393Identity.app R).hom s)) \u22a2 _root_.IsReduced ((fun x => \u2191((Spec.toLocallyRingedSpace.rightOp \u22d9 LocallyRingedSpace.\u0393).obj R)) (\u2191(Spec\u0393Identity.app R).hom s)) ** exact @IsReduced.component_reduced _ hX \u22a4 ** Qed", + "informal": "" + }, + { + "formal": "HNNExtension.lift_t ** G : Type u_1 inst\u271d\u00b2 : Group G A B : Subgroup G \u03c6 : { x // x \u2208 A } \u2243* { x // x \u2208 B } H : Type u_2 inst\u271d\u00b9 : Group H M : Type u_3 inst\u271d : Monoid M f : G \u2192* H x : H hx : \u2200 (a : { x // x \u2208 A }), x * \u2191f \u2191a = \u2191f \u2191(\u2191\u03c6 a) * x \u22a2 \u2191(lift f x hx) t = x ** simp [lift, t] ** Qed", + "informal": "" + }, + { + "formal": "HallMarriageTheorem.hall_hard_inductive_step_A ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** haveI : Nonempty \u03b9 := Fintype.card_pos_iff.mp (hn.symm \u25b8 Nat.succ_pos _) ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this : Nonempty \u03b9 \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** haveI := Classical.decEq \u03b9 ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** let x := Classical.arbitrary \u03b9 ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** have tx_ne : (t x).Nonempty := by\n rw [\u2190 Finset.card_pos]\n calc\n 0 < 1 := Nat.one_pos\n _ \u2264 (Finset.biUnion {x} t).card := ht {x}\n _ = (t x).card := by rw [Finset.singleton_biUnion] ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 tx_ne : Finset.Nonempty (t x) \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** choose y hy using tx_ne ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** let \u03b9' := { x' : \u03b9 | x' \u2260 x } ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** let t' : \u03b9' \u2192 Finset \u03b1 := fun x' => (t x').erase y ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** have card_\u03b9' : Fintype.card \u03b9' = n :=\n calc\n Fintype.card \u03b9' = Fintype.card \u03b9 - 1 := Set.card_ne_eq _\n _ = n := by rw [hn, Nat.add_succ_sub_one, add_zero] ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** rcases ih t' card_\u03b9'.le (hall_cond_of_erase y ha) with \u27e8f', hfinj, hfr\u27e9 ** case intro.intro \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x \u22a2 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9), f x \u2208 t x ** refine' \u27e8fun z => if h : z = x then y else f' \u27e8z, h\u27e9, _, _\u27e9 ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 \u22a2 Finset.Nonempty (t x) ** rw [\u2190 Finset.card_pos] ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 \u22a2 0 < card (t x) ** calc\n 0 < 1 := Nat.one_pos\n _ \u2264 (Finset.biUnion {x} t).card := ht {x}\n _ = (t x).card := by rw [Finset.singleton_biUnion] ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 \u22a2 card (Finset.biUnion {x} t) = card (t x) ** rw [Finset.singleton_biUnion] ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y \u22a2 Fintype.card \u03b9 - 1 = n ** rw [hn, Nat.add_succ_sub_one, add_zero] ** case intro.intro.refine'_1 \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x \u22a2 Function.Injective fun z => if h : z = x then y else f' { val := z, property := h } ** rintro z\u2081 z\u2082 ** case intro.intro.refine'_1 \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z\u2081 z\u2082 : \u03b9 \u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 = (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192 z\u2081 = z\u2082 ** have key : \u2200 {x}, y \u2260 f' x := by\n intro x h\n simpa [\u2190 h] using hfr x ** case intro.intro.refine'_1 \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z\u2081 z\u2082 : \u03b9 key : \u2200 {x : \u2191\u03b9'}, y \u2260 f' x \u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2081 = (fun z => if h : z = x then y else f' { val := z, property := h }) z\u2082 \u2192 z\u2081 = z\u2082 ** by_cases h\u2081 : z\u2081 = x <;> by_cases h\u2082 : z\u2082 = x <;> simp [h\u2081, h\u2082, hfinj.eq_iff, key, key.symm] ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z\u2081 z\u2082 : \u03b9 \u22a2 \u2200 {x : \u2191\u03b9'}, y \u2260 f' x ** intro x h ** \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x\u271d : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x\u271d \u03b9' : Set \u03b9 := {x' | x' \u2260 x\u271d} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z\u2081 z\u2082 : \u03b9 x : \u2191\u03b9' h : y = f' x \u22a2 False ** simpa [\u2190 h] using hfr x ** case intro.intro.refine'_2 \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x \u22a2 \u2200 (x_1 : \u03b9), (fun z => if h : z = x then y else f' { val := z, property := h }) x_1 \u2208 t x_1 ** intro z ** case intro.intro.refine'_2 \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z : \u03b9 \u22a2 (fun z => if h : z = x then y else f' { val := z, property := h }) z \u2208 t z ** simp only [ne_eq, Set.mem_setOf_eq] ** case intro.intro.refine'_2 \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z : \u03b9 \u22a2 (if h : z = Classical.arbitrary \u03b9 then y else f' { val := z, property := h }) \u2208 t z ** split_ifs with hz ** case pos \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z : \u03b9 hz : z = Classical.arbitrary \u03b9 \u22a2 y \u2208 t z ** rwa [hz] ** case neg \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' hfr : \u2200 (x : \u2191\u03b9'), f' x \u2208 t' x z : \u03b9 hz : \u00acz = Classical.arbitrary \u03b9 \u22a2 f' { val := z, property := hz } \u2208 t z ** specialize hfr \u27e8z, hz\u27e9 ** case neg \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' z : \u03b9 hz : \u00acz = Classical.arbitrary \u03b9 hfr : f' { val := z, property := hz } \u2208 t' { val := z, property := hz } \u22a2 f' { val := z, property := hz } \u2208 t z ** rw [mem_erase] at hfr ** case neg \u03b9 : Type u \u03b1 : Type v inst\u271d\u00b9 : DecidableEq \u03b1 t : \u03b9 \u2192 Finset \u03b1 inst\u271d : Fintype \u03b9 n : \u2115 hn : Fintype.card \u03b9 = n + 1 ht : \u2200 (s : Finset \u03b9), card s \u2264 card (Finset.biUnion s t) ih : \u2200 {\u03b9' : Type u} [inst : Fintype \u03b9'] (t' : \u03b9' \u2192 Finset \u03b1), Fintype.card \u03b9' \u2264 n \u2192 (\u2200 (s' : Finset \u03b9'), card s' \u2264 card (Finset.biUnion s' t')) \u2192 \u2203 f, Function.Injective f \u2227 \u2200 (x : \u03b9'), f x \u2208 t' x ha : \u2200 (s : Finset \u03b9), Finset.Nonempty s \u2192 s \u2260 univ \u2192 card s < card (Finset.biUnion s t) this\u271d : Nonempty \u03b9 this : DecidableEq \u03b9 x : \u03b9 := Classical.arbitrary \u03b9 y : \u03b1 hy : y \u2208 t x \u03b9' : Set \u03b9 := {x' | x' \u2260 x} t' : \u2191\u03b9' \u2192 Finset \u03b1 := fun x' => erase (t \u2191x') y card_\u03b9' : Fintype.card \u2191\u03b9' = n f' : \u2191\u03b9' \u2192 \u03b1 hfinj : Function.Injective f' z : \u03b9 hz : \u00acz = Classical.arbitrary \u03b9 hfr : f' { val := z, property := hz } \u2260 y \u2227 f' { val := z, property := hz } \u2208 t \u2191{ val := z, property := hz } \u22a2 f' { val := z, property := hz } \u2208 t z ** exact hfr.2 ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.is_greatest' ** \u03b1 : Type u_1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : \u03b1 m : \u2115 h : Finite a b hm : Part.get (multiplicity a b) h < m \u22a2 multiplicity a b < \u2191m ** rwa [\u2190 PartENat.coe_lt_coe, PartENat.natCast_get] at hm ** Qed", + "informal": "" + }, + { + "formal": "Finset.sdiff_union_erase_cancel ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t u v : Finset \u03b1 a b : \u03b1 hts : t \u2286 s ha : a \u2208 t \u22a2 s \\ t \u222a erase t a = erase s a ** simp_rw [erase_eq, sdiff_union_sdiff_cancel hts (singleton_subset_iff.2 ha)] ** Qed", + "informal": "" + }, + { + "formal": "List.chain_iff_forall\u2082 ** \u03b1 : Type u \u03b2 : Type v R r : \u03b1 \u2192 \u03b1 \u2192 Prop l l\u2081 l\u2082 : List \u03b1 a\u271d b a : \u03b1 \u22a2 Chain R a [] \u2194 [] = [] \u2228 Forall\u2082 R (a :: dropLast []) [] ** simp ** \u03b1 : Type u \u03b2 : Type v R r : \u03b1 \u2192 \u03b1 \u2192 Prop l\u271d l\u2081 l\u2082 : List \u03b1 a\u271d b\u271d a b : \u03b1 l : List \u03b1 \u22a2 Chain R a (b :: l) \u2194 b :: l = [] \u2228 Forall\u2082 R (a :: dropLast (b :: l)) (b :: l) ** by_cases h : l = [] <;>\nsimp [@chain_iff_forall\u2082 b l, dropLast, *] ** Qed", + "informal": "" + }, + { + "formal": "List.reduceOption_singleton ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 x : Option \u03b1 \u22a2 reduceOption [x] = Option.toList x ** cases x <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.isIso_of_epi_of_isSplitMono ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X Y : C f : X \u27f6 Y inst\u271d\u00b9 : IsSplitMono f inst\u271d : Epi f \u22a2 f \u226b retraction f = \ud835\udfd9 X ** simp ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C X Y : C f : X \u27f6 Y inst\u271d\u00b9 : IsSplitMono f inst\u271d : Epi f \u22a2 retraction f \u226b f = \ud835\udfd9 Y ** simp [\u2190 cancel_epi f] ** Qed", + "informal": "" + }, + { + "formal": "even_neg_two ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 R : Type u_4 inst\u271d : Ring \u03b1 a b : \u03b1 n : \u2115 \u22a2 Even (-2) ** simp only [even_neg, even_two] ** Qed", + "informal": "" + }, + { + "formal": "measurable_liminf' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** have : Countable (Subtype p) := Encodable.nonempty_encodable.1 hv.countable ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** rcases isEmpty_or_nonempty (Subtype p) with hp|hp ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** by_cases H : \u2203 (j : Subtype p), s j = \u2205 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 \u22a2 Measurable fun x => liminf (fun i => f i x) v ** simp_rw [hv.liminf_eq_ite, if_neg H] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** have : \u2200 i, Countable (s i) := fun i \u21a6 countable_coe_iff.2 (hs i) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** let m : Subtype p \u2192 Set \u03b4 := fun j \u21a6 {x | BddBelow (range (fun (i : s j) \u21a6 f i x))} ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** have m_meas : \u2200 j, MeasurableSet (m j) :=\n fun j \u21a6 measurableSet_bddBelow_range (fun (i : s j) \u21a6 hf i) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** have mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), x \u2209 m j} := by\n rw [setOf_forall]\n exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} \u22a2 Measurable fun x => if \u2200 (j : Subtype p), \u00acBddBelow (range fun i => f (\u2191i) x) then sSup \u2205 else \u2a06 j, \u2a05 i, f (\u2191i) x ** apply Measurable.piecewise mc_meas measurable_const ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} \u22a2 Measurable fun x => \u2a06 j, \u2a05 i, f (\u2191i) x ** apply measurable_iSup (fun j \u21a6 ?_) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** let reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x \u21a6 liminf_reparam (fun i \u21a6 f i x) s p ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** let F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x \u21a6 \u2a05 (i : s j), f i x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** have F0_meas : \u2200 j, Measurable (F0 j) := fun j \u21a6 measurable_iInf (fun (i : s j) \u21a6 hf i) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) \u22a2 Measurable fun b => \u2a05 i, f (\u2191i) b ** set F1 : \u03b4 \u2192 \u03b1 := fun x \u21a6 F0 (reparam x j) x with hF1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x \u22a2 Measurable F1 ** let g : \u2115 \u2192 Subtype p := choose (exists_surjective_nat (Subtype p)) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this : F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x \u22a2 Measurable F1 ** rw [this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this : F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x \u22a2 Measurable fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** apply Measurable.piecewise (m_meas j) (F0_meas j) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this : F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x \u22a2 Measurable fun x => F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** apply Measurable.find (fun n \u21a6 F0_meas (g n)) (fun n \u21a6 ?_) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k this : F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x n : \u2115 \u22a2 MeasurableSet {x | x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k} ** exact (m_meas (g n)).union mc_meas ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : IsEmpty (Subtype p) \u22a2 Measurable fun x => liminf (fun i => f i x) v ** simp [hv.liminf_eq_sSup_iUnion_iInter] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u2203 j, s \u2191j = \u2205 \u22a2 Measurable fun x => liminf (fun i => f i x) v ** simp_rw [hv.liminf_eq_ite, if_pos H, measurable_const] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) \u22a2 MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} ** rw [setOf_forall] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) \u22a2 MeasurableSet (\u22c2 i, {x | \u00acx \u2208 m i}) ** exact MeasurableSet.iInter (fun j \u21a6 (m_meas j).compl) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) \u22a2 \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** intro x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** by_cases H : \u2203 k, x \u2208 m k ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H\u271d : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 H : \u2203 k, x \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** rcases H with \u27e8k, hk\u27e9 ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 k : Subtype p hk : x \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** rcases choose_spec (exists_surjective_nat (Subtype p)) k with \u27e8n, rfl\u27e9 ** case pos.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 n : \u2115 hk : x \u2208 m (choose (_ : \u2203 f, Function.Surjective f) n) \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** exact \u27e8n, Or.inl hk\u27e9 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H\u271d : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 H : \u00ac\u2203 k, x \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** push_neg at H ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H\u271d : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) x : \u03b4 H : \u2200 (k : Subtype p), \u00acx \u2208 m k \u22a2 \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k ** exact \u27e80, Or.inr H\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k \u22a2 F1 = fun x => if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** ext x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 \u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** have A : reparam x j = if x \u2208 m j then j else g (Nat.find (Z x)) := rfl ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) \u22a2 F1 x = if x \u2208 m j then F0 j x else F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** split_ifs with hjx ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : x \u2208 m j \u22a2 F1 x = F0 j x ** have : reparam x j = j := by rw [A, if_pos hjx] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : x \u2208 m j this : reparam x j = j \u22a2 F1 x = F0 j x ** simp only [hF1, this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : x \u2208 m j \u22a2 reparam x j = j ** rw [A, if_pos hjx] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : \u00acx \u2208 m j \u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** have : reparam x j = g (Nat.find (Z x)) := by rw [A, if_neg hjx] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d\u00b9 : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this\u271d : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : \u00acx \u2208 m j this : reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) \u22a2 F1 x = F0 (g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k))) x ** simp only [hF1, this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b3\u2082 : Type u_4 \u03b4 : Type u_5 \u03b9\u271d : Sort y s\u271d t u : Set \u03b1 inst\u271d\u00b9\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9\u00b9 : MeasurableSpace \u03b1 inst\u271d\u00b9\u2070 : BorelSpace \u03b1 inst\u271d\u2079 : TopologicalSpace \u03b2 inst\u271d\u2078 : MeasurableSpace \u03b2 inst\u271d\u2077 : BorelSpace \u03b2 inst\u271d\u2076 : TopologicalSpace \u03b3 inst\u271d\u2075 : MeasurableSpace \u03b3 inst\u271d\u2074 : BorelSpace \u03b3 inst\u271d\u00b3 : MeasurableSpace \u03b4 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b1 inst\u271d\u00b9 : OrderTopology \u03b1 inst\u271d : SecondCountableTopology \u03b1 \u03b9 : Type u_6 \u03b9' : Type u_7 f : \u03b9 \u2192 \u03b4 \u2192 \u03b1 v : Filter \u03b9 hf : \u2200 (i : \u03b9), Measurable (f i) p : \u03b9' \u2192 Prop s : \u03b9' \u2192 Set \u03b9 hv : HasCountableBasis v p s hs : \u2200 (j : \u03b9'), Set.Countable (s j) this\u271d : Countable (Subtype p) hp : Nonempty (Subtype p) H : \u00ac\u2203 j, s \u2191j = \u2205 this : \u2200 (i : \u03b9'), Countable \u2191(s i) m : Subtype p \u2192 Set \u03b4 := fun j => {x | BddBelow (range fun i => f (\u2191i) x)} m_meas : \u2200 (j : Subtype p), MeasurableSet (m j) mc_meas : MeasurableSet {x | \u2200 (j : Subtype p), \u00acx \u2208 m j} j : Subtype p reparam : \u03b4 \u2192 Subtype p \u2192 Subtype p := fun x => liminf_reparam (fun i => f i x) s p F0 : Subtype p \u2192 \u03b4 \u2192 \u03b1 := fun j x => \u2a05 i, f (\u2191i) x F0_meas : \u2200 (j : Subtype p), Measurable (F0 j) F1 : \u03b4 \u2192 \u03b1 := fun x => F0 (reparam x j) x hF1 : F1 = fun x => F0 (reparam x j) x g : \u2115 \u2192 Subtype p := choose (_ : \u2203 f, Function.Surjective f) Z : \u2200 (x : \u03b4), \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k x : \u03b4 A : reparam x j = if x \u2208 m j then j else g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) hjx : \u00acx \u2208 m j \u22a2 reparam x j = g (Nat.find (_ : \u2203 n, x \u2208 m (g n) \u2228 \u2200 (k : Subtype p), \u00acx \u2208 m k)) ** rw [A, if_neg hjx] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Functor.initial_comp ** C : Type u\u2081 inst\u271d\u2074 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b3 : Category.{v\u2082, u\u2082} D E : Type u\u2083 inst\u271d\u00b2 : Category.{v\u2083, u\u2083} E F : C \u2964 D G : D \u2964 E inst\u271d\u00b9 : Initial F inst\u271d : Initial G \u22a2 Initial (F \u22d9 G) ** suffices Final (F \u22d9 G).op from initial_of_final_op _ ** C : Type u\u2081 inst\u271d\u2074 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b3 : Category.{v\u2082, u\u2082} D E : Type u\u2083 inst\u271d\u00b2 : Category.{v\u2083, u\u2083} E F : C \u2964 D G : D \u2964 E inst\u271d\u00b9 : Initial F inst\u271d : Initial G \u22a2 Final (F \u22d9 G).op ** exact final_comp F.op G.op ** Qed", + "informal": "" + }, + { + "formal": "Set.image_sigmaMk_preimage_sigmaMap ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) \u22a2 Sigma.mk i '' (g i \u207b\u00b9' s) = Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) ** refine' (image_sigmaMk_preimage_sigmaMap_subset f g i s).antisymm _ ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) \u22a2 Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) \u2286 Sigma.mk i '' (g i \u207b\u00b9' s) ** rintro \u27e8j, x\u27e9 \u27e8y, hys, hxy\u27e9 ** case mk.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j\u271d : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) j : \u03b9 x : \u03b1 j y : \u03b2 (f i) hys : y \u2208 s hxy : { fst := f i, snd := y } = Sigma.map f g { fst := j, snd := x } \u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy ** case mk.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j\u271d : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) j : \u03b9 x : \u03b1 j y : \u03b2 (f i) hys : y \u2208 s hxy : i = j \u2227 HEq y (g j x) \u22a2 { fst := j, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** rcases hxy with \u27e8rfl, hxy\u27e9 ** case mk.intro.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) y : \u03b2 (f i) hys : y \u2208 s x : \u03b1 i hxy : HEq y (g i x) \u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** rw [heq_iff_eq] at hxy ** case mk.intro.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) y : \u03b2 (f i) hys : y \u2208 s x : \u03b1 i hxy : y = g i x \u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** subst y ** case mk.intro.intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 \u03b2\u271d : \u03b9 \u2192 Type u_4 s\u271d s\u2081 s\u2082 : Set \u03b9 t t\u2081 t\u2082 : (i : \u03b9) \u2192 Set (\u03b1 i) u : Set ((i : \u03b9) \u00d7 \u03b1 i) x\u271d : (i : \u03b9) \u00d7 \u03b1 i i\u271d j : \u03b9 a : \u03b1 i\u271d \u03b2 : \u03b9' \u2192 Type u_5 f : \u03b9 \u2192 \u03b9' hf : Function.Injective f g : (i : \u03b9) \u2192 \u03b1 i \u2192 \u03b2 (f i) i : \u03b9 s : Set (\u03b2 (f i)) x : \u03b1 i hys : g i x \u2208 s \u22a2 { fst := i, snd := x } \u2208 Sigma.mk i '' (g i \u207b\u00b9' s) ** exact \u27e8x, hys, rfl\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Set.Iio_diff_Iio ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : LinearOrder \u03b1 a a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1 \u22a2 Iio b \\ Iio a = Ico a b ** rw [diff_eq, compl_Iio, inter_comm, Ici_inter_Iio] ** Qed", + "informal": "" + }, + { + "formal": "contMDiffWithinAt_fst ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s\u271d s\u2081 t : Set M x : M m n : \u2115\u221e s : Set (M \u00d7 N) p : M \u00d7 N \u22a2 ContMDiffWithinAt (ModelWithCorners.prod I J) I n Prod.fst s p ** rw [contMDiffWithinAt_iff'] ** \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s\u271d s\u2081 t : Set M x : M m n : \u2115\u221e s : Set (M \u00d7 N) p : M \u00d7 N \u22a2 ContinuousWithinAt Prod.fst s p \u2227 ContDiffWithinAt \ud835\udd5c n (\u2191(extChartAt I p.1) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I J) p))) ((extChartAt (ModelWithCorners.prod I J) p).target \u2229 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I J) p)) \u207b\u00b9' (s \u2229 Prod.fst \u207b\u00b9' (extChartAt I p.1).source)) (\u2191(extChartAt (ModelWithCorners.prod I J) p) p) ** refine \u27e8continuousWithinAt_fst, contDiffWithinAt_fst.congr (fun y hy => ?_) ?_\u27e9 ** case refine_1 \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s\u271d s\u2081 t : Set M x : M m n : \u2115\u221e s : Set (M \u00d7 N) p : M \u00d7 N y : E \u00d7 F hy : y \u2208 (extChartAt (ModelWithCorners.prod I J) p).target \u2229 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I J) p)) \u207b\u00b9' (s \u2229 Prod.fst \u207b\u00b9' (extChartAt I p.1).source) \u22a2 (\u2191(extChartAt I p.1) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I J) p))) y = y.1 ** exact (extChartAt I p.1).right_inv \u27e8hy.1.1.1, hy.1.2.1\u27e9 ** case refine_2 \ud835\udd5c : Type u_1 inst\u271d\u00b3\u2077 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3\u2076 : NormedAddCommGroup E inst\u271d\u00b3\u2075 : NormedSpace \ud835\udd5c E H : Type u_3 inst\u271d\u00b3\u2074 : TopologicalSpace H I : ModelWithCorners \ud835\udd5c E H M : Type u_4 inst\u271d\u00b3\u00b3 : TopologicalSpace M inst\u271d\u00b3\u00b2 : ChartedSpace H M inst\u271d\u00b3\u00b9 : SmoothManifoldWithCorners I M E' : Type u_5 inst\u271d\u00b3\u2070 : NormedAddCommGroup E' inst\u271d\u00b2\u2079 : NormedSpace \ud835\udd5c E' H' : Type u_6 inst\u271d\u00b2\u2078 : TopologicalSpace H' I' : ModelWithCorners \ud835\udd5c E' H' M' : Type u_7 inst\u271d\u00b2\u2077 : TopologicalSpace M' inst\u271d\u00b2\u2076 : ChartedSpace H' M' inst\u271d\u00b2\u2075 : SmoothManifoldWithCorners I' M' E'' : Type u_8 inst\u271d\u00b2\u2074 : NormedAddCommGroup E'' inst\u271d\u00b2\u00b3 : NormedSpace \ud835\udd5c E'' H'' : Type u_9 inst\u271d\u00b2\u00b2 : TopologicalSpace H'' I'' : ModelWithCorners \ud835\udd5c E'' H'' M'' : Type u_10 inst\u271d\u00b2\u00b9 : TopologicalSpace M'' inst\u271d\u00b2\u2070 : ChartedSpace H'' M'' F : Type u_11 inst\u271d\u00b9\u2079 : NormedAddCommGroup F inst\u271d\u00b9\u2078 : NormedSpace \ud835\udd5c F G : Type u_12 inst\u271d\u00b9\u2077 : TopologicalSpace G J : ModelWithCorners \ud835\udd5c F G N : Type u_13 inst\u271d\u00b9\u2076 : TopologicalSpace N inst\u271d\u00b9\u2075 : ChartedSpace G N inst\u271d\u00b9\u2074 : SmoothManifoldWithCorners J N F' : Type u_14 inst\u271d\u00b9\u00b3 : NormedAddCommGroup F' inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c F' G' : Type u_15 inst\u271d\u00b9\u00b9 : TopologicalSpace G' J' : ModelWithCorners \ud835\udd5c F' G' N' : Type u_16 inst\u271d\u00b9\u2070 : TopologicalSpace N' inst\u271d\u2079 : ChartedSpace G' N' inst\u271d\u2078 : SmoothManifoldWithCorners J' N' F\u2081 : Type u_17 inst\u271d\u2077 : NormedAddCommGroup F\u2081 inst\u271d\u2076 : NormedSpace \ud835\udd5c F\u2081 F\u2082 : Type u_18 inst\u271d\u2075 : NormedAddCommGroup F\u2082 inst\u271d\u2074 : NormedSpace \ud835\udd5c F\u2082 F\u2083 : Type u_19 inst\u271d\u00b3 : NormedAddCommGroup F\u2083 inst\u271d\u00b2 : NormedSpace \ud835\udd5c F\u2083 F\u2084 : Type u_20 inst\u271d\u00b9 : NormedAddCommGroup F\u2084 inst\u271d : NormedSpace \ud835\udd5c F\u2084 e : LocalHomeomorph M H e' : LocalHomeomorph M' H' f f\u2081 : M \u2192 M' s\u271d s\u2081 t : Set M x : M m n : \u2115\u221e s : Set (M \u00d7 N) p : M \u00d7 N \u22a2 (\u2191(extChartAt I p.1) \u2218 Prod.fst \u2218 \u2191(LocalEquiv.symm (extChartAt (ModelWithCorners.prod I J) p))) (\u2191(extChartAt (ModelWithCorners.prod I J) p) p) = (\u2191(extChartAt (ModelWithCorners.prod I J) p) p).1 ** exact (extChartAt I p.1).right_inv <| (extChartAt I p.1).map_source (mem_extChartAt_source _ _) ** Qed", + "informal": "" + }, + { + "formal": "CircleDeg1Lift.map_one_add ** f g : CircleDeg1Lift x : \u211d \u22a2 \u2191f (1 + x) = 1 + \u2191f x ** rw [add_comm, map_add_one, add_comm 1] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.isIntegrallyClosed_iff' ** R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R \u22a2 IsIntegrallyClosed R \u2194 \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) ** constructor ** case mp R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R \u22a2 IsIntegrallyClosed R \u2192 \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) ** intro hR p hp ** case mp R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R hR : IsIntegrallyClosed R p : R[X] hp : Monic p \u22a2 Irreducible p \u2194 Irreducible (map (algebraMap R K) p) ** exact Monic.irreducible_iff_irreducible_map_fraction_map hp ** case mpr R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R \u22a2 (\u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p))) \u2192 IsIntegrallyClosed R ** intro H ** case mpr R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R H : \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) \u22a2 IsIntegrallyClosed R ** refine'\n (isIntegrallyClosed_iff K).mpr fun {x} hx =>\n RingHom.mem_range.mp <| minpoly.mem_range_of_degree_eq_one R x _ ** case mpr R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R H : \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) x : K hx : IsIntegral R x \u22a2 degree (minpoly R x) = 1 ** rw [\u2190 Monic.degree_map (minpoly.monic hx) (algebraMap R K)] ** case mpr R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R H : \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) x : K hx : IsIntegral R x \u22a2 degree (map (algebraMap R K) (minpoly R x)) = 1 ** apply\n degree_eq_one_of_irreducible_of_root ((H _ <| minpoly.monic hx).mp (minpoly.irreducible hx)) ** case mpr R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R H : \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) x : K hx : IsIntegral R x \u22a2 IsRoot (map (algebraMap R K) (minpoly R x)) ?m.307327 R : Type u_1 inst\u271d\u2074 : CommRing R K : Type u_2 inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Algebra R K inst\u271d\u00b9 : IsFractionRing R K inst\u271d : IsDomain R H : \u2200 (p : R[X]), Monic p \u2192 (Irreducible p \u2194 Irreducible (map (algebraMap R K) p)) x : K hx : IsIntegral R x \u22a2 K ** rw [IsRoot, eval_map, \u2190 aeval_def, minpoly.aeval R x] ** Qed", + "informal": "" + }, + { + "formal": "finEquivZpowers_symm_apply ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Group G x\u271d y : G n\u271d : \u2115 inst\u271d : Finite G x : G n : \u2115 hn : \u2203 m, x ^ m = x ^ n \u22a2 \u2191(finEquivZpowers x).symm { val := x ^ n, property := hn } = { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) } ** rw [finEquivZpowers, Equiv.symm_trans_apply] ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Group G x\u271d y : G n\u271d : \u2115 inst\u271d : Finite G x : G n : \u2115 hn : \u2203 m, x ^ m = x ^ n \u22a2 \u2191(finEquivPowers x).symm (\u2191(Equiv.Set.ofEq (_ : \u2191(Submonoid.powers x) = \u2191(zpowers x))).symm { val := x ^ n, property := hn }) = { val := n % orderOf x, isLt := (_ : n % orderOf x < orderOf x) } ** exact finEquivPowers_symm_apply x n ** Qed", + "informal": "" + }, + { + "formal": "WellFounded.eq_strictMono_iff_eq_range ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r r' : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d\u00b9 : LinearOrder \u03b2 h : WellFounded fun x x_1 => x < x_1 inst\u271d : PartialOrder \u03b3 f g : \u03b2 \u2192 \u03b3 hf : StrictMono f hg : StrictMono g hfg : range f = range g \u22a2 f = g ** funext a ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r r' : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d\u00b9 : LinearOrder \u03b2 h : WellFounded fun x x_1 => x < x_1 inst\u271d : PartialOrder \u03b3 f g : \u03b2 \u2192 \u03b3 hf : StrictMono f hg : StrictMono g hfg : range f = range g a : \u03b2 \u22a2 f a = g a ** apply h.induction a ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r r' : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d\u00b9 : LinearOrder \u03b2 h : WellFounded fun x x_1 => x < x_1 inst\u271d : PartialOrder \u03b3 f g : \u03b2 \u2192 \u03b3 hf : StrictMono f hg : StrictMono g hfg : range f = range g a : \u03b2 \u22a2 \u2200 (x : \u03b2), (\u2200 (y : \u03b2), y < x \u2192 f y = g y) \u2192 f x = g x ** exact fun b H =>\n le_antisymm (eq_strictMono_iff_eq_range_aux hf hg hfg H)\n (eq_strictMono_iff_eq_range_aux hg hf hfg.symm fun a hab => (H a hab).symm) ** Qed", + "informal": "" + }, + { + "formal": "SeparatingDual.exists_continuousLinearEquiv_apply_eq ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 \u22a2 \u2203 A, \u2191A x = y ** obtain \u27e8G, Gx, Gy\u27e9 : \u2203 G : V \u2192L[R] R, G x = 1 \u2227 G y \u2260 0 :=\n exists_eq_one_ne_zero_of_ne_zero_pair hx hy ** case intro.intro R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 \u22a2 \u2203 A, \u2191A x = y ** let A : V \u2243L[R] V :=\n{ toFun := fun z \u21a6 z + G z \u2022 (y - x)\n invFun := fun z \u21a6 z + ((G y) \u207b\u00b9 * G z) \u2022 (x - y)\n map_add' := fun a b \u21a6 by simp [add_smul]; abel\n map_smul' := by simp [smul_smul]\n left_inv := fun z \u21a6 by\n simp only [id_eq, eq_mpr_eq_cast, RingHom.id_apply, smul_eq_mul, AddHom.toFun_eq_coe,\n AddHom.coe_mk, map_add, map_smul\u209b\u2097, map_sub, Gx, mul_sub, mul_one, add_sub_cancel'_right]\n rw [mul_comm (G z), \u2190 mul_assoc, inv_mul_cancel Gy]\n simp only [smul_sub, one_mul]\n abel\n right_inv := fun z \u21a6 by\n simp only [map_add, map_smul\u209b\u2097, map_mul, map_inv\u2080, RingHom.id_apply, map_sub, Gx,\n smul_eq_mul, mul_sub, mul_one]\n rw [mul_comm _ (G y), \u2190 mul_assoc, mul_inv_cancel Gy]\n simp only [smul_sub, one_mul, add_sub_cancel'_right]\n abel\n continuous_toFun := continuous_id.add (G.continuous.smul continuous_const)\n continuous_invFun :=\n continuous_id.add ((continuous_const.mul G.continuous).smul continuous_const) } ** case intro.intro R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 A : V \u2243L[R] V := ContinuousLinearEquiv.mk { toLinearMap := { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }, invFun := fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y), left_inv := (_ : \u2200 (z : V), (fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) (AddHom.toFun { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom z) = z), right_inv := (_ : \u2200 (z : V), AddHom.toFun { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom ((fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) z) = z) } \u22a2 \u2203 A, \u2191A x = y ** exact \u27e8A, show x + G x \u2022 (y - x) = y by simp [Gx]\u27e9 ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 a b : V \u22a2 (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b ** simp [add_smul] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 a b : V \u22a2 a + b + (\u2191G a \u2022 (y - x) + \u2191G b \u2022 (y - x)) = a + \u2191G a \u2022 (y - x) + (b + \u2191G b \u2022 (y - x)) ** abel ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 \u22a2 \u2200 (r : R) (x_1 : V), AddHom.toFun { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) } (r \u2022 x_1) = \u2191(RingHom.id R) r \u2022 AddHom.toFun { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) } x_1 ** simp [smul_smul] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 (fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) (AddHom.toFun { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom z) = z ** simp only [id_eq, eq_mpr_eq_cast, RingHom.id_apply, smul_eq_mul, AddHom.toFun_eq_coe,\n AddHom.coe_mk, map_add, map_smul\u209b\u2097, map_sub, Gx, mul_sub, mul_one, add_sub_cancel'_right] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 z + \u2191G z \u2022 (y - x) + ((\u2191G y)\u207b\u00b9 * (\u2191G z * \u2191G y)) \u2022 (x - y) = z ** rw [mul_comm (G z), \u2190 mul_assoc, inv_mul_cancel Gy] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 z + \u2191G z \u2022 (y - x) + (1 * \u2191G z) \u2022 (x - y) = z ** simp only [smul_sub, one_mul] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 z + (\u2191G z \u2022 y - \u2191G z \u2022 x) + (\u2191G z \u2022 x - \u2191G z \u2022 y) = z ** abel ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 AddHom.toFun { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom ((fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) z) = z ** simp only [map_add, map_smul\u209b\u2097, map_mul, map_inv\u2080, RingHom.id_apply, map_sub, Gx,\n smul_eq_mul, mul_sub, mul_one] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y) + (\u2191G z + ((\u2191G y)\u207b\u00b9 * \u2191G z - (\u2191G y)\u207b\u00b9 * \u2191G z * \u2191G y)) \u2022 (y - x) = z ** rw [mul_comm _ (G y), \u2190 mul_assoc, mul_inv_cancel Gy] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y) + (\u2191G z + ((\u2191G y)\u207b\u00b9 * \u2191G z - 1 * \u2191G z)) \u2022 (y - x) = z ** simp only [smul_sub, one_mul, add_sub_cancel'_right] ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 z : V \u22a2 z + (((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 x - ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 y) + (((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 y - ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 x) = z ** abel ** R : Type u_1 V : Type u_2 inst\u271d\u2078 : Field R inst\u271d\u2077 : AddCommGroup V inst\u271d\u2076 : TopologicalSpace R inst\u271d\u2075 : TopologicalSpace V inst\u271d\u2074 : TopologicalRing R inst\u271d\u00b3 : TopologicalAddGroup V inst\u271d\u00b2 : Module R V inst\u271d\u00b9 : SeparatingDual R V inst\u271d : ContinuousSMul R V x y : V hx : x \u2260 0 hy : y \u2260 0 G : V \u2192L[R] R Gx : \u2191G x = 1 Gy : \u2191G y \u2260 0 A : V \u2243L[R] V := ContinuousLinearEquiv.mk { toLinearMap := { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }, invFun := fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y), left_inv := (_ : \u2200 (z : V), (fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) (AddHom.toFun { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom z) = z), right_inv := (_ : \u2200 (z : V), AddHom.toFun { toAddHom := { toFun := fun z => z + \u2191G z \u2022 (y - x), map_add' := (_ : \u2200 (a b : V), (fun z => z + \u2191G z \u2022 (y - x)) (a + b) = (fun z => z + \u2191G z \u2022 (y - x)) a + (fun z => z + \u2191G z \u2022 (y - x)) b) }, map_smul' := (_ : \u2200 (a : R) (a_1 : V), a \u2022 a_1 + \u2191G (a \u2022 a_1) \u2022 (y - x) = a \u2022 (a_1 + \u2191G a_1 \u2022 (y - x))) }.toAddHom ((fun z => z + ((\u2191G y)\u207b\u00b9 * \u2191G z) \u2022 (x - y)) z) = z) } \u22a2 x + \u2191G x \u2022 (y - x) = y ** simp [Gx] ** Qed", + "informal": "" + }, + { + "formal": "Num.natSize_to_nat ** \u03b1 : Type u_1 n : Num \u22a2 natSize n = Nat.size \u2191n ** rw [\u2190 size_eq_natSize, size_to_nat] ** Qed", + "informal": "" + }, + { + "formal": "lowerClosure_singleton ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03ba : \u03b9 \u2192 Sort u_5 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 s t : Set \u03b1 x a : \u03b1 \u22a2 lowerClosure {a} = LowerSet.Iic a ** ext ** case a.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03ba : \u03b9 \u2192 Sort u_5 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 s t : Set \u03b1 x a x\u271d : \u03b1 \u22a2 x\u271d \u2208 \u2191(lowerClosure {a}) \u2194 x\u271d \u2208 \u2191(LowerSet.Iic a) ** simp ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMultilinearMap.norm_mkPiAlgebra_of_empty ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedCommRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : IsEmpty \u03b9 \u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 = \u20161\u2016 ** apply le_antisymm ** case a \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedCommRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : IsEmpty \u03b9 \u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 \u2264 \u20161\u2016 ** have := fun f => @op_norm_le_bound \ud835\udd5c \u03b9 (fun _ => A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A)) ** case a \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedCommRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : IsEmpty \u03b9 this : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 \u20161\u2016 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 \u20161\u2016 \u22a2 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 \u2264 \u20161\u2016 ** refine' this _ _ ** case a \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedCommRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : IsEmpty \u03b9 this : \u2200 (f : ContinuousMultilinearMap \ud835\udd5c (fun x => A) A), (\u2200 (m : \u03b9 \u2192 A), \u2016\u2191f m\u2016 \u2264 \u20161\u2016 * \u220f i : \u03b9, \u2016m i\u2016) \u2192 \u2016f\u2016 \u2264 \u20161\u2016 \u22a2 \u2200 (m : \u03b9 \u2192 A), \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) m\u2016 \u2264 \u20161\u2016 * \u220f i : \u03b9, \u2016m i\u2016 ** simp ** case a \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedCommRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : IsEmpty \u03b9 \u22a2 \u20161\u2016 \u2264 \u2016ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A\u2016 ** convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) fun _ => (1 : A) ** case h.e'_3 \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2077 : Fintype \u03b9 inst\u271d\u00b9\u2076 : Fintype \u03b9' inst\u271d\u00b9\u2075 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u2074 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u00b3 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u00b9\u00b2 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u00b9\u2070 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2079 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2078 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2077 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u2076 : NormedAddCommGroup G inst\u271d\u2075 : NormedSpace \ud835\udd5c G inst\u271d\u2074 : NormedAddCommGroup G' inst\u271d\u00b3 : NormedSpace \ud835\udd5c G' A : Type u_1 inst\u271d\u00b2 : NormedCommRing A inst\u271d\u00b9 : NormedAlgebra \ud835\udd5c A inst\u271d : IsEmpty \u03b9 \u22a2 \u20161\u2016 = \u2016\u2191(ContinuousMultilinearMap.mkPiAlgebra \ud835\udd5c \u03b9 A) fun x => 1\u2016 / \u220f i : \u03b9, \u20161\u2016 ** simp [eq_empty_of_isEmpty (univ : Finset \u03b9)] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.measurable_measure_mul_right ** G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s \u22a2 Measurable fun x => \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s) ** suffices\n Measurable fun y =>\n \u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z : G \u00d7 G => ((1 : G), z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s))\n by convert this using 1; ext1 x; congr 1 with y : 1; simp ** G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s \u22a2 Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) ** apply measurable_measure_prod_mk_right ** case hs G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s \u22a2 MeasurableSet ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s) ** apply measurable_const.prod_mk measurable_mul (MeasurableSet.univ.prod hs) ** G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s \u22a2 MeasurableSpace G ** infer_instance ** G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s this : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) \u22a2 Measurable fun x => \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s) ** convert this using 1 ** case h.e'_5 G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s this : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) \u22a2 (fun x => \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s)) = fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) ** ext1 x ** case h.e'_5.h G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s this : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) x : G \u22a2 \u2191\u2191\u03bc ((fun y => y * x) \u207b\u00b9' s) = \u2191\u2191\u03bc ((fun x_1 => (x_1, x)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) ** congr 1 with y : 1 ** case h.e'_5.h.e_a.h G : Type u_1 inst\u271d\u2074 : MeasurableSpace G inst\u271d\u00b3 : Group G inst\u271d\u00b2 : MeasurableMul\u2082 G \u03bc \u03bd : Measure G inst\u271d\u00b9 : SigmaFinite \u03bd inst\u271d : SigmaFinite \u03bc s : Set G hs : MeasurableSet s this : Measurable fun y => \u2191\u2191\u03bc ((fun x => (x, y)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s)) x y : G \u22a2 y \u2208 (fun y => y * x) \u207b\u00b9' s \u2194 y \u2208 (fun x_1 => (x_1, x)) \u207b\u00b9' ((fun z => (1, z.1 * z.2)) \u207b\u00b9' univ \u00d7\u02e2 s) ** simp ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.inv_lt_one ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 \u22a2 a\u207b\u00b9 < 1 \u2194 1 < a ** rw [inv_lt_iff_inv_lt, inv_one] ** Qed", + "informal": "" + }, + { + "formal": "PerfectClosure.R.sound ** K : Type u inst\u271d\u00b2 : CommRing K p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP K p m n : \u2115 x y : K H : (\u2191(frobenius K p))^[m] x = y \u22a2 mk K p (n, x) = mk K p (m + n, y) ** subst H ** K : Type u inst\u271d\u00b2 : CommRing K p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP K p m n : \u2115 x : K \u22a2 mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p))^[m] x) ** induction' m with m ih ** case succ K : Type u inst\u271d\u00b2 : CommRing K p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP K p n : \u2115 x : K m : \u2115 ih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p))^[m] x) \u22a2 mk K p (n, x) = mk K p (Nat.succ m + n, (\u2191(frobenius K p))^[Nat.succ m] x) ** rw [ih, Nat.succ_add, iterate_succ'] ** case succ K : Type u inst\u271d\u00b2 : CommRing K p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP K p n : \u2115 x : K m : \u2115 ih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p))^[m] x) \u22a2 mk K p (m + n, (\u2191(frobenius K p))^[m] x) = mk K p (Nat.succ (m + n), (\u2191(frobenius K p) \u2218 (\u2191(frobenius K p))^[m]) x) ** apply Quot.sound ** case succ.a K : Type u inst\u271d\u00b2 : CommRing K p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP K p n : \u2115 x : K m : \u2115 ih : mk K p (n, x) = mk K p (m + n, (\u2191(frobenius K p))^[m] x) \u22a2 R K p (m + n, (\u2191(frobenius K p))^[m] x) (Nat.succ (m + n), (\u2191(frobenius K p) \u2218 (\u2191(frobenius K p))^[m]) x) ** apply R.intro ** case zero K : Type u inst\u271d\u00b2 : CommRing K p : \u2115 inst\u271d\u00b9 : Fact (Nat.Prime p) inst\u271d : CharP K p n : \u2115 x : K \u22a2 mk K p (n, x) = mk K p (Nat.zero + n, (\u2191(frobenius K p))^[Nat.zero] x) ** simp only [Nat.zero_eq, zero_add, iterate_zero_apply] ** Qed", + "informal": "" + }, + { + "formal": "HasCompactSupport.continuous_convolution_left ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : NormedAddCommGroup E' inst\u271d\u00b9\u2076 : NormedAddCommGroup E'' inst\u271d\u00b9\u2075 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u2079 : MeasurableSpace G \u03bc \u03bd : Measure G inst\u271d\u2078 : NormedSpace \u211d F inst\u271d\u2077 : CompleteSpace F inst\u271d\u2076 : AddCommGroup G inst\u271d\u2075 : IsAddLeftInvariant \u03bc inst\u271d\u2074 : IsNegInvariant \u03bc inst\u271d\u00b3 : TopologicalSpace G inst\u271d\u00b2 : TopologicalAddGroup G inst\u271d\u00b9 : BorelSpace G inst\u271d : FirstCountableTopology G hcf : HasCompactSupport f hf : Continuous f hg : LocallyIntegrable g \u22a2 Continuous (convolution f g L) ** rw [\u2190 convolution_flip] ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b9\u2078 : NormedAddCommGroup E inst\u271d\u00b9\u2077 : NormedAddCommGroup E' inst\u271d\u00b9\u2076 : NormedAddCommGroup E'' inst\u271d\u00b9\u2075 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b9\u2074 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b3 : NormedSpace \ud835\udd5c E inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E' inst\u271d\u00b9\u00b9 : NormedSpace \ud835\udd5c E'' inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u2079 : MeasurableSpace G \u03bc \u03bd : Measure G inst\u271d\u2078 : NormedSpace \u211d F inst\u271d\u2077 : CompleteSpace F inst\u271d\u2076 : AddCommGroup G inst\u271d\u2075 : IsAddLeftInvariant \u03bc inst\u271d\u2074 : IsNegInvariant \u03bc inst\u271d\u00b3 : TopologicalSpace G inst\u271d\u00b2 : TopologicalAddGroup G inst\u271d\u00b9 : BorelSpace G inst\u271d : FirstCountableTopology G hcf : HasCompactSupport f hf : Continuous f hg : LocallyIntegrable g \u22a2 Continuous (convolution g f (ContinuousLinearMap.flip L)) ** exact hcf.continuous_convolution_right L.flip hg hf ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.ofReal_toReal_eq_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 h : ENNReal.ofReal (ENNReal.toReal a) = a \u22a2 a \u2260 \u22a4 ** rw [\u2190 h] ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 h : ENNReal.ofReal (ENNReal.toReal a) = a \u22a2 ENNReal.ofReal (ENNReal.toReal a) \u2260 \u22a4 ** exact ofReal_ne_top ** Qed", + "informal": "" + }, + { + "formal": "Real.Gamma_mul_add_mul_le_rpow_Gamma_mul_rpow_Gamma ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** let f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => exp (-c * x) * x ^ (c * (u - 1)) ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** have e : IsConjugateExponent (1 / a) (1 / b) := Real.isConjugateExponent_one_div ha hb hab ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** have hab' : b = 1 - a := by linarith ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** have hst : 0 < a * s + b * t := add_pos (mul_pos ha hs) (mul_pos hb ht) ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** have posf : \u2200 c u x : \u211d, x \u2208 Ioi (0 : \u211d) \u2192 0 \u2264 f c u x := fun c u x hx =>\n mul_nonneg (exp_pos _).le (rpow_pos_of_pos hx _).le ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** have posf' : \u2200 c u : \u211d, \u2200\u1d50 x : \u211d \u2202volume.restrict (Ioi 0), 0 \u2264 f c u x := fun c u =>\n (ae_restrict_iff' measurableSet_Ioi).mpr (ae_of_all _ (posf c u)) ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) \u22a2 Gamma (a * s + b * t) \u2264 Gamma s ^ a * Gamma t ^ b ** rw [Gamma_eq_integral hs, Gamma_eq_integral ht, Gamma_eq_integral hst] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) \u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (a * s + b * t - 1) \u2264 (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b ** convert\n MeasureTheory.integral_mul_le_Lp_mul_Lq_of_nonneg e (posf' a s) (posf' b t) (f_mem_Lp ha hs)\n (f_mem_Lp hb ht) using\n 1 ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) \u22a2 b = 1 - a ** linarith ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x \u22a2 \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) ** intro c x hc u hx ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x c x : \u211d hc : 0 < c u : \u211d hx : 0 < x \u22a2 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) ** dsimp only ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x c x : \u211d hc : 0 < c u : \u211d hx : 0 < x \u22a2 rexp (-x) * x ^ (u - 1) = (rexp (-c * x) * x ^ (c * (u - 1))) ^ (1 / c) ** rw [mul_rpow (exp_pos _).le ((rpow_nonneg_of_nonneg hx.le) _), \u2190 exp_mul, \u2190 rpow_mul hx.le] ** case e_a.e_a s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x c x : \u211d hc : 0 < c u : \u211d hx : 0 < x \u22a2 u - 1 = c * (u - 1) * (1 / c) ** field_simp [hc.ne'] ** case e_a.e_a s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x c x : \u211d hc : 0 < c u : \u211d hx : 0 < x \u22a2 (u - 1) * c = c * (u - 1) ** ring ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) \u22a2 \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) ** intro c u hc hu ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u \u22a2 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) ** have A : ENNReal.ofReal (1 / c) \u2260 0 := by\n rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le, one_div_pos] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 \u22a2 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) ** have B : ENNReal.ofReal (1 / c) \u2260 \u221e := ENNReal.ofReal_ne_top ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 \u22a2 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) ** rw [\u2190 mem\u2112p_norm_rpow_iff _ A B, ENNReal.toReal_ofReal (one_div_nonneg.mpr hc.le),\n ENNReal.div_self A B, mem\u2112p_one_iff_integrable] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u \u22a2 ENNReal.ofReal (1 / c) \u2260 0 ** rwa [Ne.def, ENNReal.ofReal_eq_zero, not_le, one_div_pos] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 \u22a2 Integrable fun x => \u2016f c u x\u2016 ^ (1 / c) ** apply Integrable.congr (GammaIntegral_convergent hu) ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 \u22a2 (fun x => rexp (-x) * x ^ (u - 1)) =\u1d50[Measure.restrict volume (Ioi 0)] fun x => \u2016f c u x\u2016 ^ (1 / c) ** refine' eventuallyEq_of_mem (self_mem_ae_restrict measurableSet_Ioi) fun x hx => _ ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 x : \u211d hx : x \u2208 Ioi 0 \u22a2 rexp (-x) * x ^ (u - 1) = \u2016f c u x\u2016 ^ (1 / c) ** dsimp only ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 x : \u211d hx : x \u2208 Ioi 0 \u22a2 rexp (-x) * x ^ (u - 1) = \u2016rexp (-c * x) * x ^ (c * (u - 1))\u2016 ^ (1 / c) ** rw [fpow hc u hx] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 x : \u211d hx : x \u2208 Ioi 0 \u22a2 f c u x ^ (1 / c) = \u2016rexp (-c * x) * x ^ (c * (u - 1))\u2016 ^ (1 / c) ** congr 1 ** case e_a s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 x : \u211d hx : x \u2208 Ioi 0 \u22a2 f c u x = \u2016rexp (-c * x) * x ^ (c * (u - 1))\u2016 ** exact (norm_of_nonneg (posf _ _ x hx)).symm ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 \u22a2 AEStronglyMeasurable (f c u) (Measure.restrict volume (Ioi 0)) ** refine' ContinuousOn.aestronglyMeasurable _ measurableSet_Ioi ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 \u22a2 ContinuousOn (f c u) (Ioi 0) ** refine' (Continuous.continuousOn _).mul (ContinuousAt.continuousOn fun x hx => _) ** case refine'_1 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 \u22a2 Continuous fun x => rexp (-c * x) ** exact continuous_exp.comp (continuous_const.mul continuous_id') ** case refine'_2 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) c u : \u211d hc : 0 < c hu : 0 < u A : ENNReal.ofReal (1 / c) \u2260 0 B : ENNReal.ofReal (1 / c) \u2260 \u22a4 x : \u211d hx : x \u2208 Ioi 0 \u22a2 ContinuousAt (fun x => x ^ (c * (u - 1))) x ** exact continuousAt_rpow_const _ _ (Or.inl (mem_Ioi.mp hx).ne') ** case h.e'_3 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) \u22a2 \u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (a * s + b * t - 1) = \u222b (a_1 : \u211d) in Ioi 0, f a s a_1 * f b t a_1 ** refine' set_integral_congr measurableSet_Ioi fun x hx => _ ** case h.e'_3 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 \u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = f a s x * f b t x ** dsimp only ** case h.e'_3 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 \u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1))) ** have A : exp (-x) = exp (-a * x) * exp (-b * x) := by\n rw [\u2190 exp_add, \u2190 add_mul, \u2190 neg_add, hab, neg_one_mul] ** case h.e'_3 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 A : rexp (-x) = rexp (-a * x) * rexp (-b * x) \u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1))) ** have B : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) := by\n rw [\u2190 rpow_add hx, hab']; congr 1; ring ** case h.e'_3 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 A : rexp (-x) = rexp (-a * x) * rexp (-b * x) B : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) \u22a2 rexp (-x) * x ^ (a * s + b * t - 1) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1))) ** rw [A, B] ** case h.e'_3 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 A : rexp (-x) = rexp (-a * x) * rexp (-b * x) B : x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) \u22a2 rexp (-a * x) * rexp (-b * x) * (x ^ (a * (s - 1)) * x ^ (b * (t - 1))) = rexp (-a * x) * x ^ (a * (s - 1)) * (rexp (-b * x) * x ^ (b * (t - 1))) ** ring ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 \u22a2 rexp (-x) = rexp (-a * x) * rexp (-b * x) ** rw [\u2190 exp_add, \u2190 add_mul, \u2190 neg_add, hab, neg_one_mul] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 A : rexp (-x) = rexp (-a * x) * rexp (-b * x) \u22a2 x ^ (a * s + b * t - 1) = x ^ (a * (s - 1)) * x ^ (b * (t - 1)) ** rw [\u2190 rpow_add hx, hab'] ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 A : rexp (-x) = rexp (-a * x) * rexp (-b * x) \u22a2 x ^ (a * s + (1 - a) * t - 1) = x ^ (a * (s - 1) + (1 - a) * (t - 1)) ** congr 1 ** case e_a s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 A : rexp (-x) = rexp (-a * x) * rexp (-b * x) \u22a2 a * s + (1 - a) * t - 1 = a * (s - 1) + (1 - a) * (t - 1) ** ring ** case h.e'_4 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) \u22a2 (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b = (\u222b (a_1 : \u211d) in Ioi 0, f a s a_1 ^ (1 / a)) ^ (1 / (1 / a)) * (\u222b (a : \u211d) in Ioi 0, f b t a ^ (1 / b)) ^ (1 / (1 / b)) ** rw [one_div_one_div, one_div_one_div] ** case h.e'_4 s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) \u22a2 (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (s - 1)) ^ a * (\u222b (x : \u211d) in Ioi 0, rexp (-x) * x ^ (t - 1)) ^ b = (\u222b (a_1 : \u211d) in Ioi 0, f a s a_1 ^ (1 / a)) ^ a * (\u222b (a : \u211d) in Ioi 0, f b t a ^ (1 / b)) ^ b ** congr 2 <;> exact set_integral_congr measurableSet_Ioi fun x hx => fpow (by assumption) _ hx ** s t a b : \u211d hs : 0 < s ht : 0 < t ha : 0 < a hb : 0 < b hab : a + b = 1 f : \u211d \u2192 \u211d \u2192 \u211d \u2192 \u211d := fun c u x => rexp (-c * x) * x ^ (c * (u - 1)) e : IsConjugateExponent (1 / a) (1 / b) hab' : b = 1 - a hst : 0 < a * s + b * t posf : \u2200 (c u x : \u211d), x \u2208 Ioi 0 \u2192 0 \u2264 f c u x posf' : \u2200 (c u : \u211d), \u2200\u1d50 (x : \u211d) \u2202Measure.restrict volume (Ioi 0), 0 \u2264 f c u x fpow : \u2200 {c x : \u211d}, 0 < c \u2192 \u2200 (u : \u211d), 0 < x \u2192 rexp (-x) * x ^ (u - 1) = f c u x ^ (1 / c) f_mem_Lp : \u2200 {c u : \u211d}, 0 < c \u2192 0 < u \u2192 Mem\u2112p (f c u) (ENNReal.ofReal (1 / c)) x : \u211d hx : x \u2208 Ioi 0 \u22a2 0 < b ** assumption ** Qed", + "informal": "" + }, + { + "formal": "NatOrdinal.mul_le_nmul ** a b : Ordinal.{u} \u22a2 a * b \u2264 a \u2a33 b ** refine b.limitRecOn ?_ ?_ ?_ ** case refine_1 a b : Ordinal.{u} \u22a2 a * 0 \u2264 a \u2a33 0 ** simp ** case refine_2 a b : Ordinal.{u} \u22a2 \u2200 (o : Ordinal.{u}), a * o \u2264 a \u2a33 o \u2192 a * succ o \u2264 a \u2a33 succ o ** intro c h ** case refine_2 a b c : Ordinal.{u} h : a * c \u2264 a \u2a33 c \u22a2 a * succ c \u2264 a \u2a33 succ c ** rw [mul_succ, nmul_succ] ** case refine_2 a b c : Ordinal.{u} h : a * c \u2264 a \u2a33 c \u22a2 a * c + a \u2264 a \u2a33 c \u266f a ** exact (add_le_nadd _ a).trans (nadd_le_nadd_right h a) ** case refine_3 a b : Ordinal.{u} \u22a2 \u2200 (o : Ordinal.{u}), IsLimit o \u2192 (\u2200 (o' : Ordinal.{u}), o' < o \u2192 a * o' \u2264 a \u2a33 o') \u2192 a * o \u2264 a \u2a33 o ** intro c hc H ** case refine_3 a b c : Ordinal.{u} hc : IsLimit c H : \u2200 (o' : Ordinal.{u}), o' < c \u2192 a * o' \u2264 a \u2a33 o' \u22a2 a * c \u2264 a \u2a33 c ** rcases eq_zero_or_pos a with (rfl | ha) ** case refine_3.inl b c : Ordinal.{u} hc : IsLimit c H : \u2200 (o' : Ordinal.{u}), o' < c \u2192 0 * o' \u2264 0 \u2a33 o' \u22a2 0 * c \u2264 0 \u2a33 c ** simp ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : \u2200 (o' : Ordinal.{u}), o' < c \u2192 a * o' \u2264 a \u2a33 o' ha : 0 < a \u22a2 a * c \u2264 a \u2a33 c ** have := IsNormal.blsub_eq.{u, u} (mul_isNormal ha) hc ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : \u2200 (o' : Ordinal.{u}), o' < c \u2192 a * o' \u2264 a \u2a33 o' ha : 0 < a this : (blsub c fun x x_1 => (fun x x_2 => x * x_2) a x) = (fun x x_1 => x * x_1) a c \u22a2 a * c \u2264 a \u2a33 c ** dsimp at this ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : \u2200 (o' : Ordinal.{u}), o' < c \u2192 a * o' \u2264 a \u2a33 o' ha : 0 < a this : (blsub c fun x x_1 => a * x) = a * c \u22a2 a * c \u2264 a \u2a33 c ** rw [\u2190 this, blsub_le_iff] ** case refine_3.inr a b c : Ordinal.{u} hc : IsLimit c H : \u2200 (o' : Ordinal.{u}), o' < c \u2192 a * o' \u2264 a \u2a33 o' ha : 0 < a this : (blsub c fun x x_1 => a * x) = a * c \u22a2 \u2200 (i : Ordinal.{u}), i < c \u2192 a * i < a \u2a33 c ** exact fun i hi => (H i hi).trans_lt (nmul_lt_nmul_of_pos_left hi ha) ** Qed", + "informal": "" + }, + { + "formal": "padicNorm.div ** p : \u2115 hp : Fact (Nat.Prime p) q r : \u211a hr : r = 0 \u22a2 padicNorm p (q / r) = padicNorm p q / padicNorm p r ** simp [hr] ** p : \u2115 hp : Fact (Nat.Prime p) q r : \u211a hr : \u00acr = 0 \u22a2 padicNorm p (q / r) * padicNorm p r = padicNorm p q ** rw [\u2190 padicNorm.mul, div_mul_cancel _ hr] ** Qed", + "informal": "" + }, + { + "formal": "RatFunc.eval_eq_zero_of_eval\u2082_denom_eq_zero ** K : Type u inst\u271d\u00b9 : Field K L : Type u_1 inst\u271d : Field L f : K \u2192+* L a : L x : RatFunc K h : Polynomial.eval\u2082 f a (denom x) = 0 \u22a2 eval f a x = 0 ** rw [eval, h, div_zero] ** Qed", + "informal": "" + }, + { + "formal": "Asymptotics.isBigO_top ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 G : Type u_5 E' : Type u_6 F' : Type u_7 G' : Type u_8 E'' : Type u_9 F'' : Type u_10 G'' : Type u_11 E''' : Type u_12 R : Type u_13 R' : Type u_14 \ud835\udd5c : Type u_15 \ud835\udd5c' : Type u_16 inst\u271d\u00b9\u00b3 : Norm E inst\u271d\u00b9\u00b2 : Norm F inst\u271d\u00b9\u00b9 : Norm G inst\u271d\u00b9\u2070 : SeminormedAddCommGroup E' inst\u271d\u2079 : SeminormedAddCommGroup F' inst\u271d\u2078 : SeminormedAddCommGroup G' inst\u271d\u2077 : NormedAddCommGroup E'' inst\u271d\u2076 : NormedAddCommGroup F'' inst\u271d\u2075 : NormedAddCommGroup G'' inst\u271d\u2074 : SeminormedRing R inst\u271d\u00b3 : SeminormedAddGroup E''' inst\u271d\u00b2 : SeminormedRing R' inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedField \ud835\udd5c' c c' c\u2081 c\u2082 : \u211d f : \u03b1 \u2192 E g : \u03b1 \u2192 F k : \u03b1 \u2192 G f' : \u03b1 \u2192 E' g' : \u03b1 \u2192 F' k' : \u03b1 \u2192 G' f'' : \u03b1 \u2192 E'' g'' : \u03b1 \u2192 F'' k'' : \u03b1 \u2192 G'' l l' : Filter \u03b1 \u22a2 f =O[\u22a4] g \u2194 \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C * \u2016g x\u2016 ** rw [isBigO_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 G : Type u_5 E' : Type u_6 F' : Type u_7 G' : Type u_8 E'' : Type u_9 F'' : Type u_10 G'' : Type u_11 E''' : Type u_12 R : Type u_13 R' : Type u_14 \ud835\udd5c : Type u_15 \ud835\udd5c' : Type u_16 inst\u271d\u00b9\u00b3 : Norm E inst\u271d\u00b9\u00b2 : Norm F inst\u271d\u00b9\u00b9 : Norm G inst\u271d\u00b9\u2070 : SeminormedAddCommGroup E' inst\u271d\u2079 : SeminormedAddCommGroup F' inst\u271d\u2078 : SeminormedAddCommGroup G' inst\u271d\u2077 : NormedAddCommGroup E'' inst\u271d\u2076 : NormedAddCommGroup F'' inst\u271d\u2075 : NormedAddCommGroup G'' inst\u271d\u2074 : SeminormedRing R inst\u271d\u00b3 : SeminormedAddGroup E''' inst\u271d\u00b2 : SeminormedRing R' inst\u271d\u00b9 : NormedField \ud835\udd5c inst\u271d : NormedField \ud835\udd5c' c c' c\u2081 c\u2082 : \u211d f : \u03b1 \u2192 E g : \u03b1 \u2192 F k : \u03b1 \u2192 G f' : \u03b1 \u2192 E' g' : \u03b1 \u2192 F' k' : \u03b1 \u2192 G' f'' : \u03b1 \u2192 E'' g'' : \u03b1 \u2192 F'' k'' : \u03b1 \u2192 G'' l l' : Filter \u03b1 \u22a2 (\u2203 c, \u2200\u1da0 (x : \u03b1) in \u22a4, \u2016f x\u2016 \u2264 c * \u2016g x\u2016) \u2194 \u2203 C, \u2200 (x : \u03b1), \u2016f x\u2016 \u2264 C * \u2016g x\u2016 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.set_integral_eq_zero_of_ae_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = 0 ** by_cases hf : AEStronglyMeasurable f (\u03bc.restrict t) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = 0 case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : \u00acAEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = 0 ** swap ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = 0 ** have : \u222b x in t, hf.mk f x \u2202\u03bc = 0 := by\n refine' integral_eq_zero_of_ae _\n rw [EventuallyEq,\n ae_restrict_iff (hf.stronglyMeasurable_mk.measurableSet_eq_fun stronglyMeasurable_zero)]\n filter_upwards [ae_imp_of_ae_restrict hf.ae_eq_mk, ht_eq] with x hx h'x h''x\n rw [\u2190 hx h''x]\n exact h'x h''x ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) this : \u222b (x : \u03b1) in t, AEStronglyMeasurable.mk f hf x \u2202\u03bc = 0 \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = 0 ** rw [\u2190 this] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) this : \u222b (x : \u03b1) in t, AEStronglyMeasurable.mk f hf x \u2202\u03bc = 0 \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = \u222b (x : \u03b1) in t, AEStronglyMeasurable.mk f hf x \u2202\u03bc ** exact integral_congr_ae hf.ae_eq_mk ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : \u00acAEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u222b (x : \u03b1) in t, f x \u2202\u03bc = 0 ** rw [integral_undef] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : \u00acAEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u00acIntegrable fun x => f x ** contrapose! hf ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : Integrable fun x => f x \u22a2 AEStronglyMeasurable f (Measure.restrict \u03bc t) ** exact hf.1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u222b (x : \u03b1) in t, AEStronglyMeasurable.mk f hf x \u2202\u03bc = 0 ** refine' integral_eq_zero_of_ae _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 (fun x => AEStronglyMeasurable.mk f hf x) =\u1d50[Measure.restrict \u03bc t] 0 ** rw [EventuallyEq,\n ae_restrict_iff (hf.stronglyMeasurable_mk.measurableSet_eq_fun stronglyMeasurable_zero)] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 AEStronglyMeasurable.mk f hf x = OfNat.ofNat 0 x ** filter_upwards [ae_imp_of_ae_restrict hf.ae_eq_mk, ht_eq] with x hx h'x h''x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) x : \u03b1 hx : x \u2208 t \u2192 f x = AEStronglyMeasurable.mk f hf x h'x : x \u2208 t \u2192 f x = 0 h''x : x \u2208 t \u22a2 AEStronglyMeasurable.mk f hf x = OfNat.ofNat 0 x ** rw [\u2190 hx h''x] ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b2 : MeasurableSpace \u03b1 inst\u271d\u00b9 : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 inst\u271d : NormedSpace \u211d E ht_eq : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 t \u2192 f x = 0 hf : AEStronglyMeasurable f (Measure.restrict \u03bc t) x : \u03b1 hx : x \u2208 t \u2192 f x = AEStronglyMeasurable.mk f hf x h'x : x \u2208 t \u2192 f x = 0 h''x : x \u2208 t \u22a2 f x = OfNat.ofNat 0 x ** exact h'x h''x ** Qed", + "informal": "" + }, + { + "formal": "Finset.weightedVSub_apply ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b2 : Ring k inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082 : Finset \u03b9\u2082 w : \u03b9 \u2192 k p : \u03b9 \u2192 P \u22a2 \u2191(weightedVSub s p) w = \u2211 i in s, w i \u2022 (p i -\u1d65 Classical.choice (_ : Nonempty P)) ** simp [weightedVSub, LinearMap.sum_apply] ** Qed", + "informal": "" + }, + { + "formal": "StarConvex.union ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : SMul \ud835\udd5c E inst\u271d : SMul \ud835\udd5c F x : E s t : Set E hs : StarConvex \ud835\udd5c x s ht : StarConvex \ud835\udd5c x t \u22a2 StarConvex \ud835\udd5c x (s \u222a t) ** rintro y (hy | hy) a b ha hb hab ** case inl \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : SMul \ud835\udd5c E inst\u271d : SMul \ud835\udd5c F x : E s t : Set E hs : StarConvex \ud835\udd5c x s ht : StarConvex \ud835\udd5c x t y : E hy : y \u2208 s a b : \ud835\udd5c ha : 0 \u2264 a hb : 0 \u2264 b hab : a + b = 1 \u22a2 a \u2022 x + b \u2022 y \u2208 s \u222a t ** exact Or.inl (hs hy ha hb hab) ** case inr \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : SMul \ud835\udd5c E inst\u271d : SMul \ud835\udd5c F x : E s t : Set E hs : StarConvex \ud835\udd5c x s ht : StarConvex \ud835\udd5c x t y : E hy : y \u2208 t a b : \ud835\udd5c ha : 0 \u2264 a hb : 0 \u2264 b hab : a + b = 1 \u22a2 a \u2022 x + b \u2022 y \u2208 s \u222a t ** exact Or.inr (ht hy ha hb hab) ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.StructuredArrow.map_id ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D S S' S'' : D Y Y' Y'' : C T T' : C \u2964 D f : StructuredArrow S T \u22a2 (map (\ud835\udfd9 S)).obj f = f ** rw [eq_mk f] ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D S S' S'' : D Y Y' Y'' : C T T' : C \u2964 D f : StructuredArrow S T \u22a2 (map (\ud835\udfd9 S)).obj (mk f.hom) = mk f.hom ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SignedMeasure.of_symmDiff_compl_positive_negative ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c hj' : restrict 0 j \u2264 restrict s j \u2227 restrict s j\u1d9c \u2264 restrict 0 j\u1d9c \u22a2 \u2191s (i \u2206 j) = 0 \u2227 \u2191s (i\u1d9c \u2206 j\u1d9c) = 0 ** rw [restrict_le_restrict_iff s 0, restrict_le_restrict_iff 0 s] at hi' hj' ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 \u2191s (i \u2206 j) = 0 \u2227 \u2191s (i\u1d9c \u2206 j\u1d9c) = 0 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : restrict 0 j \u2264 restrict s j \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet j \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : restrict 0 j \u2264 restrict s j \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet i \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : restrict 0 j \u2264 restrict s j \u2227 restrict s j\u1d9c \u2264 restrict 0 j\u1d9c \u22a2 MeasurableSet j\u1d9c \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c hj' : restrict 0 j \u2264 restrict s j \u2227 restrict s j\u1d9c \u2264 restrict 0 j\u1d9c \u22a2 MeasurableSet i\u1d9c ** constructor ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : restrict 0 j \u2264 restrict s j \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet j \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : restrict 0 j \u2264 restrict s j \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet i \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : restrict 0 j \u2264 restrict s j \u2227 restrict s j\u1d9c \u2264 restrict 0 j\u1d9c \u22a2 MeasurableSet j\u1d9c \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c hj' : restrict 0 j \u2264 restrict s j \u2227 restrict s j\u1d9c \u2264 restrict 0 j\u1d9c \u22a2 MeasurableSet i\u1d9c ** all_goals measurability ** case left \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 \u2191s (i \u2206 j) = 0 ** rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, of_union,\n le_antisymm (hi'.2 (hi.compl.inter hj) (Set.inter_subset_left _ _))\n (hj'.1 (hi.compl.inter hj) (Set.inter_subset_right _ _)),\n le_antisymm (hj'.2 (hj.compl.inter hi) (Set.inter_subset_left _ _))\n (hi'.1 (hj.compl.inter hi) (Set.inter_subset_right _ _)),\n zero_apply, zero_apply, zero_add] ** case left.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 Disjoint (j\u1d9c \u2229 i) (i\u1d9c \u2229 j) ** exact\n Set.disjoint_of_subset_left (Set.inter_subset_left _ _)\n (Set.disjoint_of_subset_right (Set.inter_subset_right _ _)\n (disjoint_comm.1 (IsCompl.disjoint isCompl_compl))) ** case left.hA \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet (j\u1d9c \u2229 i) ** exact hj.compl.inter hi ** case left.hB \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet (i\u1d9c \u2229 j) ** exact hi.compl.inter hj ** case right \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 \u2191s (i\u1d9c \u2206 j\u1d9c) = 0 ** rw [Set.symmDiff_def, Set.diff_eq_compl_inter, Set.diff_eq_compl_inter, compl_compl,\n compl_compl, of_union,\n le_antisymm (hi'.2 (hj.inter hi.compl) (Set.inter_subset_right _ _))\n (hj'.1 (hj.inter hi.compl) (Set.inter_subset_left _ _)),\n le_antisymm (hj'.2 (hi.inter hj.compl) (Set.inter_subset_right _ _))\n (hi'.1 (hi.inter hj.compl) (Set.inter_subset_left _ _)),\n zero_apply, zero_apply, zero_add] ** case right.h \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 Disjoint (j \u2229 i\u1d9c) (i \u2229 j\u1d9c) ** exact\n Set.disjoint_of_subset_left (Set.inter_subset_left _ _)\n (Set.disjoint_of_subset_right (Set.inter_subset_right _ _)\n (IsCompl.disjoint isCompl_compl)) ** case right.hA \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet (j \u2229 i\u1d9c) ** exact hj.inter hi.compl ** case right.hB \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : (\u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i \u2192 \u21910 j \u2264 \u2191s j) \u2227 \u2200 \u2983j : Set \u03b1\u2984, MeasurableSet j \u2192 j \u2286 i\u1d9c \u2192 \u2191s j \u2264 \u21910 j hj' : (\u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j \u2192 \u21910 j_1 \u2264 \u2191s j_1) \u2227 \u2200 \u2983j_1 : Set \u03b1\u2984, MeasurableSet j_1 \u2192 j_1 \u2286 j\u1d9c \u2192 \u2191s j_1 \u2264 \u21910 j_1 \u22a2 MeasurableSet (i \u2229 j\u1d9c) ** exact hi.inter hj.compl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M inst\u271d : OrderedAddCommMonoid M s\u271d : SignedMeasure \u03b1 i\u271d j\u271d : Set \u03b1 s : SignedMeasure \u03b1 i j : Set \u03b1 hi : MeasurableSet i hj : MeasurableSet j hi' : restrict 0 i \u2264 restrict s i \u2227 restrict s i\u1d9c \u2264 restrict 0 i\u1d9c hj' : restrict 0 j \u2264 restrict s j \u2227 restrict s j\u1d9c \u2264 restrict 0 j\u1d9c \u22a2 MeasurableSet i\u1d9c ** measurability ** Qed", + "informal": "" + }, + { + "formal": "Seminorm.finset_sup_apply ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E s : Finset \u03b9 x : E \u22a2 \u2191(Finset.sup s p) x = \u2191(Finset.sup s fun i => { val := \u2191(p i) x, property := (_ : 0 \u2264 \u2191(p i) x) }) ** induction' s using Finset.cons_induction_on with a s ha ih ** case h\u2081 R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E x : E \u22a2 \u2191(Finset.sup \u2205 p) x = \u2191(Finset.sup \u2205 fun i => { val := \u2191(p i) x, property := (_ : 0 \u2264 \u2191(p i) x) }) ** rw [Finset.sup_empty, Finset.sup_empty, coe_bot, _root_.bot_eq_zero, Pi.zero_apply] ** case h\u2081 R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E x : E \u22a2 0 = \u21910 ** norm_cast ** case h\u2082 R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E x : E a : \u03b9 s : Finset \u03b9 ha : \u00aca \u2208 s ih : \u2191(Finset.sup s p) x = \u2191(Finset.sup s fun i => { val := \u2191(p i) x, property := (_ : 0 \u2264 \u2191(p i) x) }) \u22a2 \u2191(Finset.sup (Finset.cons a s ha) p) x = \u2191(Finset.sup (Finset.cons a s ha) fun i => { val := \u2191(p i) x, property := (_ : 0 \u2264 \u2191(p i) x) }) ** rw [Finset.sup_cons, Finset.sup_cons, coe_sup, sup_eq_max, Pi.sup_apply, sup_eq_max,\n NNReal.coe_max, coe_mk, ih] ** Qed", + "informal": "" + }, + { + "formal": "Subfield.coe_sInf ** K : Type u L : Type v M : Type w inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Field M S : Set (Subfield K) \u22a2 \u2191(sInf (toSubring '' S)) = \u22c2 s \u2208 S, \u2191s ** ext x ** case h K : Type u L : Type v M : Type w inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Field M S : Set (Subfield K) x : K \u22a2 x \u2208 \u2191(sInf (toSubring '' S)) \u2194 x \u2208 \u22c2 s \u2208 S, \u2191s ** rw [Subring.coe_sInf, Set.mem_iInter, Set.mem_iInter] ** case h K : Type u L : Type v M : Type w inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Field M S : Set (Subfield K) x : K \u22a2 (\u2200 (i : Subring K), x \u2208 \u22c2 (_ : i \u2208 toSubring '' S), \u2191i) \u2194 \u2200 (i : Subfield K), x \u2208 \u22c2 (_ : i \u2208 S), \u2191i ** exact\n \u27e8fun h s s' \u27e8s_mem, s'_eq\u27e9 => h s.toSubring _ \u27e8\u27e8s, s_mem, rfl\u27e9, s'_eq\u27e9,\n fun h s s' \u27e8\u27e8s'', s''_mem, s_eq\u27e9, (s'_eq : \u2191s = s')\u27e9 =>\n h s'' _ \u27e8s''_mem, by simp [\u2190 s_eq, \u2190 s'_eq]\u27e9\u27e9 ** K : Type u L : Type v M : Type w inst\u271d\u00b2 : Field K inst\u271d\u00b9 : Field L inst\u271d : Field M S : Set (Subfield K) x : K h : \u2200 (i : Subfield K), x \u2208 \u22c2 (_ : i \u2208 S), \u2191i s : Subring K s' : Set K x\u271d : s' \u2208 Set.range fun h => \u2191s s'' : Subfield K s''_mem : s'' \u2208 S s_eq : s''.toSubring = s s'_eq : \u2191s = s' \u22a2 (fun h => \u2191s'') s''_mem = s' ** simp [\u2190 s_eq, \u2190 s'_eq] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.iterate_derivative_C_mul ** R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a\u271d b : R n : \u2115 inst\u271d : Semiring R a : R p : R[X] k : \u2115 \u22a2 (\u2191derivative)^[k] (\u2191C a * p) = \u2191C a * (\u2191derivative)^[k] p ** simp_rw [\u2190 smul_eq_C_mul, iterate_derivative_smul] ** Qed", + "informal": "" + }, + { + "formal": "Rel.interedges_biUnion_right ** \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b3 : LinearOrderedField \ud835\udd5c r : \u03b1 \u2192 \u03b2 \u2192 Prop inst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a) s\u271d s\u2081 s\u2082 : Finset \u03b1 t\u271d t\u2081 t\u2082 : Finset \u03b2 a : \u03b1 b : \u03b2 \u03b4 : \ud835\udd5c inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b9 f : \u03b9 \u2192 Finset \u03b2 \u22a2 interedges r s (Finset.biUnion t f) = Finset.biUnion t fun b => interedges r s (f b) ** ext a ** case a \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b3 : LinearOrderedField \ud835\udd5c r : \u03b1 \u2192 \u03b2 \u2192 Prop inst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a) s\u271d s\u2081 s\u2082 : Finset \u03b1 t\u271d t\u2081 t\u2082 : Finset \u03b2 a\u271d : \u03b1 b : \u03b2 \u03b4 : \ud835\udd5c inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b9 f : \u03b9 \u2192 Finset \u03b2 a : \u03b1 \u00d7 \u03b2 \u22a2 a \u2208 interedges r s (Finset.biUnion t f) \u2194 a \u2208 Finset.biUnion t fun b => interedges r s (f b) ** simp only [mem_interedges_iff, mem_biUnion] ** case a \ud835\udd5c : Type u_1 \u03b9 : Type u_2 \u03ba : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b3 : LinearOrderedField \ud835\udd5c r : \u03b1 \u2192 \u03b2 \u2192 Prop inst\u271d\u00b2 : (a : \u03b1) \u2192 DecidablePred (r a) s\u271d s\u2081 s\u2082 : Finset \u03b1 t\u271d t\u2081 t\u2082 : Finset \u03b2 a\u271d : \u03b1 b : \u03b2 \u03b4 : \ud835\udd5c inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 s : Finset \u03b1 t : Finset \u03b9 f : \u03b9 \u2192 Finset \u03b2 a : \u03b1 \u00d7 \u03b2 \u22a2 a.1 \u2208 s \u2227 (\u2203 a_1, a_1 \u2208 t \u2227 a.2 \u2208 f a_1) \u2227 r a.1 a.2 \u2194 \u2203 a_1, a_1 \u2208 t \u2227 a.1 \u2208 s \u2227 a.2 \u2208 f a_1 \u2227 r a.1 a.2 ** exact \u27e8fun \u27e8x\u2081, \u27e8x\u2082, x\u2083, x\u2084\u27e9, x\u2085\u27e9 \u21a6 \u27e8x\u2082, x\u2083, x\u2081, x\u2084, x\u2085\u27e9,\n fun \u27e8x\u2082, x\u2083, x\u2081, x\u2084, x\u2085\u27e9 \u21a6 \u27e8x\u2081, \u27e8x\u2082, x\u2083, x\u2084\u27e9, x\u2085\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "NNReal.iInf_empty ** \u03b9 : Sort u_1 f\u271d : \u03b9 \u2192 \u211d\u22650 inst\u271d : IsEmpty \u03b9 f : \u03b9 \u2192 \u211d\u22650 \u22a2 \u2a05 i, f i = 0 ** rw [iInf_of_empty', sInf_empty] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.ProbabilityMeasure.nonempty ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 \u22a2 Nonempty \u03a9 ** by_contra maybe_empty ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 maybe_empty : \u00acNonempty \u03a9 \u22a2 False ** have zero : (\u03bc : Measure \u03a9) univ = 0 := by\n rw [univ_eq_empty_iff.mpr (not_nonempty_iff.mp maybe_empty), measure_empty] ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 maybe_empty : \u00acNonempty \u03a9 zero : \u2191\u2191\u2191\u03bc univ = 0 \u22a2 False ** rw [measure_univ] at zero ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 maybe_empty : \u00acNonempty \u03a9 zero : 1 = 0 \u22a2 False ** exact zero_ne_one zero.symm ** \u03a9 : Type u_1 inst\u271d : MeasurableSpace \u03a9 \u03bc : ProbabilityMeasure \u03a9 maybe_empty : \u00acNonempty \u03a9 \u22a2 \u2191\u2191\u2191\u03bc univ = 0 ** rw [univ_eq_empty_iff.mpr (not_nonempty_iff.mp maybe_empty), measure_empty] ** Qed", + "informal": "" + }, + { + "formal": "ModuleCat.mono_iff_ker_eq_bot ** R : Type u inst\u271d\u00b2 : Ring R X Y : ModuleCat R f : X \u27f6 Y M : Type v inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M hf : LinearMap.ker f = \u22a5 \u22a2 Function.Injective \u2191f ** convert LinearMap.ker_eq_bot.1 hf ** Qed", + "informal": "" + }, + { + "formal": "MonoidAlgebra.liftNC_one ** k : Type u\u2081 G : Type u\u2082 H : Type u_1 R : Type u_2 inst\u271d\u00b3 : NonAssocSemiring R inst\u271d\u00b2 : Semiring k inst\u271d\u00b9 : One G g_hom : Type u_3 inst\u271d : OneHomClass g_hom G R f : k \u2192+* R g : g_hom \u22a2 \u2191(liftNC \u2191f \u2191g) 1 = 1 ** simp [one_def] ** Qed", + "informal": "" + }, + { + "formal": "AddCommGroup.add_modEq_right ** \u03b1 : Type u_1 inst\u271d : AddCommGroup \u03b1 p a a\u2081 a\u2082 b b\u2081 b\u2082 c : \u03b1 n : \u2115 z : \u2124 \u22a2 a + b \u2261 b [PMOD p] \u2194 a \u2261 0 [PMOD p] ** simp [\u2190 modEq_sub_iff_add_modEq] ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.mapRange_zero ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 inst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i) inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i) f : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i hf : \u2200 (i : \u03b9), f i 0 = 0 \u22a2 mapRange f hf 0 = 0 ** ext ** case h \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 inst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i) inst\u271d : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i) f : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i hf : \u2200 (i : \u03b9), f i 0 = 0 i\u271d : \u03b9 \u22a2 \u2191(mapRange f hf 0) i\u271d = \u21910 i\u271d ** simp only [mapRange_apply, coe_zero, Pi.zero_apply, hf] ** Qed", + "informal": "" + }, + { + "formal": "padicNorm.dvd_iff_norm_le ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 \u22a2 \u2191(p ^ n) \u2223 z \u2194 padicNorm p \u2191z \u2264 \u2191p ^ (-\u2191n) ** unfold padicNorm ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 \u22a2 \u2191(p ^ n) \u2223 z \u2194 (if \u2191z = 0 then 0 else \u2191p ^ (-padicValRat p \u2191z)) \u2264 \u2191p ^ (-\u2191n) ** split_ifs with hz ** case pos p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u2191z = 0 \u22a2 \u2191(p ^ n) \u2223 z \u2194 0 \u2264 \u2191p ^ (-\u2191n) ** norm_cast at hz ** case pos p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : z = 0 \u22a2 \u2191(p ^ n) \u2223 z \u2194 0 \u2264 \u2191p ^ (-\u2191n) ** simp [hz] ** case neg p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u00ac\u2191z = 0 \u22a2 \u2191(p ^ n) \u2223 z \u2194 \u2191p ^ (-padicValRat p \u2191z) \u2264 \u2191p ^ (-\u2191n) ** rw [zpow_le_iff_le, neg_le_neg_iff, padicValRat.of_int,\n padicValInt.of_ne_one_ne_zero hp.1.ne_one _] ** case neg p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u00ac\u2191z = 0 \u22a2 \u2191(p ^ n) \u2223 z \u2194 \u2191n \u2264 \u2191(Part.get (multiplicity (\u2191p) z) (_ : multiplicity.Finite (\u2191p) z)) ** norm_cast ** case neg p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u00ac\u2191z = 0 \u22a2 \u2191(p ^ n) \u2223 z \u2194 n \u2264 Part.get (multiplicity (\u2191p) z) (_ : multiplicity.Finite (\u2191p) z) ** rw [\u2190 PartENat.coe_le_coe, PartENat.natCast_get, \u2190 multiplicity.pow_dvd_iff_le_multiplicity,\n Nat.cast_pow] ** p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u00ac\u2191z = 0 \u22a2 z \u2260 0 p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u00ac\u2191z = 0 \u22a2 z \u2260 0 ** exact_mod_cast hz ** case neg p : \u2115 hp : Fact (Nat.Prime p) n : \u2115 z : \u2124 hz : \u00ac\u2191z = 0 \u22a2 1 < \u2191p ** exact_mod_cast hp.1.one_lt ** Qed", + "informal": "" + }, + { + "formal": "List.cyclicPermutations_of_ne_nil ** \u03b1 : Type u l\u271d l' l : List \u03b1 h : l \u2260 [] \u22a2 cyclicPermutations l = dropLast (zipWith (fun x x_1 => x ++ x_1) (tails l) (inits l)) ** obtain \u27e8hd, tl, rfl\u27e9 := exists_cons_of_ne_nil h ** case intro.intro \u03b1 : Type u l l' : List \u03b1 hd : \u03b1 tl : List \u03b1 h : hd :: tl \u2260 [] \u22a2 cyclicPermutations (hd :: tl) = dropLast (zipWith (fun x x_1 => x ++ x_1) (tails (hd :: tl)) (inits (hd :: tl))) ** exact cyclicPermutations_cons _ _ ** Qed", + "informal": "" + }, + { + "formal": "Equiv.piCongrLeft_symm_preimage_univ_pi ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : \u03b9 \u2192 Type u_3 f : \u03b9' \u2243 \u03b9 t : (i : \u03b9) \u2192 Set (\u03b1 i) \u22a2 (\u2191(piCongrLeft \u03b1 f).symm \u207b\u00b9' pi univ fun i' => t (\u2191f i')) = pi univ t ** simpa [f.surjective.range_eq] using piCongrLeft_symm_preimage_pi f univ t ** Qed", + "informal": "" + }, + { + "formal": "Finset.strongDownwardInduction_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 s\u271d t : Finset \u03b1 f : \u03b1 \u2192 \u03b2 n : \u2115 p : Finset \u03b1 \u2192 Sort u_3 H : (t\u2081 : Finset \u03b1) \u2192 ({t\u2082 : Finset \u03b1} \u2192 card t\u2082 \u2264 n \u2192 t\u2081 \u2282 t\u2082 \u2192 p t\u2082) \u2192 card t\u2081 \u2264 n \u2192 p t\u2081 s : Finset \u03b1 \u22a2 strongDownwardInduction H s = H s fun {t} ht x => strongDownwardInduction H t ht ** rw [strongDownwardInduction] ** Qed", + "informal": "" + }, + { + "formal": "nhdsWithin_Ici_eq'' ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : OrderTopology \u03b1 a : \u03b1 \u22a2 \ud835\udcdd[Ici a] a = (\u2a05 u, \u2a05 (_ : a < u), \ud835\udcdf (Iio u)) \u2293 \ud835\udcdf (Ici a) ** rw [nhdsWithin, nhds_eq_order] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : OrderTopology \u03b1 a : \u03b1 \u22a2 ((\u2a05 b \u2208 Iio a, \ud835\udcdf (Ioi b)) \u2293 \u2a05 b \u2208 Ioi a, \ud835\udcdf (Iio b)) \u2293 \ud835\udcdf (Ici a) = (\u2a05 u, \u2a05 (_ : a < u), \ud835\udcdf (Iio u)) \u2293 \ud835\udcdf (Ici a) ** refine' le_antisymm (inf_le_inf_right _ inf_le_right) (le_inf (le_inf _ inf_le_left) inf_le_right) ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : OrderTopology \u03b1 a : \u03b1 \u22a2 (\u2a05 u, \u2a05 (_ : a < u), \ud835\udcdf (Iio u)) \u2293 \ud835\udcdf (Ici a) \u2264 \u2a05 b \u2208 Iio a, \ud835\udcdf (Ioi b) ** exact inf_le_right.trans (le_iInf\u2082 fun l hl => principal_mono.2 <| Ici_subset_Ioi.2 hl) ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.biproduct.mapBiproduct_hom_desc ** C : Type u\u2081 inst\u271d\u2076 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u2075 : Category.{v\u2082, u\u2082} D inst\u271d\u2074 : HasZeroMorphisms C inst\u271d\u00b3 : HasZeroMorphisms D F : C \u2964 D inst\u271d\u00b2 : PreservesZeroMorphisms F J : Type w\u2081 f : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : PreservesBiproduct f F W : C g : (j : J) \u2192 f j \u27f6 W \u22a2 ((mapBiproduct F f).hom \u226b desc fun j => F.map (g j)) = F.map (desc g) ** rw [\u2190 biproduct.mapBiproduct_inv_map_desc, Iso.hom_inv_id_assoc] ** Qed", + "informal": "" + }, + { + "formal": "BoxIntegral.integrable_of_continuousOn ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc \u22a2 Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) ** have huc := I.isCompact_Icc.uniformContinuousOn_of_continuous hc ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : UniformContinuousOn f (\u2191Box.Icc I) \u22a2 Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) ** rw [Metric.uniformContinuousOn_iff_le] at huc ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u22a2 Integrable I l f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) ** refine' integrable_iff_cauchy_basis.2 fun \u03b5 \u03b50 => _ ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u22a2 \u2203 r, (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227 \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I), MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192 IsPartition \u03c0\u2081 \u2192 MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081) (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264 \u03b5 ** rcases exists_pos_mul_lt \u03b50 (\u03bc.toBoxAdditive I) with \u27e8\u03b5', \u03b50', h\u03b5\u27e9 ** case intro.intro \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u22a2 \u2203 r, (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227 \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I), MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192 IsPartition \u03c0\u2081 \u2192 MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081) (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264 \u03b5 ** rcases huc \u03b5' \u03b50' with \u27e8\u03b4, \u03b40 : 0 < \u03b4, H\u03b4\u27e9 ** case intro.intro.intro.intro \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' \u22a2 \u2203 r, (\u2200 (c : \u211d\u22650), RCond l (r c)) \u2227 \u2200 (c\u2081 c\u2082 : \u211d\u22650) (\u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I), MemBaseSet l I c\u2081 (r c\u2081) \u03c0\u2081 \u2192 IsPartition \u03c0\u2081 \u2192 MemBaseSet l I c\u2082 (r c\u2082) \u03c0\u2082 \u2192 IsPartition \u03c0\u2082 \u2192 dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081) (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264 \u03b5 ** refine' \u27e8fun _ _ => \u27e8\u03b4 / 2, half_pos \u03b40\u27e9, fun _ _ _ => rfl, fun c\u2081 c\u2082 \u03c0\u2081 \u03c0\u2082 h\u2081 h\u2081p h\u2082 h\u2082p => _\u27e9 ** case intro.intro.intro.intro \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 \u22a2 dist (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2081) (integralSum f (BoxAdditiveMap.toSMul (Measure.toBoxAdditive \u03bc)) \u03c0\u2082) \u2264 \u03b5 ** simp only [dist_eq_norm, integralSum_sub_partitions _ _ h\u2081p h\u2082p, BoxAdditiveMap.toSMul_apply,\n \u2190 smul_sub] ** case intro.intro.intro.intro \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 this : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' \u22a2 \u2016\u2211 x in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes, \u2191(Measure.toBoxAdditive \u03bc) x \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) x) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) x))\u2016 \u2264 \u03b5 ** refine' (norm_sum_le_of_le _ this).trans _ ** case intro.intro.intro.intro \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 this : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' \u22a2 \u2211 b in (\u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition).boxes, \u2191(Measure.toBoxAdditive \u03bc) b * \u03b5' \u2264 \u03b5 ** rw [\u2190 Finset.sum_mul, \u03bc.toBoxAdditive.sum_partition_boxes le_top (h\u2081p.inf h\u2082p)] ** case intro.intro.intro.intro \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 this : \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' \u22a2 \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' \u2264 \u03b5 ** exact h\u03b5.le ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 \u22a2 \u2200 (J : Box \u03b9), J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u2192 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' ** intro J hJ ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition \u22a2 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' ** have : 0 \u2264 \u03bc.toBoxAdditive J := ENNReal.toReal_nonneg ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 \u2016\u2191(Measure.toBoxAdditive \u03bc) J \u2022 (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) - f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J))\u2016 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' ** rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg this, \u2190 dist_eq_norm] ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 \u2191(Measure.toBoxAdditive \u03bc) J * dist (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J)) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J)) \u2264 \u2191(Measure.toBoxAdditive \u03bc) J * \u03b5' ** refine' mul_le_mul_of_nonneg_left _ this ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 dist (f (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J)) (f (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J)) \u2264 \u03b5' ** refine' H\u03b4 _ (TaggedPrepartition.tag_mem_Icc _ _) _ (TaggedPrepartition.tag_mem_Icc _ _) _ ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 dist (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J) \u2264 \u03b4 ** rw [\u2190 add_halves \u03b4] ** \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 dist (tag (infPrepartition \u03c0\u2081 \u03c0\u2082.toPrepartition) J) (tag (infPrepartition \u03c0\u2082 \u03c0\u2081.toPrepartition) J) \u2264 \u03b4 / 2 + \u03b4 / 2 ** refine' (dist_triangle_left _ _ J.upper).trans (add_le_add (h\u2081.1 _ _ _) (h\u2082.1 _ _ _)) ** case refine'_1 \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 Prepartition.biUnionIndex \u03c0\u2081.toPrepartition (fun J => Prepartition.restrict \u03c0\u2082.toPrepartition J) J \u2208 \u03c0\u2081 ** exact Prepartition.biUnionIndex_mem _ hJ ** case refine'_2 \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 J.upper \u2208 \u2191Box.Icc (Prepartition.biUnionIndex \u03c0\u2081.toPrepartition (fun J => Prepartition.restrict \u03c0\u2082.toPrepartition J) J) ** exact Box.le_iff_Icc.1 (Prepartition.le_biUnionIndex _ hJ) J.upper_mem_Icc ** case refine'_3 \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J \u2208 \u03c0\u2082 ** rw [_root_.inf_comm] at hJ ** case refine'_3 \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2082.toPrepartition \u2293 \u03c0\u2081.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J \u2208 \u03c0\u2082 ** exact Prepartition.biUnionIndex_mem _ hJ ** case refine'_4 \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2081.toPrepartition \u2293 \u03c0\u2082.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 J.upper \u2208 \u2191Box.Icc (Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J) ** rw [_root_.inf_comm] at hJ ** case refine'_4 \u03b9 : Type u E : Type v F : Type w inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F I\u271d J\u271d : Box \u03b9 \u03c0 : TaggedPrepartition I\u271d inst\u271d\u00b2 : Fintype \u03b9 l : IntegrationParams f\u271d g : (\u03b9 \u2192 \u211d) \u2192 E vol : \u03b9 \u2192\u1d47\u1d43[\u22a4] E \u2192L[\u211d] F y y' : F inst\u271d\u00b9 : CompleteSpace E I : Box \u03b9 f : (\u03b9 \u2192 \u211d) \u2192 E hc : ContinuousOn f (\u2191Box.Icc I) \u03bc : Measure (\u03b9 \u2192 \u211d) inst\u271d : IsLocallyFiniteMeasure \u03bc huc : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5 \u03b5 : \u211d \u03b50 : \u03b5 > 0 \u03b5' : \u211d \u03b50' : 0 < \u03b5' h\u03b5 : \u2191(Measure.toBoxAdditive \u03bc) I * \u03b5' < \u03b5 \u03b4 : \u211d \u03b40 : 0 < \u03b4 H\u03b4 : \u2200 (x : \u03b9 \u2192 \u211d), x \u2208 \u2191Box.Icc I \u2192 \u2200 (y : \u03b9 \u2192 \u211d), y \u2208 \u2191Box.Icc I \u2192 dist x y \u2264 \u03b4 \u2192 dist (f x) (f y) \u2264 \u03b5' c\u2081 c\u2082 : \u211d\u22650 \u03c0\u2081 \u03c0\u2082 : TaggedPrepartition I h\u2081 : MemBaseSet l I c\u2081 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2081) \u03c0\u2081 h\u2081p : IsPartition \u03c0\u2081 h\u2082 : MemBaseSet l I c\u2082 ((fun x x => { val := \u03b4 / 2, property := (_ : 0 < \u03b4 / 2) }) c\u2082) \u03c0\u2082 h\u2082p : IsPartition \u03c0\u2082 J : Box \u03b9 hJ : J \u2208 \u03c0\u2082.toPrepartition \u2293 \u03c0\u2081.toPrepartition this : 0 \u2264 \u2191(Measure.toBoxAdditive \u03bc) J \u22a2 J.upper \u2208 \u2191Box.Icc (Prepartition.biUnionIndex \u03c0\u2082.toPrepartition (fun J => Prepartition.restrict \u03c0\u2081.toPrepartition J) J) ** exact Box.le_iff_Icc.1 (Prepartition.le_biUnionIndex _ hJ) J.upper_mem_Icc ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.colimit.map_desc ** J : Type u\u2081 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} J K : Type u\u2082 inst\u271d\u00b2 : Category.{v\u2082, u\u2082} K C : Type u inst\u271d\u00b9 : Category.{v, u} C F : J \u2964 C inst\u271d : HasColimitsOfShape J C G : J \u2964 C \u03b1 : F \u27f6 G c : Cocone G \u22a2 colimMap \u03b1 \u226b desc G c = desc F ((Cocones.precompose \u03b1).obj c) ** ext j ** case w J : Type u\u2081 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} J K : Type u\u2082 inst\u271d\u00b2 : Category.{v\u2082, u\u2082} K C : Type u inst\u271d\u00b9 : Category.{v, u} C F : J \u2964 C inst\u271d : HasColimitsOfShape J C G : J \u2964 C \u03b1 : F \u27f6 G c : Cocone G j : J \u22a2 \u03b9 F j \u226b colimMap \u03b1 \u226b desc G c = \u03b9 F j \u226b desc F ((Cocones.precompose \u03b1).obj c) ** simp [\u2190 assoc, colimit.\u03b9_map, assoc, colimit.\u03b9_desc, colimit.\u03b9_desc] ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.sup_typein_limit ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop o : Ordinal.{u} ho : \u2200 (a : Ordinal.{u}), a < o \u2192 succ a < o \u22a2 \u2200 (a : Ordinal.{u}), a < o \u2192 succ a < o ** assumption ** Qed", + "informal": "" + }, + { + "formal": "Finset.erase_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t\u271d u v : Finset \u03b1 a\u271d b a : \u03b1 s t : Finset \u03b1 \u22a2 erase s a \u2229 t = erase (s \u2229 t) a ** simpa only [inter_comm t] using inter_erase a t s ** Qed", + "informal": "" + }, + { + "formal": "IntervalIntegrable.comp_sub_right ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedRing A f g : \u211d \u2192 E a b : \u211d \u03bc : Measure \u211d hf : IntervalIntegrable f volume a b c : \u211d \u22a2 IntervalIntegrable (fun x => f (x - c)) volume (a + c) (b + c) ** simpa only [sub_neg_eq_add] using IntervalIntegrable.comp_add_right hf (-c) ** Qed", + "informal": "" + }, + { + "formal": "List.tfae_cons_of_mem ** a b : Prop l : List Prop h : b \u2208 l H : TFAE (a :: l) \u22a2 a \u2208 a :: l ** simp ** a b : Prop l : List Prop h : b \u2208 l \u22a2 (a \u2194 b) \u2227 TFAE l \u2192 TFAE (a :: l) ** rintro \u27e8ab, H\u27e9 p (_ | \u27e8_, hp\u27e9) q (_ | \u27e8_, hq\u27e9) ** case intro.head.head a b : Prop l : List Prop h : b \u2208 l ab : a \u2194 b H : TFAE l \u22a2 a \u2194 a ** rfl ** case intro.head.tail a b : Prop l : List Prop h : b \u2208 l ab : a \u2194 b H : TFAE l q : Prop hq : Mem q l \u22a2 a \u2194 q ** exact ab.trans (H _ h _ hq) ** case intro.tail.head a b : Prop l : List Prop h : b \u2208 l ab : a \u2194 b H : TFAE l p : Prop hp : Mem p l \u22a2 p \u2194 a ** exact (ab.trans (H _ h _ hp)).symm ** case intro.tail.tail a b : Prop l : List Prop h : b \u2208 l ab : a \u2194 b H : TFAE l p : Prop hp : Mem p l q : Prop hq : Mem q l \u22a2 p \u2194 q ** exact H _ hp _ hq ** Qed", + "informal": "" + }, + { + "formal": "Set.biUnion_Ico_eq_Iio_self_iff ** \u03b9 : Sort u \u03b1 : Type v \u03b2 : Type w inst\u271d : LinearOrder \u03b1 a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1 p : \u03b9 \u2192 Prop f : (i : \u03b9) \u2192 p i \u2192 \u03b1 a : \u03b1 \u22a2 \u22c3 i, \u22c3 (hi : p i), Ico (f i hi) a = Iio a \u2194 \u2200 (x : \u03b1), x < a \u2192 \u2203 i hi, f i hi \u2264 x ** simp [\u2190 Ici_inter_Iio, \u2190 iUnion_inter, subset_def] ** Qed", + "informal": "" + }, + { + "formal": "AlternatingMap.compLinearMap_injective ** R : Type u_1 inst\u271d\u00b9\u2074 : Semiring R M : Type u_2 inst\u271d\u00b9\u00b3 : AddCommMonoid M inst\u271d\u00b9\u00b2 : Module R M N : Type u_3 inst\u271d\u00b9\u00b9 : AddCommMonoid N inst\u271d\u00b9\u2070 : Module R N P : Type u_4 inst\u271d\u2079 : AddCommMonoid P inst\u271d\u2078 : Module R P M' : Type u_5 inst\u271d\u2077 : AddCommGroup M' inst\u271d\u2076 : Module R M' N' : Type u_6 inst\u271d\u2075 : AddCommGroup N' inst\u271d\u2074 : Module R N' \u03b9 : Type u_7 \u03b9' : Type u_8 \u03b9'' : Type u_9 M\u2082 : Type u_10 inst\u271d\u00b3 : AddCommMonoid M\u2082 inst\u271d\u00b2 : Module R M\u2082 M\u2083 : Type u_11 inst\u271d\u00b9 : AddCommMonoid M\u2083 inst\u271d : Module R M\u2083 f : M\u2082 \u2192\u2097[R] M hf : Function.Surjective \u2191f g\u2081 g\u2082 : AlternatingMap R M N \u03b9 h : (fun g => compLinearMap g f) g\u2081 = (fun g => compLinearMap g f) g\u2082 x : \u03b9 \u2192 M \u22a2 \u2191g\u2081 x = \u2191g\u2082 x ** simpa [Function.surjInv_eq hf] using ext_iff.mp h (Function.surjInv hf \u2218 x) ** Qed", + "informal": "" + }, + { + "formal": "QuadraticForm.associated_eq_self_apply ** S : Type u_1 T : Type u_2 R : Type u_3 M : Type u_4 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : CommSemiring S inst\u271d\u00b9 : Algebra S R inst\u271d : Invertible 2 B\u2081 : BilinForm R M Q : QuadraticForm R M x : M \u22a2 bilin (\u2191(associatedHom S) Q) x x = \u2191Q x ** rw [associated_apply, map_add_self, \u2190 three_add_one_eq_four, \u2190 two_add_one_eq_three,\n add_mul, add_mul, one_mul, add_sub_cancel, add_sub_cancel, invOf_mul_self_assoc] ** Qed", + "informal": "" + }, + { + "formal": "Array.getElem_ofFn_go ** n : Nat \u03b1 : Type ?u.103683 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } this : size acc + (n - size acc) = n \u22a2 k < size (ofFn.go f i acc) ** simp [*] ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } \u22a2 (ofFn.go f i acc)[k] = f { val := k, isLt := hki } ** unfold ofFn.go ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } \u22a2 (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc)[k] = f { val := k, isLt := hki } ** if hin : i < n then\n have : 1 + (n - (i + 1)) = n - i :=\n Nat.sub_sub .. \u25b8 Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. \u25b8 hin))\n simp only [dif_pos hin]\n rw [getElem_ofFn_go f (i+1) _ hin (by simp [*]) (fun j hj => ?hacc)]\n cases (Nat.lt_or_eq_of_le <| Nat.le_of_lt_succ (by simpa using hj)) with\n | inl hj => simp [get_push, hj, hacc j hj]\n | inr hj => simp [get_push, *]\nelse\n simp [hin, hacc k (Nat.lt_of_lt_of_le hki (Nat.le_of_not_lt (hi \u25b8 hin)))] ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n \u22a2 (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc)[k] = f { val := k, isLt := hki } ** have : 1 + (n - (i + 1)) = n - i :=\n Nat.sub_sub .. \u25b8 Nat.add_sub_cancel' (Nat.le_sub_of_add_le (Nat.add_comm .. \u25b8 hin)) ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i \u22a2 (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc)[k] = f { val := k, isLt := hki } ** simp only [dif_pos hin] ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i \u22a2 (ofFn.go f (i + 1) (push acc (f { val := i, isLt := hin })))[k] = f { val := k, isLt := hki } ** rw [getElem_ofFn_go f (i+1) _ hin (by simp [*]) (fun j hj => ?hacc)] ** case hacc n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i j : Nat hj : j < size (push acc (f { val := i, isLt := hin })) \u22a2 (push acc (f { val := i, isLt := hin }))[j] = f { val := j, isLt := (_ : j < n) } ** cases (Nat.lt_or_eq_of_le <| Nat.le_of_lt_succ (by simpa using hj)) with\n| inl hj => simp [get_push, hj, hacc j hj]\n| inr hj => simp [get_push, *] ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i \u22a2 i + 1 = size (push acc (f { val := i, isLt := hin })) ** simp [*] ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i j : Nat hj : j < size (push acc (f { val := i, isLt := hin })) \u22a2 ?m.105816 < Nat.succ ?m.105817 ** simpa using hj ** case hacc.inl n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i j : Nat hj\u271d : j < size (push acc (f { val := i, isLt := hin })) hj : j < size acc \u22a2 (push acc (f { val := i, isLt := hin }))[j] = f { val := j, isLt := (_ : j < n) } ** simp [get_push, hj, hacc j hj] ** case hacc.inr n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : i < n this : 1 + (n - (i + 1)) = n - i j : Nat hj\u271d : j < size (push acc (f { val := i, isLt := hin })) hj : j = size acc \u22a2 (push acc (f { val := i, isLt := hin }))[j] = f { val := j, isLt := (_ : j < n) } ** simp [get_push, *] ** n : Nat \u03b1 : Type u_1 f : Fin n \u2192 \u03b1 i : Nat acc : Array \u03b1 k : Nat hki : k < n hin\u271d : i \u2264 n hi : i = size acc hacc : \u2200 (j : Nat) (hj : j < size acc), acc[j] = f { val := j, isLt := (_ : j < n) } hin : \u00aci < n \u22a2 (if h : i < n then ofFn.go f (i + 1) (push acc (f { val := i, isLt := h })) else acc)[k] = f { val := k, isLt := hki } ** simp [hin, hacc k (Nat.lt_of_lt_of_le hki (Nat.le_of_not_lt (hi \u25b8 hin)))] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.withDensity\u1d65_eq_withDensity_pos_part_sub_withDensity_neg_part ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E f : \u03b1 \u2192 \u211d hfi : Integrable f \u22a2 withDensity\u1d65 \u03bc f = toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E f : \u03b1 \u2192 \u211d hfi : Integrable f this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) \u22a2 withDensity\u1d65 \u03bc f = toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E f : \u03b1 \u2192 \u211d hfi : Integrable f this\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) \u22a2 withDensity\u1d65 \u03bc f = toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x)) ** ext i hi ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 E : Type u_3 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u211d E inst\u271d : CompleteSpace E f\u271d g : \u03b1 \u2192 E f : \u03b1 \u2192 \u211d hfi : Integrable f this\u271d : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) this : IsFiniteMeasure (withDensity \u03bc fun x => ENNReal.ofReal ((-f) x)) i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(withDensity\u1d65 \u03bc f) i = \u2191(toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (f x)) - toSignedMeasure (withDensity \u03bc fun x => ENNReal.ofReal (-f x))) i ** rw [withDensity\u1d65_apply hfi hi,\n integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi.integrableOn,\n VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi,\n toSignedMeasure_apply_measurable hi, withDensity_apply _ hi, withDensity_apply _ hi] ** Qed", + "informal": "" + }, + { + "formal": "IsLowerSet.thickening' ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : NormedOrderedGroup \u03b1 s : Set \u03b1 hs : IsLowerSet s \u03b5 : \u211d \u22a2 IsLowerSet (thickening \u03b5 s) ** rw [\u2190 ball_mul_one] ** \u03b1 : Type u_1 \u03b9 : Type u_2 inst\u271d : NormedOrderedGroup \u03b1 s : Set \u03b1 hs : IsLowerSet s \u03b5 : \u211d \u22a2 IsLowerSet (ball 1 \u03b5 * s) ** exact hs.mul_left ** Qed", + "informal": "" + }, + { + "formal": "Nat.Icc_succ_left ** a b c : \u2115 \u22a2 Icc (succ a) b = Ioc a b ** ext x ** case a a b c x : \u2115 \u22a2 x \u2208 Icc (succ a) b \u2194 x \u2208 Ioc a b ** rw [mem_Icc, mem_Ioc, succ_le_iff] ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.aeval_esymm_eq_multiset_esymm ** \u03c3 : Type u_1 R : Type u_2 \u03c4 : Type u_3 S : Type u_4 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Fintype \u03c3 inst\u271d\u00b9 : Fintype \u03c4 inst\u271d : Algebra R S f : \u03c3 \u2192 S n : \u2115 \u22a2 \u2191(aeval f) (esymm \u03c3 R n) = Multiset.esymm (Multiset.map f univ.val) n ** simp_rw [esymm, aeval_sum, aeval_prod, aeval_X, Finset.esymm_map_val] ** Qed", + "informal": "" + }, + { + "formal": "AffineSubspace.direction_le ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b2 : Ring k inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module k V S : AffineSpace V P p\u2081 p\u2082 : P s1 s2 : AffineSubspace k P h : s1 \u2264 s2 \u22a2 direction s1 \u2264 direction s2 ** simp only [direction_eq_vectorSpan, vectorSpan_def] ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b2 : Ring k inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module k V S : AffineSpace V P p\u2081 p\u2082 : P s1 s2 : AffineSubspace k P h : s1 \u2264 s2 \u22a2 Submodule.span k (\u2191s1 -\u1d65 \u2191s1) \u2264 Submodule.span k (\u2191s2 -\u1d65 \u2191s2) ** exact vectorSpan_mono k h ** Qed", + "informal": "" + }, + { + "formal": "Metric.hausdorffDist_le_of_infDist ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r \u22a2 hausdorffDist s t \u2264 r ** by_cases h1 : hausdorffEdist s t = \u22a4 ** case neg \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 \u22a2 hausdorffDist s t \u2264 r ** cases' s.eq_empty_or_nonempty with hs hs ** case neg.inr \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s \u22a2 hausdorffDist s t \u2264 r ** cases' t.eq_empty_or_nonempty with ht ht ** case neg.inr.inr \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t this : hausdorffEdist s t \u2264 ENNReal.ofReal r \u22a2 hausdorffDist s t \u2264 r ** rwa [hausdorffDist, \u2190 ENNReal.toReal_ofReal hr,\n ENNReal.toReal_le_toReal h1 ENNReal.ofReal_ne_top] ** case pos \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : hausdorffEdist s t = \u22a4 \u22a2 hausdorffDist s t \u2264 r ** rwa [hausdorffDist, h1, ENNReal.top_toReal] ** case neg.inl \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : s = \u2205 \u22a2 hausdorffDist s t \u2264 r ** rwa [hs, hausdorffDist_empty'] ** case neg.inr.inl \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : t = \u2205 \u22a2 hausdorffDist s t \u2264 r ** rwa [ht, hausdorffDist_empty] ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t \u22a2 hausdorffEdist s t \u2264 ENNReal.ofReal r ** apply hausdorffEdist_le_of_infEdist _ _ ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 infEdist x t \u2264 ENNReal.ofReal r ** intro x hx ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x\u271d y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t x : \u03b1 hx : x \u2208 s \u22a2 infEdist x t \u2264 ENNReal.ofReal r ** have I := H1 x hx ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x\u271d y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t x : \u03b1 hx : x \u2208 s I : infDist x t \u2264 r \u22a2 infEdist x t \u2264 ENNReal.ofReal r ** rwa [infDist, \u2190 ENNReal.toReal_ofReal hr,\n ENNReal.toReal_le_toReal (infEdist_ne_top ht) ENNReal.ofReal_ne_top] at I ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t \u22a2 \u2200 (x : \u03b1), x \u2208 t \u2192 infEdist x s \u2264 ENNReal.ofReal r ** intro x hx ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x\u271d y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t x : \u03b1 hx : x \u2208 t \u22a2 infEdist x s \u2264 ENNReal.ofReal r ** have I := H2 x hx ** \u03b9 : Sort u_1 \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b9 : PseudoMetricSpace \u03b1 inst\u271d : PseudoMetricSpace \u03b2 s t u : Set \u03b1 x\u271d y : \u03b1 \u03a6 : \u03b1 \u2192 \u03b2 r : \u211d hr : 0 \u2264 r H1 : \u2200 (x : \u03b1), x \u2208 s \u2192 infDist x t \u2264 r H2 : \u2200 (x : \u03b1), x \u2208 t \u2192 infDist x s \u2264 r h1 : \u00achausdorffEdist s t = \u22a4 hs : Set.Nonempty s ht : Set.Nonempty t x : \u03b1 hx : x \u2208 t I : infDist x s \u2264 r \u22a2 infEdist x s \u2264 ENNReal.ofReal r ** rwa [infDist, \u2190 ENNReal.toReal_ofReal hr,\n ENNReal.toReal_le_toReal (infEdist_ne_top hs) ENNReal.ofReal_ne_top] at I ** Qed", + "informal": "" + }, + { + "formal": "archimedean_iff_rat_lt ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 H : \u2200 (x : \u03b1), \u2203 q, x < \u2191q x : \u03b1 q : \u211a h : x < \u2191q \u22a2 \u2191q \u2264 \u2191\u2308q\u2309\u208a ** simpa only [Rat.cast_coe_nat] using (@Rat.cast_le \u03b1 _ _ _).2 (Nat.le_ceil _) ** Qed", + "informal": "" + }, + { + "formal": "Finset.offDiag_inter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 x : \u03b1 \u00d7 \u03b1 \u22a2 \u2191(offDiag (s \u2229 t)) = \u2191(offDiag s \u2229 offDiag t) ** push_cast ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 x : \u03b1 \u00d7 \u03b1 \u22a2 Set.offDiag (\u2191s \u2229 \u2191t) = Set.offDiag \u2191s \u2229 Set.offDiag \u2191t ** exact Set.offDiag_inter _ _ ** Qed", + "informal": "" + }, + { + "formal": "AEMeasurable.subtype_mk ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 h : AEMeasurable f s : Set \u03b2 hfs : \u2200 (x : \u03b1), f x \u2208 s \u22a2 AEMeasurable (codRestrict f s hfs) ** nontriviality \u03b1 ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 h : AEMeasurable f s : Set \u03b2 hfs : \u2200 (x : \u03b1), f x \u2208 s \u271d : Nontrivial \u03b1 \u22a2 AEMeasurable (codRestrict f s hfs) ** inhabit \u03b1 ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 h : AEMeasurable f s : Set \u03b2 hfs : \u2200 (x : \u03b1), f x \u2208 s \u271d : Nontrivial \u03b1 inhabited_h : Inhabited \u03b1 \u22a2 AEMeasurable (codRestrict f s hfs) ** obtain \u27e8g, g_meas, hg, fg\u27e9 : \u2203 g : \u03b1 \u2192 \u03b2, Measurable g \u2227 range g \u2286 s \u2227 f =\u1d50[\u03bc] g :=\n h.exists_ae_eq_range_subset (eventually_of_forall hfs) \u27e8_, hfs default\u27e9 ** case intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g\u271d : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 h : AEMeasurable f s : Set \u03b2 hfs : \u2200 (x : \u03b1), f x \u2208 s \u271d : Nontrivial \u03b1 inhabited_h : Inhabited \u03b1 g : \u03b1 \u2192 \u03b2 g_meas : Measurable g hg : range g \u2286 s fg : f =\u1da0[ae \u03bc] g \u22a2 AEMeasurable (codRestrict f s hfs) ** refine' \u27e8codRestrict g s fun x => hg (mem_range_self _), Measurable.subtype_mk g_meas, _\u27e9 ** case intro.intro.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g\u271d : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 h : AEMeasurable f s : Set \u03b2 hfs : \u2200 (x : \u03b1), f x \u2208 s \u271d : Nontrivial \u03b1 inhabited_h : Inhabited \u03b1 g : \u03b1 \u2192 \u03b2 g_meas : Measurable g hg : range g \u2286 s fg : f =\u1da0[ae \u03bc] g \u22a2 codRestrict f s hfs =\u1da0[ae \u03bc] codRestrict g s (_ : \u2200 (x : \u03b1), g x \u2208 s) ** filter_upwards [fg] with x hx ** case h \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b4 : Type u_5 R : Type u_6 m0 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 f g\u271d : \u03b1 \u2192 \u03b2 \u03bc \u03bd : Measure \u03b1 h : AEMeasurable f s : Set \u03b2 hfs : \u2200 (x : \u03b1), f x \u2208 s \u271d : Nontrivial \u03b1 inhabited_h : Inhabited \u03b1 g : \u03b1 \u2192 \u03b2 g_meas : Measurable g hg : range g \u2286 s fg : f =\u1da0[ae \u03bc] g x : \u03b1 hx : f x = g x \u22a2 codRestrict f s hfs x = codRestrict g s (_ : \u2200 (x : \u03b1), g x \u2208 s) x ** simpa [Subtype.ext_iff] ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.inner_weightedVSub ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P \u03b9\u2081 : Type u_3 s\u2081 : Finset \u03b9\u2081 w\u2081 : \u03b9\u2081 \u2192 \u211d p\u2081 : \u03b9\u2081 \u2192 P h\u2081 : \u2211 i in s\u2081, w\u2081 i = 0 \u03b9\u2082 : Type u_4 s\u2082 : Finset \u03b9\u2082 w\u2082 : \u03b9\u2082 \u2192 \u211d p\u2082 : \u03b9\u2082 \u2192 P h\u2082 : \u2211 i in s\u2082, w\u2082 i = 0 \u22a2 inner (\u2191(Finset.weightedVSub s\u2081 p\u2081) w\u2081) (\u2191(Finset.weightedVSub s\u2082 p\u2082) w\u2082) = (-\u2211 i\u2081 in s\u2081, \u2211 i\u2082 in s\u2082, w\u2081 i\u2081 * w\u2082 i\u2082 * (dist (p\u2081 i\u2081) (p\u2082 i\u2082) * dist (p\u2081 i\u2081) (p\u2082 i\u2082))) / 2 ** rw [Finset.weightedVSub_apply, Finset.weightedVSub_apply,\n inner_sum_smul_sum_smul_of_sum_eq_zero _ h\u2081 _ h\u2082] ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P \u03b9\u2081 : Type u_3 s\u2081 : Finset \u03b9\u2081 w\u2081 : \u03b9\u2081 \u2192 \u211d p\u2081 : \u03b9\u2081 \u2192 P h\u2081 : \u2211 i in s\u2081, w\u2081 i = 0 \u03b9\u2082 : Type u_4 s\u2082 : Finset \u03b9\u2082 w\u2082 : \u03b9\u2082 \u2192 \u211d p\u2082 : \u03b9\u2082 \u2192 P h\u2082 : \u2211 i in s\u2082, w\u2082 i = 0 \u22a2 (-\u2211 i\u2081 in s\u2081, \u2211 i\u2082 in s\u2082, w\u2081 i\u2081 * w\u2082 i\u2082 * (\u2016p\u2081 i\u2081 -\u1d65 choice (_ : Nonempty P) - (p\u2082 i\u2082 -\u1d65 choice (_ : Nonempty P))\u2016 * \u2016p\u2081 i\u2081 -\u1d65 choice (_ : Nonempty P) - (p\u2082 i\u2082 -\u1d65 choice (_ : Nonempty P))\u2016)) / 2 = (-\u2211 i\u2081 in s\u2081, \u2211 i\u2082 in s\u2082, w\u2081 i\u2081 * w\u2082 i\u2082 * (dist (p\u2081 i\u2081) (p\u2082 i\u2082) * dist (p\u2081 i\u2081) (p\u2082 i\u2082))) / 2 ** simp_rw [vsub_sub_vsub_cancel_right] ** V : Type u_1 P : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P \u03b9\u2081 : Type u_3 s\u2081 : Finset \u03b9\u2081 w\u2081 : \u03b9\u2081 \u2192 \u211d p\u2081 : \u03b9\u2081 \u2192 P h\u2081 : \u2211 i in s\u2081, w\u2081 i = 0 \u03b9\u2082 : Type u_4 s\u2082 : Finset \u03b9\u2082 w\u2082 : \u03b9\u2082 \u2192 \u211d p\u2082 : \u03b9\u2082 \u2192 P h\u2082 : \u2211 i in s\u2082, w\u2082 i = 0 \u22a2 (-\u2211 x in s\u2081, \u2211 x_1 in s\u2082, w\u2081 x * w\u2082 x_1 * (\u2016p\u2081 x -\u1d65 p\u2082 x_1\u2016 * \u2016p\u2081 x -\u1d65 p\u2082 x_1\u2016)) / 2 = (-\u2211 i\u2081 in s\u2081, \u2211 i\u2082 in s\u2082, w\u2081 i\u2081 * w\u2082 i\u2082 * (dist (p\u2081 i\u2081) (p\u2082 i\u2082) * dist (p\u2081 i\u2081) (p\u2082 i\u2082))) / 2 ** rcongr (i\u2081 i\u2082) <;> rw [dist_eq_norm_vsub V (p\u2081 i\u2081) (p\u2082 i\u2082)] ** Qed", + "informal": "" + }, + { + "formal": "Finset.map_add_right_Icc ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b9 : ExistsAddOfLE \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 map (addRightEmbedding c) (Icc a b) = Icc (a + c) (b + c) ** rw [\u2190 coe_inj, coe_map, coe_Icc, coe_Icc] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b9 : ExistsAddOfLE \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 \u2191(addRightEmbedding c) '' Set.Icc a b = Set.Icc (a + c) (b + c) ** exact Set.image_add_const_Icc _ _ _ ** Qed", + "informal": "" + }, + { + "formal": "mul_eq_one_iff_inv_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 G : Type u_3 inst\u271d : Group G a b c d : G \u22a2 a * b = 1 \u2194 a\u207b\u00b9 = b ** rw [mul_eq_one_iff_eq_inv, inv_eq_iff_eq_inv] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_mul_mirror ** R : Type u_1 inst\u271d : Semiring R p q : R[X] \u22a2 coeff (p * mirror p) (natDegree p + natTrailingDegree p) = sum p fun n x => x ^ 2 ** rw [coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk] ** R : Type u_1 inst\u271d : Semiring R p q : R[X] n : \u2115 hn : n \u2208 Finset.range (Nat.succ (natDegree p + natTrailingDegree p)) \u22a2 coeff p (n, natDegree p + natTrailingDegree p - n).1 * coeff (mirror p) (n, natDegree p + natTrailingDegree p - n).2 = coeff p n ^ 2 ** rw [coeff_mirror, \u2190 revAt_le (Finset.mem_range_succ_iff.mp hn), revAt_invol, \u2190 sq] ** Qed", + "informal": "" + }, + { + "formal": "continuousOn_clm_apply ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X \u22a2 ContinuousOn f s \u2194 \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s ** refine' \u27e8fun h y => (ContinuousLinearMap.apply \ud835\udd5c F y).continuous.comp_continuousOn h, fun h => _\u27e9 ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X h : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s \u22a2 ContinuousOn f s ** let d := finrank \ud835\udd5c E ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X h : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s d : \u2115 := finrank \ud835\udd5c E \u22a2 ContinuousOn f s ** have hd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c) := (finrank_fin_fun \ud835\udd5c).symm ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X h : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s d : \u2115 := finrank \ud835\udd5c E hd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c) \u22a2 ContinuousOn f s ** let e\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X h : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s d : \u2115 := finrank \ud835\udd5c E hd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c) e\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd \u22a2 ContinuousOn f s ** let e\u2082 : (E \u2192L[\ud835\udd5c] F) \u2243L[\ud835\udd5c] Fin d \u2192 F :=\n (e\u2081.arrowCongr (1 : F \u2243L[\ud835\udd5c] F)).trans (ContinuousLinearEquiv.piRing (Fin d)) ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X h : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s d : \u2115 := finrank \ud835\udd5c E hd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c) e\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd e\u2082 : (E \u2192L[\ud835\udd5c] F) \u2243L[\ud835\udd5c] Fin d \u2192 F := ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e\u2081 1) (ContinuousLinearEquiv.piRing (Fin d)) \u22a2 ContinuousOn f s ** rw [\u2190 Function.comp.left_id f, \u2190 e\u2082.symm_comp_self] ** \ud835\udd5c : Type u inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type v inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type w inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F F' : Type x inst\u271d\u2077 : AddCommGroup F' inst\u271d\u2076 : Module \ud835\udd5c F' inst\u271d\u2075 : TopologicalSpace F' inst\u271d\u2074 : TopologicalAddGroup F' inst\u271d\u00b3 : ContinuousSMul \ud835\udd5c F' inst\u271d\u00b2 : CompleteSpace \ud835\udd5c X : Type u_1 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FiniteDimensional \ud835\udd5c E f : X \u2192 E \u2192L[\ud835\udd5c] F s : Set X h : \u2200 (y : E), ContinuousOn (fun x => \u2191(f x) y) s d : \u2115 := finrank \ud835\udd5c E hd : d = finrank \ud835\udd5c (Fin d \u2192 \ud835\udd5c) e\u2081 : E \u2243L[\ud835\udd5c] Fin d \u2192 \ud835\udd5c := ContinuousLinearEquiv.ofFinrankEq hd e\u2082 : (E \u2192L[\ud835\udd5c] F) \u2243L[\ud835\udd5c] Fin d \u2192 F := ContinuousLinearEquiv.trans (ContinuousLinearEquiv.arrowCongr e\u2081 1) (ContinuousLinearEquiv.piRing (Fin d)) \u22a2 ContinuousOn ((\u2191(ContinuousLinearEquiv.symm e\u2082) \u2218 \u2191e\u2082) \u2218 f) s ** exact e\u2082.symm.continuous.comp_continuousOn (continuousOn_pi.mpr fun i => h _) ** Qed", + "informal": "" + }, + { + "formal": "OrderIso.map_sSup ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d\u00b9 : CompleteLattice \u03b1 f\u271d g s\u271d t : \u03b9 \u2192 \u03b1 a b : \u03b1 inst\u271d : CompleteLattice \u03b2 f : \u03b1 \u2243o \u03b2 s : Set \u03b1 \u22a2 \u2191f (sSup s) = \u2a06 a \u2208 s, \u2191f a ** simp only [sSup_eq_iSup, OrderIso.map_iSup] ** Qed", + "informal": "" + }, + { + "formal": "Option.join_eq_some ** \u03b1\u271d : Type u_1 a : \u03b1\u271d x : Option (Option \u03b1\u271d) \u22a2 join x = some a \u2194 x = some (some a) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Filter.bliminf_false ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 inst\u271d : CompleteLattice \u03b1 f : Filter \u03b2 u : \u03b2 \u2192 \u03b1 \u22a2 (bliminf u f fun x => False) = \u22a4 ** simp [bliminf_eq] ** Qed", + "informal": "" + }, + { + "formal": "toIcoDiv_add_left' ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c : \u03b1 n : \u2124 a b : \u03b1 \u22a2 toIcoDiv hp (p + a) b = toIcoDiv hp a b - 1 ** rw [add_comm, toIcoDiv_add_right'] ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.BoundedFormula.relabel_ex ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b2 : Structure L M inst\u271d\u00b9 : Structure L N inst\u271d : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 g : \u03b1 \u2192 \u03b2 \u2295 Fin n k : \u2115 \u03c6 : BoundedFormula L \u03b1 (k + 1) \u22a2 relabel g (BoundedFormula.ex \u03c6) = BoundedFormula.ex (relabel g \u03c6) ** simp [BoundedFormula.ex] ** Qed", + "informal": "" + }, + { + "formal": "InnerProductGeometry.angle_eq_pi_iff ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V \u22a2 angle x y = \u03c0 \u2194 x \u2260 0 \u2227 \u2203 r, r < 0 \u2227 y = r \u2022 x ** rw [angle, \u2190 real_inner_div_norm_mul_norm_eq_neg_one_iff, Real.arccos_eq_pi, LE.le.le_iff_eq] ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x\u271d y\u271d x y : V \u22a2 -1 \u2264 inner x y / (\u2016x\u2016 * \u2016y\u2016) ** exact (abs_le.mp (abs_real_inner_div_norm_mul_norm_le_one x y)).1 ** Qed", + "informal": "" + }, + { + "formal": "StarConvex.sub ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedRing \ud835\udd5c inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y : E s t : Set E hs : StarConvex \ud835\udd5c x s ht : StarConvex \ud835\udd5c y t \u22a2 StarConvex \ud835\udd5c (x - y) (s - t) ** simp_rw [sub_eq_add_neg] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : OrderedRing \ud835\udd5c inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : AddCommGroup F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F x y : E s t : Set E hs : StarConvex \ud835\udd5c x s ht : StarConvex \ud835\udd5c y t \u22a2 StarConvex \ud835\udd5c (x + -y) (s + -t) ** exact hs.add ht.neg ** Qed", + "informal": "" + }, + { + "formal": "IsUpperSet.image ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03ba : \u03b9 \u2192 Sort u_5 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 s : Set \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 hs : IsUpperSet s f : \u03b1 \u2243o \u03b2 \u22a2 IsUpperSet (\u2191f '' s) ** change IsUpperSet ((f : \u03b1 \u2243 \u03b2) '' s) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03ba : \u03b9 \u2192 Sort u_5 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 s : Set \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 hs : IsUpperSet s f : \u03b1 \u2243o \u03b2 \u22a2 IsUpperSet (\u2191\u2191f '' s) ** rw [Set.image_equiv_eq_preimage_symm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03ba : \u03b9 \u2192 Sort u_5 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 s : Set \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 hs : IsUpperSet s f : \u03b1 \u2243o \u03b2 \u22a2 IsUpperSet (\u2191(\u2191f).symm \u207b\u00b9' s) ** exact hs.preimage f.symm.monotone ** Qed", + "informal": "" + }, + { + "formal": "PowerSeries.X_pow_dvd_iff ** R : Type u_1 inst\u271d : Semiring R n : \u2115 \u03c6 : R\u27e6X\u27e7 \u22a2 X ^ n \u2223 \u03c6 \u2194 \u2200 (m : \u2115), m < n \u2192 \u2191(coeff R m) \u03c6 = 0 ** convert@MvPowerSeries.X_pow_dvd_iff Unit R _ () n \u03c6 ** case h.e'_2.a R : Type u_1 inst\u271d : Semiring R n : \u2115 \u03c6 : R\u27e6X\u27e7 \u22a2 (\u2200 (m : \u2115), m < n \u2192 \u2191(coeff R m) \u03c6 = 0) \u2194 \u2200 (m : Unit \u2192\u2080 \u2115), \u2191m () < n \u2192 \u2191(MvPowerSeries.coeff R m) \u03c6 = 0 ** constructor <;> intro h m hm ** case h.e'_2.a.mp R : Type u_1 inst\u271d : Semiring R n : \u2115 \u03c6 : R\u27e6X\u27e7 h : \u2200 (m : \u2115), m < n \u2192 \u2191(coeff R m) \u03c6 = 0 m : Unit \u2192\u2080 \u2115 hm : \u2191m () < n \u22a2 \u2191(MvPowerSeries.coeff R m) \u03c6 = 0 ** rw [Finsupp.unique_single m] ** case h.e'_2.a.mp R : Type u_1 inst\u271d : Semiring R n : \u2115 \u03c6 : R\u27e6X\u27e7 h : \u2200 (m : \u2115), m < n \u2192 \u2191(coeff R m) \u03c6 = 0 m : Unit \u2192\u2080 \u2115 hm : \u2191m () < n \u22a2 \u2191(MvPowerSeries.coeff R fun\u2080 | default => \u2191m default) \u03c6 = 0 ** convert h _ hm ** case h.e'_2.a.mpr R : Type u_1 inst\u271d : Semiring R n : \u2115 \u03c6 : R\u27e6X\u27e7 h : \u2200 (m : Unit \u2192\u2080 \u2115), \u2191m () < n \u2192 \u2191(MvPowerSeries.coeff R m) \u03c6 = 0 m : \u2115 hm : m < n \u22a2 \u2191(coeff R m) \u03c6 = 0 ** apply h ** case h.e'_2.a.mpr.a R : Type u_1 inst\u271d : Semiring R n : \u2115 \u03c6 : R\u27e6X\u27e7 h : \u2200 (m : Unit \u2192\u2080 \u2115), \u2191m () < n \u2192 \u2191(MvPowerSeries.coeff R m) \u03c6 = 0 m : \u2115 hm : m < n \u22a2 (\u2191fun\u2080 | () => m) () < n ** simpa only [Finsupp.single_eq_same] using hm ** Qed", + "informal": "" + }, + { + "formal": "Nat.minSqFac_dvd ** n d : \u2115 h : minSqFac n = some d \u22a2 d * d \u2223 n ** have := minSqFac_has_prop n ** n d : \u2115 h : minSqFac n = some d this : MinSqFacProp n (minSqFac n) \u22a2 d * d \u2223 n ** rw [h] at this ** n d : \u2115 h : minSqFac n = some d this : MinSqFacProp n (some d) \u22a2 d * d \u2223 n ** exact this.2.1 ** Qed", + "informal": "" + }, + { + "formal": "contDiffOn_succ_iff_fderiv_of_open ** \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n\u271d : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F n : \u2115 hs : IsOpen s \u22a2 ContDiffOn \ud835\udd5c (\u2191(n + 1)) f s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c f y) s ** rw [contDiffOn_succ_iff_fderivWithin hs.uniqueDiffOn] ** \ud835\udd5c : Type u inst\u271d\u2078 : NontriviallyNormedField \ud835\udd5c E : Type uE inst\u271d\u2077 : NormedAddCommGroup E inst\u271d\u2076 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G X : Type uX inst\u271d\u00b9 : NormedAddCommGroup X inst\u271d : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f f\u2081 : E \u2192 F g : F \u2192 G x x\u2080 : E c : F m n\u271d : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F n : \u2115 hs : IsOpen s \u22a2 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderivWithin \ud835\udd5c f s y) s \u2194 DifferentiableOn \ud835\udd5c f s \u2227 ContDiffOn \ud835\udd5c (\u2191n) (fun y => fderiv \ud835\udd5c f y) s ** exact Iff.rfl.and (contDiffOn_congr fun x hx \u21a6 fderivWithin_of_open hs hx) ** Qed", + "informal": "" + }, + { + "formal": "QPF.Fix.rec_unique ** F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 h : Fix F \u2192 \u03b1 hyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x) \u22a2 rec g = h ** ext x ** case h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 h : Fix F \u2192 \u03b1 hyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x) x : Fix F \u22a2 rec g x = h x ** apply Fix.ind_rec ** case h.h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 h : Fix F \u2192 \u03b1 hyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x) x : Fix F \u22a2 \u2200 (x : F (Fix F)), rec g <$> x = (fun x => h x) <$> x \u2192 rec g (mk x) = h (mk x) ** intro x hyp' ** case h.h F : Type u \u2192 Type u inst\u271d : Functor F q : QPF F \u03b1 : Type u g : F \u03b1 \u2192 \u03b1 h : Fix F \u2192 \u03b1 hyp : \u2200 (x : F (Fix F)), h (mk x) = g (h <$> x) x\u271d : Fix F x : F (Fix F) hyp' : rec g <$> x = (fun x => h x) <$> x \u22a2 rec g (mk x) = h (mk x) ** rw [hyp, \u2190 hyp', Fix.rec_eq] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.ofReal_coe_nat ** \u03b1 : Type u_1 \u03b2 : Type u_2 a b c d : \u211d\u22650\u221e r p q : \u211d\u22650 n : \u2115 \u22a2 ENNReal.ofReal \u2191n = \u2191n ** simp [ENNReal.ofReal] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.Types.binaryCofan_isColimit_iff ** X Y : Type u c : BinaryCofan X Y \u22a2 Nonempty (IsColimit c) \u2194 Injective (BinaryCofan.inl c) \u2227 Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) ** constructor ** case mp X Y : Type u c : BinaryCofan X Y \u22a2 Nonempty (IsColimit c) \u2192 Injective (BinaryCofan.inl c) \u2227 Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) ** rintro \u27e8h\u27e9 ** case mp.intro X Y : Type u c : BinaryCofan X Y h : IsColimit c \u22a2 Injective (BinaryCofan.inl c) \u2227 Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) ** rw [\u2190 show _ = c.inl from\n h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) \u27e8WalkingPair.left\u27e9,\n \u2190 show _ = c.inr from\n h.comp_coconePointUniqueUpToIso_inv (binaryCoproductColimit X Y) \u27e8WalkingPair.right\u27e9] ** case mp.intro X Y : Type u c : BinaryCofan X Y h : IsColimit c \u22a2 Injective ((binaryCoproductCocone X Y).\u03b9.app { as := left } \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227 Injective ((binaryCoproductCocone X Y).\u03b9.app { as := right } \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227 IsCompl (Set.range ((binaryCoproductCocone X Y).\u03b9.app { as := left } \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) (Set.range ((binaryCoproductCocone X Y).\u03b9.app { as := right } \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) ** dsimp [binaryCoproductCocone] ** case mp.intro X Y : Type u c : BinaryCofan X Y h : IsColimit c \u22a2 Injective (Sum.inl \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227 Injective (Sum.inr \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv) \u2227 IsCompl (Set.range (Sum.inl \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) (Set.range (Sum.inr \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) ** refine'\n \u27e8(h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp\n Sum.inl_injective,\n (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.injective.comp\n Sum.inr_injective, _\u27e9 ** case mp.intro X Y : Type u c : BinaryCofan X Y h : IsColimit c \u22a2 IsCompl (Set.range (Sum.inl \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) (Set.range (Sum.inr \u226b (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv)) ** erw [Set.range_comp, \u2190 eq_compl_iff_isCompl, Set.range_comp _ Sum.inr, \u2190\n Set.image_compl_eq\n (h.coconePointUniqueUpToIso (binaryCoproductColimit X Y)).symm.toEquiv.bijective] ** case mp.intro X Y : Type u c : BinaryCofan X Y h : IsColimit c \u22a2 (IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).inv '' Set.range Sum.inl = \u2191(IsColimit.coconePointUniqueUpToIso h (binaryCoproductColimit X Y)).symm.toEquiv '' (Set.range Sum.inr)\u1d9c ** simp ** case mpr X Y : Type u c : BinaryCofan X Y \u22a2 Injective (BinaryCofan.inl c) \u2227 Injective (BinaryCofan.inr c) \u2227 IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) \u2192 Nonempty (IsColimit c) ** rintro \u27e8h\u2081, h\u2082, h\u2083\u27e9 ** case mpr.intro.intro X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) \u22a2 Nonempty (IsColimit c) ** have : \u2200 x, x \u2208 Set.range c.inl \u2228 x \u2208 Set.range c.inr := by\n rw [eq_compl_iff_isCompl.mpr h\u2083.symm]\n exact fun _ => or_not ** case mpr.intro.intro X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) \u22a2 Nonempty (IsColimit c) ** refine' \u27e8BinaryCofan.IsColimit.mk _ _ _ _ _\u27e9 ** X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) \u22a2 \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) ** rw [eq_compl_iff_isCompl.mpr h\u2083.symm] ** X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) \u22a2 \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 (Set.range (BinaryCofan.inl c))\u1d9c ** exact fun _ => or_not ** case mpr.intro.intro.refine'_1 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) \u22a2 {T : Type u} \u2192 (X \u27f6 T) \u2192 (Y \u27f6 T) \u2192 (c.pt \u27f6 T) ** intro T f g x ** case mpr.intro.intro.refine'_1 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : c.pt \u22a2 T ** exact\n if h : x \u2208 Set.range c.inl then f ((Equiv.ofInjective _ h\u2081).symm \u27e8x, h\u27e9)\n else g ((Equiv.ofInjective _ h\u2082).symm \u27e8x, (this x).resolve_left h\u27e9) ** case mpr.intro.intro.refine'_2 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) \u22a2 \u2200 {T : Type u} (f : X \u27f6 T) (g : Y \u27f6 T), (BinaryCofan.inl c \u226b fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) = f ** intro T f g ** case mpr.intro.intro.refine'_2 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T \u22a2 (BinaryCofan.inl c \u226b fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) = f ** funext x ** case mpr.intro.intro.refine'_2.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := left } \u22a2 (BinaryCofan.inl c \u226b fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) x = f x ** dsimp ** case mpr.intro.intro.refine'_2.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := left } \u22a2 (if h : BinaryCofan.inl c x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inl c x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := BinaryCofan.inl c x, property := (_ : BinaryCofan.inl c x \u2208 Set.range (BinaryCofan.inr c)) })) = f x ** simp [h\u2081.eq_iff] ** case mpr.intro.intro.refine'_3 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) \u22a2 \u2200 {T : Type u} (f : X \u27f6 T) (g : Y \u27f6 T), (BinaryCofan.inr c \u226b fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) = g ** intro T f g ** case mpr.intro.intro.refine'_3 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T \u22a2 (BinaryCofan.inr c \u226b fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) = g ** funext x ** case mpr.intro.intro.refine'_3.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := right } \u22a2 (BinaryCofan.inr c \u226b fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) x = g x ** dsimp ** case mpr.intro.intro.refine'_3.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := right } \u22a2 (if h : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inr c x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inr c)) })) = g x ** simp only [Set.mem_range, Equiv.ofInjective_symm_apply,\n dite_eq_right_iff, forall_exists_index] ** case mpr.intro.intro.refine'_3.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := right } \u22a2 \u2200 (x_1 : X) (h : BinaryCofan.inl c x_1 = BinaryCofan.inr c x), f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) = g x ** intro y e ** case mpr.intro.intro.refine'_3.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := right } y : X e : BinaryCofan.inl c y = BinaryCofan.inr c x \u22a2 f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) = g x ** have : c.inr x \u2208 Set.range c.inl \u2293 Set.range c.inr := \u27e8\u27e8_, e\u27e9, \u27e8_, rfl\u27e9\u27e9 ** case mpr.intro.intro.refine'_3.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this\u271d : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := right } y : X e : BinaryCofan.inl c y = BinaryCofan.inr c x this : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c) \u2293 Set.range (BinaryCofan.inr c) \u22a2 f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) = g x ** rw [disjoint_iff.mp h\u2083.1] at this ** case mpr.intro.intro.refine'_3.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this\u271d : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u f : X \u27f6 T g : Y \u27f6 T x : (pair X Y).obj { as := right } y : X e : BinaryCofan.inl c y = BinaryCofan.inr c x this : BinaryCofan.inr c x \u2208 \u22a5 \u22a2 f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := BinaryCofan.inr c x, property := (_ : BinaryCofan.inr c x \u2208 Set.range (BinaryCofan.inl c)) }) = g x ** exact this.elim ** case mpr.intro.intro.refine'_4 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) \u22a2 \u2200 {T : Type u} (f : X \u27f6 T) (g : Y \u27f6 T) (m : c.pt \u27f6 T), BinaryCofan.inl c \u226b m = f \u2192 BinaryCofan.inr c \u226b m = g \u2192 m = fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then f (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else g (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }) ** rintro T _ _ m rfl rfl ** case mpr.intro.intro.refine'_4 X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u m : c.pt \u27f6 T \u22a2 m = fun x => if h : x \u2208 Set.range (BinaryCofan.inl c) then (BinaryCofan.inl c \u226b m) (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else (BinaryCofan.inr c \u226b m) (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }) ** funext x ** case mpr.intro.intro.refine'_4.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u m : c.pt \u27f6 T x : c.pt \u22a2 m x = if h : x \u2208 Set.range (BinaryCofan.inl c) then (BinaryCofan.inl c \u226b m) (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h }) else (BinaryCofan.inr c \u226b m) (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) }) ** dsimp ** case mpr.intro.intro.refine'_4.h X Y : Type u c : BinaryCofan X Y h\u2081 : Injective (BinaryCofan.inl c) h\u2082 : Injective (BinaryCofan.inr c) h\u2083 : IsCompl (Set.range (BinaryCofan.inl c)) (Set.range (BinaryCofan.inr c)) this : \u2200 (x : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left }), x \u2208 Set.range (BinaryCofan.inl c) \u2228 x \u2208 Set.range (BinaryCofan.inr c) T : Type u m : c.pt \u27f6 T x : c.pt \u22a2 m x = if h : x \u2208 Set.range (BinaryCofan.inl c) then m (BinaryCofan.inl c (\u2191(Equiv.ofInjective (BinaryCofan.inl c) h\u2081).symm { val := x, property := h })) else m (BinaryCofan.inr c (\u2191(Equiv.ofInjective (BinaryCofan.inr c) h\u2082).symm { val := x, property := (_ : x \u2208 Set.range (BinaryCofan.inr c)) })) ** split_ifs <;> exact congr_arg _ (Equiv.apply_ofInjective_symm _ \u27e8_, _\u27e9).symm ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.frange_single ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : Zero M x : \u03b1 y r : M hr : r \u2208 frange fun\u2080 | x => y t : r \u2260 0 ht1 : \u03b1 ht2 : (\u2191fun\u2080 | x => y) ht1 = r \u22a2 (\u2191fun\u2080 | x => y) ht1 \u2208 {y} ** rw [single_apply] at ht2 \u22a2 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : Zero M x : \u03b1 y r : M hr : r \u2208 frange fun\u2080 | x => y t : r \u2260 0 ht1 : \u03b1 ht2\u271d : (\u2191fun\u2080 | x => y) ht1 = r ht2 : (if x = ht1 then y else 0) = r \u22a2 (if x = ht1 then y else 0) \u2208 {y} ** split_ifs at ht2 \u22a2 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : Zero M x : \u03b1 y r : M hr : r \u2208 frange fun\u2080 | x => y t : r \u2260 0 ht1 : \u03b1 ht2\u271d : (\u2191fun\u2080 | x => y) ht1 = r h\u271d : x = ht1 ht2 : y = r \u22a2 y \u2208 {y} ** exact Finset.mem_singleton_self _ ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d : Zero M x : \u03b1 y r : M hr : r \u2208 frange fun\u2080 | x => y t : r \u2260 0 ht1 : \u03b1 ht2\u271d : (\u2191fun\u2080 | x => y) ht1 = r h\u271d : \u00acx = ht1 ht2 : 0 = r \u22a2 0 \u2208 {y} ** exact (t ht2.symm).elim ** Qed", + "informal": "" + }, + { + "formal": "Finset.centerMass_ite_eq ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i \u2208 t \u22a2 centerMass t (fun j => if i = j then 1 else 0) z = z i ** rw [Finset.centerMass_eq_of_sum_1] ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i \u2208 t \u22a2 \u2211 i_1 in t, (if i = i_1 then 1 else 0) \u2022 z i_1 = z i case hw R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i \u2208 t \u22a2 (\u2211 i_1 in t, if i = i_1 then 1 else 0) = 1 ** trans \u2211 j in t, if i = j then z i else 0 ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i \u2208 t \u22a2 \u2211 i_1 in t, (if i = i_1 then 1 else 0) \u2022 z i_1 = \u2211 j in t, if i = j then z i else 0 ** congr with i ** case e_f.h R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i\u271d j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i\u271d \u2208 t i : \u03b9 \u22a2 (if i\u271d = i then 1 else 0) \u2022 z i = if i\u271d = i then z i\u271d else 0 ** split_ifs with h ** case pos R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i\u271d j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i\u271d \u2208 t i : \u03b9 h : i\u271d = i \u22a2 1 \u2022 z i = z i\u271d case neg R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i\u271d j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i\u271d \u2208 t i : \u03b9 h : \u00aci\u271d = i \u22a2 0 \u2022 z i = 0 ** exacts [h \u25b8 one_smul _ _, zero_smul _ _] ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i \u2208 t \u22a2 (\u2211 j in t, if i = j then z i else 0) = z i ** rw [sum_ite_eq, if_pos hi] ** case hw R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E hi : i \u2208 t \u22a2 (\u2211 i_1 in t, if i = i_1 then 1 else 0) = 1 ** rw [sum_ite_eq, if_pos hi] ** Qed", + "informal": "" + }, + { + "formal": "AlgHom.ofLinearMap_toLinearMap ** R : Type u A : Type v B : Type w C : Type u\u2081 D : Type v\u2081 inst\u271d\u2078 : CommSemiring R inst\u271d\u2077 : Semiring A inst\u271d\u2076 : Semiring B inst\u271d\u2075 : Semiring C inst\u271d\u2074 : Semiring D inst\u271d\u00b3 : Algebra R A inst\u271d\u00b2 : Algebra R B inst\u271d\u00b9 : Algebra R C inst\u271d : Algebra R D \u03c6 : A \u2192\u2090[R] B map_one : \u2191(toLinearMap \u03c6) 1 = 1 map_mul : \u2200 (x y : A), \u2191(toLinearMap \u03c6) (x * y) = \u2191(toLinearMap \u03c6) x * \u2191(toLinearMap \u03c6) y \u22a2 ofLinearMap (toLinearMap \u03c6) map_one map_mul = \u03c6 ** ext ** case H R : Type u A : Type v B : Type w C : Type u\u2081 D : Type v\u2081 inst\u271d\u2078 : CommSemiring R inst\u271d\u2077 : Semiring A inst\u271d\u2076 : Semiring B inst\u271d\u2075 : Semiring C inst\u271d\u2074 : Semiring D inst\u271d\u00b3 : Algebra R A inst\u271d\u00b2 : Algebra R B inst\u271d\u00b9 : Algebra R C inst\u271d : Algebra R D \u03c6 : A \u2192\u2090[R] B map_one : \u2191(toLinearMap \u03c6) 1 = 1 map_mul : \u2200 (x y : A), \u2191(toLinearMap \u03c6) (x * y) = \u2191(toLinearMap \u03c6) x * \u2191(toLinearMap \u03c6) y x\u271d : A \u22a2 \u2191(ofLinearMap (toLinearMap \u03c6) map_one map_mul) x\u271d = \u2191\u03c6 x\u271d ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Matrix.Fin.transpose_circulant ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : Type u_3 n : Type u_4 R : Type u_5 \u22a2 \u2200 (v : Fin 0 \u2192 \u03b1), (circulant v)\u1d40 = circulant fun i => v (-i) ** simp [Injective] ** Qed", + "informal": "" + }, + { + "formal": "Monotone.rightLim_le_leftLim ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y \u22a2 rightLim f x \u2264 leftLim f y ** letI : TopologicalSpace \u03b1 := Preorder.topology \u03b1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y this : TopologicalSpace \u03b1 := Preorder.topology \u03b1 \u22a2 rightLim f x \u2264 leftLim f y ** haveI : OrderTopology \u03b1 := \u27e8rfl\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y this\u271d : TopologicalSpace \u03b1 := Preorder.topology \u03b1 this : OrderTopology \u03b1 \u22a2 rightLim f x \u2264 leftLim f y ** rcases eq_or_ne (\ud835\udcdd[<] y) \u22a5 with (h' | h') ** case inr \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y this\u271d : TopologicalSpace \u03b1 := Preorder.topology \u03b1 this : OrderTopology \u03b1 h' : \ud835\udcdd[Iio y] y \u2260 \u22a5 \u22a2 rightLim f x \u2264 leftLim f y ** obtain \u27e8a, \u27e8xa, ay\u27e9\u27e9 : (Ioo x y).Nonempty :=\n forall_mem_nonempty_iff_neBot.2 (neBot_iff.2 h') (Ioo x y)\n (Ioo_mem_nhdsWithin_Iio \u27e8h, le_refl _\u27e9) ** case inr.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y this\u271d : TopologicalSpace \u03b1 := Preorder.topology \u03b1 this : OrderTopology \u03b1 h' : \ud835\udcdd[Iio y] y \u2260 \u22a5 a : \u03b1 xa : x < a ay : a < y \u22a2 rightLim f x \u2264 leftLim f y ** calc\n rightLim f x \u2264 f a := hf.rightLim_le xa\n _ \u2264 leftLim f y := hf.le_leftLim ay ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y this\u271d : TopologicalSpace \u03b1 := Preorder.topology \u03b1 this : OrderTopology \u03b1 h' : \ud835\udcdd[Iio y] y = \u22a5 \u22a2 rightLim f x \u2264 leftLim f y ** simp [leftLim, h'] ** case inl \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b3 : LinearOrder \u03b1 inst\u271d\u00b2 : ConditionallyCompleteLinearOrder \u03b2 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : OrderTopology \u03b2 f : \u03b1 \u2192 \u03b2 hf : Monotone f x y : \u03b1 h : x < y this\u271d : TopologicalSpace \u03b1 := Preorder.topology \u03b1 this : OrderTopology \u03b1 h' : \ud835\udcdd[Iio y] y = \u22a5 \u22a2 rightLim f x \u2264 f y ** exact rightLim_le hf h ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.ne_zero_of_trailingDegree_lt ** R : Type u S : Type v a b : R n\u271d m : \u2115 inst\u271d : Semiring R p q : R[X] \u03b9 : Type u_1 n : \u2115\u221e h : trailingDegree p < n h\u2080 : p = 0 \u22a2 n \u2264 trailingDegree p ** simp [h\u2080] ** Qed", + "informal": "" + }, + { + "formal": "rank_fun ** K : Type u V\u271d V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7\u271d : Type u\u2081' \u03c6 : \u03b7\u271d \u2192 Type u_1 inst\u271d\u00b9\u2078 : Ring K inst\u271d\u00b9\u2077 : StrongRankCondition K inst\u271d\u00b9\u2076 : AddCommGroup V\u271d inst\u271d\u00b9\u2075 : Module K V\u271d inst\u271d\u00b9\u2074 : Module.Free K V\u271d inst\u271d\u00b9\u00b3 : AddCommGroup V' inst\u271d\u00b9\u00b2 : Module K V' inst\u271d\u00b9\u00b9 : Module.Free K V' inst\u271d\u00b9\u2070 : AddCommGroup V\u2081 inst\u271d\u2079 : Module K V\u2081 inst\u271d\u2078 : Module.Free K V\u2081 inst\u271d\u2077 : (i : \u03b7\u271d) \u2192 AddCommGroup (\u03c6 i) inst\u271d\u2076 : (i : \u03b7\u271d) \u2192 Module K (\u03c6 i) inst\u271d\u2075 : \u2200 (i : \u03b7\u271d), Module.Free K (\u03c6 i) inst\u271d\u2074 : Fintype \u03b7\u271d V \u03b7 : Type u inst\u271d\u00b3 : Fintype \u03b7 inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V inst\u271d : Module.Free K V \u22a2 Module.rank K (\u03b7 \u2192 V) = \u2191(Fintype.card \u03b7) * Module.rank K V ** rw [rank_pi, Cardinal.sum_const', Cardinal.mk_fintype] ** Qed", + "informal": "" + }, + { + "formal": "inf_sdiff_right_comm ** \u03b1 : Type u \u03b2 : Type u_1 w x y z : \u03b1 inst\u271d : GeneralizedBooleanAlgebra \u03b1 \u22a2 x \\ z \u2293 y = (x \u2293 y) \\ z ** rw [@inf_comm _ _ x, inf_comm, inf_sdiff_assoc] ** Qed", + "informal": "" + }, + { + "formal": "IsROrC.add_conj ** K : Type u_1 E : Type u_2 inst\u271d : IsROrC K z : K \u22a2 z + \u2191(starRingEnd K) z = \u2191(\u2191re z) + \u2191(\u2191im z) * I + (\u2191(\u2191re z) - \u2191(\u2191im z) * I) ** rw [re_add_im, conj_eq_re_sub_im] ** K : Type u_1 E : Type u_2 inst\u271d : IsROrC K z : K \u22a2 \u2191(\u2191re z) + \u2191(\u2191im z) * I + (\u2191(\u2191re z) - \u2191(\u2191im z) * I) = 2 * \u2191(\u2191re z) ** rw [add_add_sub_cancel, two_mul] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.toLaurent_C_mul_X_pow ** R : Type u_1 inst\u271d : Semiring R n : \u2115 r : R \u22a2 \u2191toLaurent (\u2191Polynomial.C r * X ^ n) = \u2191C r * T \u2191n ** simp only [_root_.map_mul, Polynomial.toLaurent_C, Polynomial.toLaurent_X_pow] ** Qed", + "informal": "" + }, + { + "formal": "Real.Angle.cos_eq_iff_coe_eq_or_eq_neg ** \u03b8 \u03c8 : \u211d \u22a2 cos \u03b8 = cos \u03c8 \u2194 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 ** constructor ** case mp \u03b8 \u03c8 : \u211d \u22a2 cos \u03b8 = cos \u03c8 \u2192 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 ** intro Hcos ** case mp \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 \u22a2 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 ** rw [\u2190 sub_eq_zero, cos_sub_cos, mul_eq_zero, mul_eq_zero, neg_eq_zero,\n eq_false (two_ne_zero' \u211d), false_or_iff, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos ** case mp \u03b8 \u03c8 : \u211d Hcos\u271d : cos \u03b8 = cos \u03c8 Hcos : (\u2203 n, \u2191n * \u03c0 = (\u03b8 + \u03c8) / 2) \u2228 \u2203 n, \u2191n * \u03c0 = (\u03b8 - \u03c8) / 2 \u22a2 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 ** rcases Hcos with (\u27e8n, hn\u27e9 | \u27e8n, hn\u27e9) ** case mp.inl.intro \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 n : \u2124 hn : \u2191n * \u03c0 = (\u03b8 + \u03c8) / 2 \u22a2 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 ** right ** case mp.inl.intro.h \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 n : \u2124 hn : \u2191n * \u03c0 = (\u03b8 + \u03c8) / 2 \u22a2 \u2191\u03b8 = -\u2191\u03c8 ** rw [eq_div_iff_mul_eq (two_ne_zero' \u211d), \u2190 sub_eq_iff_eq_add] at hn ** case mp.inl.intro.h \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 n : \u2124 hn : \u2191n * \u03c0 * 2 - \u03c8 = \u03b8 \u22a2 \u2191\u03b8 = -\u2191\u03c8 ** rw [\u2190 hn, coe_sub, eq_neg_iff_add_eq_zero, sub_add_cancel, mul_assoc, coe_int_mul_eq_zsmul,\n mul_comm, coe_two_pi, zsmul_zero] ** case mp.inr.intro \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 n : \u2124 hn : \u2191n * \u03c0 = (\u03b8 - \u03c8) / 2 \u22a2 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 ** left ** case mp.inr.intro.h \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 n : \u2124 hn : \u2191n * \u03c0 = (\u03b8 - \u03c8) / 2 \u22a2 \u2191\u03b8 = \u2191\u03c8 ** rw [eq_div_iff_mul_eq (two_ne_zero' \u211d), eq_sub_iff_add_eq] at hn ** case mp.inr.intro.h \u03b8 \u03c8 : \u211d Hcos : cos \u03b8 = cos \u03c8 n : \u2124 hn : \u2191n * \u03c0 * 2 + \u03c8 = \u03b8 \u22a2 \u2191\u03b8 = \u2191\u03c8 ** rw [\u2190 hn, coe_add, mul_assoc, coe_int_mul_eq_zsmul, mul_comm, coe_two_pi, zsmul_zero,\n zero_add] ** case mpr \u03b8 \u03c8 : \u211d \u22a2 \u2191\u03b8 = \u2191\u03c8 \u2228 \u2191\u03b8 = -\u2191\u03c8 \u2192 cos \u03b8 = cos \u03c8 ** rw [angle_eq_iff_two_pi_dvd_sub, \u2190 coe_neg, angle_eq_iff_two_pi_dvd_sub] ** case mpr \u03b8 \u03c8 : \u211d \u22a2 ((\u2203 k, \u03b8 - \u03c8 = 2 * \u03c0 * \u2191k) \u2228 \u2203 k, \u03b8 - -\u03c8 = 2 * \u03c0 * \u2191k) \u2192 cos \u03b8 = cos \u03c8 ** rintro (\u27e8k, H\u27e9 | \u27e8k, H\u27e9) ** case mpr.inl.intro \u03b8 \u03c8 : \u211d k : \u2124 H : \u03b8 - \u03c8 = 2 * \u03c0 * \u2191k \u22a2 cos \u03b8 = cos \u03c8 case mpr.inr.intro \u03b8 \u03c8 : \u211d k : \u2124 H : \u03b8 - -\u03c8 = 2 * \u03c0 * \u2191k \u22a2 cos \u03b8 = cos \u03c8 ** rw [\u2190 sub_eq_zero, cos_sub_cos, H, mul_assoc 2 \u03c0 k, mul_div_cancel_left _ (two_ne_zero' \u211d),\n mul_comm \u03c0 _, sin_int_mul_pi, mul_zero] ** case mpr.inr.intro \u03b8 \u03c8 : \u211d k : \u2124 H : \u03b8 - -\u03c8 = 2 * \u03c0 * \u2191k \u22a2 cos \u03b8 = cos \u03c8 ** rw [\u2190 sub_eq_zero, cos_sub_cos, \u2190 sub_neg_eq_add, H, mul_assoc 2 \u03c0 k,\n mul_div_cancel_left _ (two_ne_zero' \u211d), mul_comm \u03c0 _, sin_int_mul_pi, mul_zero,\n zero_mul] ** Qed", + "informal": "" + }, + { + "formal": "linearIndependent_iff_card_eq_finrank_span ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V \u22a2 LinearIndependent K b \u2194 Fintype.card \u03b9 = Set.finrank K (Set.range b) ** constructor ** case mp K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V \u22a2 LinearIndependent K b \u2192 Fintype.card \u03b9 = Set.finrank K (Set.range b) ** intro h ** case mp K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V h : LinearIndependent K b \u22a2 Fintype.card \u03b9 = Set.finrank K (Set.range b) ** exact (finrank_span_eq_card h).symm ** case mpr K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V \u22a2 Fintype.card \u03b9 = Set.finrank K (Set.range b) \u2192 LinearIndependent K b ** intro hc ** case mpr K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) \u22a2 LinearIndependent K b ** let f := Submodule.subtype (span K (Set.range b)) ** case mpr K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) \u22a2 LinearIndependent K b ** let b' : \u03b9 \u2192 span K (Set.range b) := fun i =>\n \u27e8b i, mem_span.2 fun p hp => hp (Set.mem_range_self _)\u27e9 ** case mpr K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } \u22a2 LinearIndependent K b ** have hs : \u22a4 \u2264 span K (Set.range b') := by\n intro x\n have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f\n have hf : f '' Set.range b' = Set.range b := by\n ext x\n simp [Set.mem_image, Set.mem_range]\n rw [hf] at h\n have hx : (x : V) \u2208 span K (Set.range b) := x.property\n conv at hx =>\n arg 2\n rw [h]\n simpa [mem_map] using hx ** case mpr K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } hs : \u22a4 \u2264 span K (Set.range b') \u22a2 LinearIndependent K b ** have hi : LinearMap.ker f = \u22a5 := ker_subtype _ ** case mpr K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } hs : \u22a4 \u2264 span K (Set.range b') hi : LinearMap.ker f = \u22a5 \u22a2 LinearIndependent K b ** convert (linearIndependent_of_top_le_span_of_card_eq_finrank hs hc).map' _ hi ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } \u22a2 \u22a4 \u2264 span K (Set.range b') ** intro x ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } \u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b') ** have h : span K (f '' Set.range b') = map f (span K (Set.range b')) := span_image f ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } h : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b')) \u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b') ** have hf : f '' Set.range b' = Set.range b := by\n ext x\n simp [Set.mem_image, Set.mem_range] ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } h : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b')) hf : \u2191f '' Set.range b' = Set.range b \u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b') ** rw [hf] at h ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } h : span K (Set.range b) = Submodule.map f (span K (Set.range b')) hf : \u2191f '' Set.range b' = Set.range b \u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b') ** have hx : (x : V) \u2208 span K (Set.range b) := x.property ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } h : span K (Set.range b) = Submodule.map f (span K (Set.range b')) hf : \u2191f '' Set.range b' = Set.range b hx : \u2191x \u2208 span K (Set.range b) \u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b') ** conv at hx =>\n arg 2\n rw [h] ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } h : span K (Set.range b) = Submodule.map f (span K (Set.range b')) hf : \u2191f '' Set.range b' = Set.range b hx : \u2191x \u2208 Submodule.map f (span K (Set.range b')) \u22a2 x \u2208 \u22a4 \u2192 x \u2208 span K (Set.range b') ** simpa [mem_map] using hx ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x : { x // x \u2208 span K (Set.range b) } h : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b')) \u22a2 \u2191f '' Set.range b' = Set.range b ** ext x ** case h K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V \u03b9 : Type u_1 inst\u271d : Fintype \u03b9 b : \u03b9 \u2192 V hc : Fintype.card \u03b9 = Set.finrank K (Set.range b) f : { x // x \u2208 span K (Set.range b) } \u2192\u2097[K] V := Submodule.subtype (span K (Set.range b)) b' : \u03b9 \u2192 { x // x \u2208 span K (Set.range b) } := fun i => { val := b i, property := (_ : b i \u2208 span K (Set.range b)) } x\u271d : { x // x \u2208 span K (Set.range b) } h : span K (\u2191f '' Set.range b') = Submodule.map f (span K (Set.range b')) x : V \u22a2 x \u2208 \u2191f '' Set.range b' \u2194 x \u2208 Set.range b ** simp [Set.mem_image, Set.mem_range] ** Qed", + "informal": "" + }, + { + "formal": "norm_smul_of_nonneg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u00b2 : NormedField \u03b1 inst\u271d\u00b9 : SeminormedAddCommGroup \u03b2 inst\u271d : NormedSpace \u211d \u03b2 t : \u211d ht : 0 \u2264 t x : \u03b2 \u22a2 \u2016t \u2022 x\u2016 = t * \u2016x\u2016 ** rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg ht] ** Qed", + "informal": "" + }, + { + "formal": "Int.min_eq_right ** a b : Int h : b \u2264 a \u22a2 min a b = b ** rw [Int.min_comm a b] ** a b : Int h : b \u2264 a \u22a2 min b a = b ** exact Int.min_eq_left h ** Qed", + "informal": "" + }, + { + "formal": "ContDiffWithinAt.div ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u2076 : NontriviallyNormedField \ud835\udd5c D : Type uD inst\u271d\u00b9\u2075 : NormedAddCommGroup D inst\u271d\u00b9\u2074 : NormedSpace \ud835\udd5c D E : Type uE inst\u271d\u00b9\u00b3 : NormedAddCommGroup E inst\u271d\u00b9\u00b2 : NormedSpace \ud835\udd5c E F : Type uF inst\u271d\u00b9\u00b9 : NormedAddCommGroup F inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c F G : Type uG inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedSpace \ud835\udd5c G X : Type u_2 inst\u271d\u2077 : NormedAddCommGroup X inst\u271d\u2076 : NormedSpace \ud835\udd5c X s s\u2081 t u : Set E f\u271d f\u2081 : E \u2192 F g\u271d : F \u2192 G x x\u2080 : E c : F b : E \u00d7 F \u2192 G m n\u271d : \u2115\u221e p : E \u2192 FormalMultilinearSeries \ud835\udd5c E F R : Type u_3 inst\u271d\u2075 : NormedRing R inst\u271d\u2074 : NormedAlgebra \ud835\udd5c R \ud835\udd5c' : Type u_4 inst\u271d\u00b3 : NormedField \ud835\udd5c' inst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c' inst\u271d\u00b9 : CompleteSpace \ud835\udd5c' inst\u271d : CompleteSpace \ud835\udd5c f g : E \u2192 \ud835\udd5c n : \u2115\u221e hf : ContDiffWithinAt \ud835\udd5c n f s x hg : ContDiffWithinAt \ud835\udd5c n g s x hx : g x \u2260 0 \u22a2 ContDiffWithinAt \ud835\udd5c n (fun x => f x / g x) s x ** simpa only [div_eq_mul_inv] using hf.mul (hg.inv hx) ** Qed", + "informal": "" + }, + { + "formal": "Filter.lift'_pure ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 f f\u2081 f\u2082 : Filter \u03b1 h h\u2081 h\u2082 : Set \u03b1 \u2192 Set \u03b2 a : \u03b1 hh : Monotone h \u22a2 Filter.lift' (pure a) h = \ud835\udcdf (h {a}) ** rw [\u2190 principal_singleton, lift'_principal hh] ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.iSup_range_stdBasis ** R : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : Semiring R \u03c6 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (\u03c6 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Finite \u03b9 \u22a2 \u2a06 i, range (stdBasis R \u03c6 i) = \u22a4 ** cases nonempty_fintype \u03b9 ** case intro R : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : Semiring R \u03c6 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (\u03c6 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Finite \u03b9 val\u271d : Fintype \u03b9 \u22a2 \u2a06 i, range (stdBasis R \u03c6 i) = \u22a4 ** convert top_unique (iInf_emptyset.ge.trans <| iInf_ker_proj_le_iSup_range_stdBasis R \u03c6 _) ** case h.e'_2.h.e'_4.h R : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : Semiring R \u03c6 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (\u03c6 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Finite \u03b9 val\u271d : Fintype \u03b9 x\u271d : \u03b9 \u22a2 range (stdBasis R \u03c6 x\u271d) = \u2a06 (_ : x\u271d \u2208 ?intro.convert_1), range (stdBasis R \u03c6 x\u271d) ** rename_i i ** case h.e'_2.h.e'_4.h R : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : Semiring R \u03c6 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (\u03c6 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Finite \u03b9 val\u271d : Fintype \u03b9 i : \u03b9 \u22a2 range (stdBasis R \u03c6 i) = \u2a06 (_ : i \u2208 ?intro.convert_1), range (stdBasis R \u03c6 i) ** exact ((@iSup_pos _ _ _ fun _ => range <| stdBasis R \u03c6 i) <| Finset.mem_univ i).symm ** case intro.convert_2 R : Type u_1 \u03b9 : Type u_2 inst\u271d\u2074 : Semiring R \u03c6 : \u03b9 \u2192 Type u_3 inst\u271d\u00b3 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Module R (\u03c6 i) inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : Finite \u03b9 val\u271d : Fintype \u03b9 \u22a2 univ \u2286 \u2191Finset.univ \u222a \u2205 ** rw [Finset.coe_univ, Set.union_empty] ** Qed", + "informal": "" + }, + { + "formal": "PowerSeries.degree_trunc_lt ** R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n : \u2115 \u22a2 degree (trunc n f) < \u2191n ** rw [degree_lt_iff_coeff_zero] ** R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n : \u2115 \u22a2 \u2200 (m : \u2115), n \u2264 m \u2192 Polynomial.coeff (trunc n f) m = 0 ** intros ** R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n m\u271d : \u2115 a\u271d : n \u2264 m\u271d \u22a2 Polynomial.coeff (trunc n f) m\u271d = 0 ** rw [coeff_trunc] ** R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n m\u271d : \u2115 a\u271d : n \u2264 m\u271d \u22a2 (if m\u271d < n then \u2191(coeff R m\u271d) f else 0) = 0 ** split_ifs with h ** case pos R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n m\u271d : \u2115 a\u271d : n \u2264 m\u271d h : m\u271d < n \u22a2 \u2191(coeff R m\u271d) f = 0 ** rw [\u2190not_le] at h ** case pos R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n m\u271d : \u2115 a\u271d : n \u2264 m\u271d h : \u00acn \u2264 m\u271d \u22a2 \u2191(coeff R m\u271d) f = 0 ** contradiction ** case neg R : Type u_1 inst\u271d : Semiring R f : R\u27e6X\u27e7 n m\u271d : \u2115 a\u271d : n \u2264 m\u271d h : \u00acm\u271d < n \u22a2 0 = 0 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "isLittleO_pow_exp_pos_mul_atTop ** k : \u2115 b : \u211d hb : 0 < b \u22a2 (fun x => x ^ \u2191k) =o[atTop] fun x => rexp (b * x) ** simpa using isLittleO_zpow_exp_pos_mul_atTop k hb ** Qed", + "informal": "" + }, + { + "formal": "mulSupport_comp_inv_smul\u2080 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : GroupWithZero \u03b1 inst\u271d\u00b9 : MulAction \u03b1 \u03b2 inst\u271d : One \u03b3 c : \u03b1 hc : c \u2260 0 f : \u03b2 \u2192 \u03b3 \u22a2 (mulSupport fun x => f (c\u207b\u00b9 \u2022 x)) = c \u2022 mulSupport f ** ext x ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : GroupWithZero \u03b1 inst\u271d\u00b9 : MulAction \u03b1 \u03b2 inst\u271d : One \u03b3 c : \u03b1 hc : c \u2260 0 f : \u03b2 \u2192 \u03b3 x : \u03b2 \u22a2 (x \u2208 mulSupport fun x => f (c\u207b\u00b9 \u2022 x)) \u2194 x \u2208 c \u2022 mulSupport f ** simp only [mem_smul_set_iff_inv_smul_mem\u2080 hc, mem_mulSupport] ** Qed", + "informal": "" + }, + { + "formal": "ofBoolAlg_symmDiff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : BooleanRing \u03b1 inst\u271d\u00b9 : BooleanRing \u03b2 inst\u271d : BooleanRing \u03b3 a b : AsBoolAlg \u03b1 \u22a2 \u2191ofBoolAlg (a \u2206 b) = \u2191ofBoolAlg a + \u2191ofBoolAlg b ** rw [symmDiff_eq_sup_sdiff_inf] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : BooleanRing \u03b1 inst\u271d\u00b9 : BooleanRing \u03b2 inst\u271d : BooleanRing \u03b3 a b : AsBoolAlg \u03b1 \u22a2 \u2191ofBoolAlg ((a \u2294 b) \\ (a \u2293 b)) = \u2191ofBoolAlg a + \u2191ofBoolAlg b ** exact of_boolalg_symmDiff_aux _ _ ** Qed", + "informal": "" + }, + { + "formal": "LocalEquiv.refl_restr_target ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 e : LocalEquiv \u03b1 \u03b2 e' : LocalEquiv \u03b2 \u03b3 s : Set \u03b1 \u22a2 (LocalEquiv.restr (LocalEquiv.refl \u03b1) s).target = s ** change univ \u2229 id \u207b\u00b9' s = s ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 e : LocalEquiv \u03b1 \u03b2 e' : LocalEquiv \u03b2 \u03b3 s : Set \u03b1 \u22a2 univ \u2229 id \u207b\u00b9' s = s ** simp ** Qed", + "informal": "" + }, + { + "formal": "Finset.map_add_right_Ioo ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b9 : ExistsAddOfLE \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 map (addRightEmbedding c) (Ioo a b) = Ioo (a + c) (b + c) ** rw [\u2190 coe_inj, coe_map, coe_Ioo, coe_Ioo] ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d\u00b2 : OrderedCancelAddCommMonoid \u03b1 inst\u271d\u00b9 : ExistsAddOfLE \u03b1 inst\u271d : LocallyFiniteOrder \u03b1 a b c : \u03b1 \u22a2 \u2191(addRightEmbedding c) '' Set.Ioo a b = Set.Ioo (a + c) (b + c) ** exact Set.image_add_const_Ioo _ _ _ ** Qed", + "informal": "" + }, + { + "formal": "Submonoid.LocalizationMap.mk'_eq_iff_eq_mul ** M : Type u_1 inst\u271d\u00b2 : CommMonoid M S : Submonoid M N : Type u_2 inst\u271d\u00b9 : CommMonoid N P : Type u_3 inst\u271d : CommMonoid P f : LocalizationMap S N x : M y : { x // x \u2208 S } z : N \u22a2 mk' f x y = z \u2194 \u2191(toMap f) x = z * \u2191(toMap f) \u2191y ** rw [eq_comm, eq_mk'_iff_mul_eq, eq_comm] ** Qed", + "informal": "" + }, + { + "formal": "LocalEquiv.refl_trans ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 e : LocalEquiv \u03b1 \u03b2 e' : LocalEquiv \u03b2 \u03b3 \u22a2 (LocalEquiv.trans (LocalEquiv.refl \u03b1) e).source = e.source ** simp [trans_source, preimage_id] ** Qed", + "informal": "" + }, + { + "formal": "List.cons_prefix_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 \u22a2 a :: l\u2081 <+: b :: l\u2082 \u2194 a = b \u2227 l\u2081 <+: l\u2082 ** constructor ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 \u22a2 a :: l\u2081 <+: b :: l\u2082 \u2192 a = b \u2227 l\u2081 <+: l\u2082 ** rintro \u27e8L, hL\u27e9 ** case mp.intro \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 L : List \u03b1 hL : a :: l\u2081 ++ L = b :: l\u2082 \u22a2 a = b \u2227 l\u2081 <+: l\u2082 ** simp only [cons_append] at hL ** case mp.intro \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 L : List \u03b1 hL : a :: (l\u2081 ++ L) = b :: l\u2082 \u22a2 a = b \u2227 l\u2081 <+: l\u2082 ** injection hL with hLLeft hLRight ** case mp.intro \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 L : List \u03b1 hLLeft : a = b hLRight : l\u2081 ++ L = l\u2082 \u22a2 a = b \u2227 l\u2081 <+: l\u2082 ** exact \u27e8hLLeft, \u27e8L, hLRight\u27e9\u27e9 ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a b : \u03b1 m n : \u2115 \u22a2 a = b \u2227 l\u2081 <+: l\u2082 \u2192 a :: l\u2081 <+: b :: l\u2082 ** rintro \u27e8rfl, h\u27e9 ** case mpr.intro \u03b1 : Type u_1 \u03b2 : Type u_2 l l\u2081 l\u2082 l\u2083 : List \u03b1 a : \u03b1 m n : \u2115 h : l\u2081 <+: l\u2082 \u22a2 a :: l\u2081 <+: a :: l\u2082 ** rwa [prefix_cons_inj] ** Qed", + "informal": "" + }, + { + "formal": "deriv_mem_iff ** \ud835\udd5c : Type u inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c F : Type v inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F E : Type w inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E f\u271d f\u2080 f\u2081 g : \ud835\udd5c \u2192 F f' f\u2080' f\u2081' g' : F x\u271d : \ud835\udd5c s\u271d t : Set \ud835\udd5c L L\u2081 L\u2082 : Filter \ud835\udd5c f : \ud835\udd5c \u2192 F s : Set F x : \ud835\udd5c \u22a2 deriv f x \u2208 s \u2194 DifferentiableAt \ud835\udd5c f x \u2227 deriv f x \u2208 s \u2228 \u00acDifferentiableAt \ud835\udd5c f x \u2227 0 \u2208 s ** by_cases hx : DifferentiableAt \ud835\udd5c f x <;> simp [deriv_zero_of_not_differentiableAt, *] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.hasStrictInitial_of_isUniversal ** J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) \u22a2 \u2200 (A : C) (f : A \u27f6 \u22a5_ C), IsIso f ** intro A f ** J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C \u22a2 IsIso f ** suffices IsColimit (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) by\n obtain \u27e8l, h\u2081, h\u2082\u27e9 := Limits.BinaryCofan.IsColimit.desc' this (f \u226b initial.to A) (\ud835\udfd9 A)\n rcases (Category.id_comp _).symm.trans h\u2082 with rfl\n exact \u27e8\u27e8_, ((Category.id_comp _).symm.trans h\u2081).symm, initialIsInitial.hom_ext _ _\u27e9\u27e9 ** J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C \u22a2 IsColimit (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) ** refine' (H (BinaryCofan.mk (\ud835\udfd9 _) (\ud835\udfd9 _)) (mapPair f f) f (by ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> simp)\n (mapPair_equifibered _) _).some ** J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C \u22a2 \u2200 (j : Discrete WalkingPair), IsPullback ((BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)).\u03b9.app j) ((mapPair f f).app j) f ((BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))).\u03b9.app j) ** rintro \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;>\n exact IsPullback.of_horiz_isIso \u27e8(Category.id_comp _).trans (Category.comp_id _).symm\u27e9 ** J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C this : IsColimit (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) \u22a2 IsIso f ** obtain \u27e8l, h\u2081, h\u2082\u27e9 := Limits.BinaryCofan.IsColimit.desc' this (f \u226b initial.to A) (\ud835\udfd9 A) ** case mk.intro J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C this : IsColimit (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) l : (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)).pt \u27f6 A h\u2081 : BinaryCofan.inl (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) \u226b l = f \u226b initial.to A h\u2082 : BinaryCofan.inr (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) \u226b l = \ud835\udfd9 A \u22a2 IsIso f ** rcases (Category.id_comp _).symm.trans h\u2082 with rfl ** case mk.intro J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C this : IsColimit (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) h\u2081 : BinaryCofan.inl (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) \u226b \ud835\udfd9 A = f \u226b initial.to A h\u2082 : BinaryCofan.inr (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)) \u226b \ud835\udfd9 A = \ud835\udfd9 A \u22a2 IsIso f ** exact \u27e8\u27e8_, ((Category.id_comp _).symm.trans h\u2081).symm, initialIsInitial.hom_ext _ _\u27e9\u27e9 ** J : Type v' inst\u271d\u00b2 : Category.{u', v'} J C : Type u inst\u271d\u00b9 : Category.{v, u} C X Y : C inst\u271d : HasInitial C H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))) A : C f : A \u27f6 \u22a5_ C \u22a2 mapPair f f \u226b (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C))).\u03b9 = (BinaryCofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A)).\u03b9 \u226b (Functor.const (Discrete WalkingPair)).map f ** ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> simp ** Qed", + "informal": "" + }, + { + "formal": "List.IsSuffix.filter ** \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l\u2081 l\u2082 : List \u03b1 h : l\u2081 <:+ l\u2082 \u22a2 List.filter p l\u2081 <:+ List.filter p l\u2082 ** obtain \u27e8xs, rfl\u27e9 := h ** case intro \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l\u2081 xs : List \u03b1 \u22a2 List.filter p l\u2081 <:+ List.filter p (xs ++ l\u2081) ** rw [filter_append] ** case intro \u03b1 : Type u_1 p : \u03b1 \u2192 Bool l\u2081 xs : List \u03b1 \u22a2 List.filter p l\u2081 <:+ List.filter p xs ++ List.filter p l\u2081 ** apply suffix_append ** Qed", + "informal": "" + }, + { + "formal": "VitaliFamily.measure_le_of_frequently_le ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191\u03bd s ** apply ENNReal.le_of_forall_pos_le_add fun \u03b5 \u03b5pos _ => ?_ ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 ** obtain \u27e8U, sU, U_open, \u03bdU\u27e9 : \u2203 (U : Set \u03b1), s \u2286 U \u2227 IsOpen U \u2227 \u03bd U \u2264 \u03bd s + \u03b5 :=\n exists_isOpen_le_add s \u03bd (ENNReal.coe_pos.2 \u03b5pos).ne' ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 ** let f : \u03b1 \u2192 Set (Set \u03b1) := fun _ => {a | \u03c1 a \u2264 \u03bd a \u2227 a \u2286 U} ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 ** have h : v.FineSubfamilyOn f s := by\n apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_\n have :=\n (hs x hx).and_eventually\n ((v.eventually_filterAt_mem_sets x).and\n (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx))))\n apply Frequently.mono this\n rintro a \u27e8\u03c1a, _, aU\u27e9\n exact \u27e8\u03c1a, aU\u27e9 ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} h : FineSubfamilyOn v f s \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 ** haveI : Encodable h.index := h.index_countable.toEncodable ** case intro.intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} h : FineSubfamilyOn v f s this : Encodable \u2191(FineSubfamilyOn.index h) \u22a2 \u2191\u2191\u03c1 s \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 ** calc\n \u03c1 s \u2264 \u2211' x : h.index, \u03c1 (h.covering x) := h.measure_le_tsum_of_absolutelyContinuous h\u03c1\n _ \u2264 \u2211' x : h.index, \u03bd (h.covering x) := (ENNReal.tsum_le_tsum fun x => (h.covering_mem x.2).1)\n _ = \u03bd (\u22c3 x : h.index, h.covering x) := by\n rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2]\n _ \u2264 \u03bd U := (measure_mono (iUnion_subset fun i => (h.covering_mem i.2).2))\n _ \u2264 \u03bd s + \u03b5 := \u03bdU ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} \u22a2 FineSubfamilyOn v f s ** apply v.fineSubfamilyOn_of_frequently f s fun x hx => ?_ ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} x : \u03b1 hx : x \u2208 s \u22a2 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x ** have :=\n (hs x hx).and_eventually\n ((v.eventually_filterAt_mem_sets x).and\n (v.eventually_filterAt_subset_of_nhds (U_open.mem_nhds (sU hx)))) ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} x : \u03b1 hx : x \u2208 s this : \u2203\u1da0 (x_1 : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 x_1 \u2264 \u2191\u2191\u03bd x_1 \u2227 x_1 \u2208 setsAt v x \u2227 x_1 \u2286 U \u22a2 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, a \u2208 f x ** apply Frequently.mono this ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} x : \u03b1 hx : x \u2208 s this : \u2203\u1da0 (x_1 : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 x_1 \u2264 \u2191\u2191\u03bd x_1 \u2227 x_1 \u2208 setsAt v x \u2227 x_1 \u2286 U \u22a2 \u2200 (x_1 : Set \u03b1), \u2191\u2191\u03c1 x_1 \u2264 \u2191\u2191\u03bd x_1 \u2227 x_1 \u2208 setsAt v x \u2227 x_1 \u2286 U \u2192 x_1 \u2208 f x ** rintro a \u27e8\u03c1a, _, aU\u27e9 ** case intro.intro \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} x : \u03b1 hx : x \u2208 s this : \u2203\u1da0 (x_1 : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 x_1 \u2264 \u2191\u2191\u03bd x_1 \u2227 x_1 \u2208 setsAt v x \u2227 x_1 \u2286 U a : Set \u03b1 \u03c1a : \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a left\u271d : a \u2208 setsAt v x aU : a \u2286 U \u22a2 a \u2208 f x ** exact \u27e8\u03c1a, aU\u27e9 ** \u03b1 : Type u_1 inst\u271d\u2074 : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : SecondCountableTopology \u03b1 inst\u271d\u00b9 : BorelSpace \u03b1 \u03c1 \u03bd : Measure \u03b1 inst\u271d : IsLocallyFiniteMeasure \u03bd h\u03c1 : \u03c1 \u226a \u03bc s : Set \u03b1 hs : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203\u1da0 (a : Set \u03b1) in filterAt v x, \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u03b5 : \u211d\u22650 \u03b5pos : 0 < \u03b5 x\u271d : \u2191\u2191\u03bd s < \u22a4 U : Set \u03b1 sU : s \u2286 U U_open : IsOpen U \u03bdU : \u2191\u2191\u03bd U \u2264 \u2191\u2191\u03bd s + \u2191\u03b5 f : \u03b1 \u2192 Set (Set \u03b1) := fun x => {a | \u2191\u2191\u03c1 a \u2264 \u2191\u2191\u03bd a \u2227 a \u2286 U} h : FineSubfamilyOn v f s this : Encodable \u2191(FineSubfamilyOn.index h) \u22a2 \u2211' (x : \u2191(FineSubfamilyOn.index h)), \u2191\u2191\u03bd (FineSubfamilyOn.covering h \u2191x) = \u2191\u2191\u03bd (\u22c3 x, FineSubfamilyOn.covering h \u2191x) ** rw [measure_iUnion h.covering_disjoint_subtype fun i => h.measurableSet_u i.2] ** Qed", + "informal": "" + }, + { + "formal": "Function.update_apply ** \u03b1 : Sort u \u03b2\u271d : \u03b1 \u2192 Sort v \u03b1' : Sort w inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b1' f\u271d g : (a : \u03b1) \u2192 \u03b2\u271d a a\u271d : \u03b1 b\u271d : \u03b2\u271d a\u271d \u03b2 : Sort u_1 f : \u03b1 \u2192 \u03b2 a' : \u03b1 b : \u03b2 a : \u03b1 \u22a2 update f a' b a = if a = a' then b else f a ** rcases Decidable.eq_or_ne a a' with rfl | hne <;> simp [*] ** Qed", + "informal": "" + }, + { + "formal": "Wbtw.left_ne_right_of_ne_left ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : OrderedRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' x y z : P h : Wbtw R x y z hne : y \u2260 x \u22a2 x \u2260 z ** rintro rfl ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : OrderedRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' x y : P hne : y \u2260 x h : Wbtw R x y x \u22a2 False ** rw [wbtw_self_iff] at h ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : OrderedRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' x y : P hne : y \u2260 x h : y = x \u22a2 False ** exact hne h ** Qed", + "informal": "" + }, + { + "formal": "List.pmap_eq_map_attach ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a \u22a2 pmap f l H = map (fun x => f \u2191x (_ : p \u2191x)) (attach l) ** rw [attach, map_pmap] ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 l : List \u03b1 H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a \u22a2 pmap f l H = pmap (fun a h => f \u2191{ val := a, property := h } (_ : p \u2191{ val := a, property := h })) l (_ : \u2200 (x : \u03b1), x \u2208 l \u2192 x \u2208 l) ** exact pmap_congr l fun _ _ _ _ => rfl ** Qed", + "informal": "" + }, + { + "formal": "DistribMulActionHom.id_apply ** M' : Type u_1 X : Type u_2 inst\u271d\u00b2\u00b3 : SMul M' X Y : Type u_3 inst\u271d\u00b2\u00b2 : SMul M' Y Z : Type u_4 inst\u271d\u00b2\u00b9 : SMul M' Z M : Type u_5 inst\u271d\u00b2\u2070 : Monoid M A : Type u_6 inst\u271d\u00b9\u2079 : AddMonoid A inst\u271d\u00b9\u2078 : DistribMulAction M A A' : Type u_7 inst\u271d\u00b9\u2077 : AddGroup A' inst\u271d\u00b9\u2076 : DistribMulAction M A' B : Type u_8 inst\u271d\u00b9\u2075 : AddMonoid B inst\u271d\u00b9\u2074 : DistribMulAction M B B' : Type u_9 inst\u271d\u00b9\u00b3 : AddGroup B' inst\u271d\u00b9\u00b2 : DistribMulAction M B' C : Type u_10 inst\u271d\u00b9\u00b9 : AddMonoid C inst\u271d\u00b9\u2070 : DistribMulAction M C R : Type u_11 inst\u271d\u2079 : Semiring R inst\u271d\u2078 : MulSemiringAction M R R' : Type u_12 inst\u271d\u2077 : Ring R' inst\u271d\u2076 : MulSemiringAction M R' S : Type u_13 inst\u271d\u2075 : Semiring S inst\u271d\u2074 : MulSemiringAction M S S' : Type u_14 inst\u271d\u00b3 : Ring S' inst\u271d\u00b2 : MulSemiringAction M S' T : Type u_15 inst\u271d\u00b9 : Semiring T inst\u271d : MulSemiringAction M T x : A \u22a2 \u2191(DistribMulActionHom.id M) x = x ** rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Center.leftUnitor_inv_f ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C X : Center C \u22a2 (\u03bb_ X).inv.f = (\u03bb_ X.fst).inv ** apply Iso.inv_ext' ** case hom_inv_id C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C X : Center C \u22a2 (\u03bb_ X.fst).hom \u226b (\u03bb_ X).inv.f = \ud835\udfd9 (tensorUnit' \u2297 X).fst ** rw [\u2190 leftUnitor_hom_f, \u2190 comp_f, Iso.hom_inv_id] ** case hom_inv_id C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C inst\u271d : MonoidalCategory C X : Center C \u22a2 (\ud835\udfd9 (tensorUnit' \u2297 X)).f = \ud835\udfd9 (tensorUnit' \u2297 X).fst ** rfl ** Qed", + "informal": "" + }, + { + "formal": "ext_inner_map ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : InnerProductSpace \u211d F dec_E : DecidableEq E V : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u2102 V S T : V \u2192\u2097[\u2102] V \u22a2 (\u2200 (x : V), inner (\u2191S x) x = inner (\u2191T x) x) \u2194 S = T ** rw [\u2190 sub_eq_zero, \u2190 inner_map_self_eq_zero] ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : InnerProductSpace \u211d F dec_E : DecidableEq E V : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u2102 V S T : V \u2192\u2097[\u2102] V \u22a2 (\u2200 (x : V), inner (\u2191S x) x = inner (\u2191T x) x) \u2194 \u2200 (x : V), inner (\u2191(S - T) x) x = 0 ** refine' forall_congr' fun x => _ ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : InnerProductSpace \u211d F dec_E : DecidableEq E V : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u2102 V S T : V \u2192\u2097[\u2102] V x : V \u22a2 inner (\u2191S x) x = inner (\u2191T x) x \u2194 inner (\u2191(S - T) x) x = 0 ** rw [LinearMap.sub_apply, inner_sub_left, sub_eq_zero] ** Qed", + "informal": "" + }, + { + "formal": "List.card_cons_of_not_mem ** \u03b1 : Type u_1 \u03b2 : Sort ?u.14118 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 a : \u03b1 as : List \u03b1 h : \u00aca \u2208 as \u22a2 card (a :: as) = card as + 1 ** simp [card, h] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.NatTrans.id_hcomp_app ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D E : Type u\u2083 inst\u271d : Category.{v\u2083, u\u2083} E F G H\u271d I : C \u2964 D H : E \u2964 C \u03b1 : F \u27f6 G X : E \u22a2 (\ud835\udfd9 H \u25eb \u03b1).app X = \u03b1.app (H.obj X) ** simp ** Qed", + "informal": "" + }, + { + "formal": "PNat.exists_prime_and_dvd ** n : \u2115+ hn : n \u2260 1 \u22a2 \u2203 p, Prime p \u2227 p \u2223 n ** obtain \u27e8p, hp\u27e9 := Nat.exists_prime_and_dvd (mt coe_eq_one_iff.mp hn) ** case intro n : \u2115+ hn : n \u2260 1 p : \u2115 hp : Nat.Prime p \u2227 p \u2223 \u2191n \u22a2 \u2203 p, Prime p \u2227 p \u2223 n ** exists (\u27e8p, Nat.Prime.pos hp.left\u27e9 : \u2115+) ** case intro n : \u2115+ hn : n \u2260 1 p : \u2115 hp : Nat.Prime p \u2227 p \u2223 \u2191n \u22a2 Prime { val := p, property := (_ : 0 < p) } \u2227 { val := p, property := (_ : 0 < p) } \u2223 n ** rw [dvd_iff] ** case intro n : \u2115+ hn : n \u2260 1 p : \u2115 hp : Nat.Prime p \u2227 p \u2223 \u2191n \u22a2 Prime { val := p, property := (_ : 0 < p) } \u2227 \u2191{ val := p, property := (_ : 0 < p) } \u2223 \u2191n ** apply hp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.AECover.integrable_of_lintegral_nnnorm_bounded' ** \u03b1 : Type u_1 \u03b9 : Type u_2 E : Type u_3 inst\u271d\u00b3 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 l : Filter \u03b9 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NeBot l inst\u271d : IsCountablyGenerated l \u03c6 : \u03b9 \u2192 Set \u03b1 h\u03c6 : AECover \u03bc l \u03c6 f : \u03b1 \u2192 E I : \u211d\u22650 hfm : AEStronglyMeasurable f \u03bc hbounded : \u2200\u1da0 (i : \u03b9) in l, \u222b\u207b (x : \u03b1) in \u03c6 i, \u2191\u2016f x\u2016\u208a \u2202\u03bc \u2264 \u2191I \u22a2 \u2200\u1da0 (i : \u03b9) in l, \u222b\u207b (x : \u03b1) in \u03c6 i, \u2191\u2016f x\u2016\u208a \u2202\u03bc \u2264 ENNReal.ofReal \u2191I ** simpa only [ENNReal.ofReal_coe_nnreal] using hbounded ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.NonPreadditiveAbelian.neg_sub' ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a b : X \u27f6 Y \u22a2 -(a - b) = -a + b ** rw [neg_def, neg_def] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a b : X \u27f6 Y \u22a2 0 - (a - b) = 0 - a + b ** conv_lhs => rw [\u2190 sub_self (0 : X \u27f6 Y)] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : NonPreadditiveAbelian C X Y : C a b : X \u27f6 Y \u22a2 0 - 0 - (a - b) = 0 - a + b ** rw [sub_sub_sub, add_def, neg_def] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.IsTrail.takeUntil ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' inst\u271d : DecidableEq V u v w : V p : Walk G v w hc : IsTrail p h : u \u2208 support p \u22a2 IsTrail (append (takeUntil p u h) (?m.293706 hc h)) ** rwa [\u2190 take_spec _ h] at hc ** Qed", + "informal": "" + }, + { + "formal": "RatFunc.liftOn_condition_of_liftOn'_condition ** K : Type u inst\u271d : CommRing K P : Sort v f : K[X] \u2192 K[X] \u2192 P H : \u2200 {p q a : K[X]}, q \u2260 0 \u2192 a \u2260 0 \u2192 f (a * p) (a * q) = f p q p q p' q' : K[X] hq : q \u2260 0 hq' : q' \u2260 0 h : q' * p = q * p' \u22a2 f (q' * p) (q' * q) = f (q * p') (q * q') ** rw [h, mul_comm q'] ** Qed", + "informal": "" + }, + { + "formal": "FiniteField.pow_card_sub_one_eq_one ** K : Type u_1 R : Type u_2 inst\u271d\u00b9 : GroupWithZero K inst\u271d : Fintype K a : K ha : a \u2260 0 \u22a2 a ^ (q - 1) = 1 ** calc\n a ^ (Fintype.card K - 1) = (Units.mk0 a ha ^ (Fintype.card K - 1) : K\u02e3).1 := by\n rw [Units.val_pow_eq_pow_val, Units.val_mk0]\n _ = 1 := by\n classical\n rw [\u2190 Fintype.card_units, pow_card_eq_one]\n rfl ** K : Type u_1 R : Type u_2 inst\u271d\u00b9 : GroupWithZero K inst\u271d : Fintype K a : K ha : a \u2260 0 \u22a2 a ^ (q - 1) = \u2191(Units.mk0 a ha ^ (q - 1)) ** rw [Units.val_pow_eq_pow_val, Units.val_mk0] ** K : Type u_1 R : Type u_2 inst\u271d\u00b9 : GroupWithZero K inst\u271d : Fintype K a : K ha : a \u2260 0 \u22a2 \u2191(Units.mk0 a ha ^ (q - 1)) = 1 ** classical\n rw [\u2190 Fintype.card_units, pow_card_eq_one]\n rfl ** K : Type u_1 R : Type u_2 inst\u271d\u00b9 : GroupWithZero K inst\u271d : Fintype K a : K ha : a \u2260 0 \u22a2 \u2191(Units.mk0 a ha ^ (q - 1)) = 1 ** rw [\u2190 Fintype.card_units, pow_card_eq_one] ** K : Type u_1 R : Type u_2 inst\u271d\u00b9 : GroupWithZero K inst\u271d : Fintype K a : K ha : a \u2260 0 \u22a2 \u21911 = 1 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.VectorMeasure.le_restrict_univ_iff_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M \u22a2 restrict v univ \u2264 restrict w univ \u2194 v \u2264 w ** constructor ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M \u22a2 restrict v univ \u2264 restrict w univ \u2192 v \u2264 w ** intro h s hs ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M h : restrict v univ \u2264 restrict w univ s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191v s \u2264 \u2191w s ** have := h s hs ** case mp \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M h : restrict v univ \u2264 restrict w univ s : Set \u03b1 hs : MeasurableSet s this : \u2191(restrict v univ) s \u2264 \u2191(restrict w univ) s \u22a2 \u2191v s \u2264 \u2191w s ** rwa [restrict_apply _ MeasurableSet.univ hs, Set.inter_univ,\n restrict_apply _ MeasurableSet.univ hs, Set.inter_univ] at this ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M \u22a2 v \u2264 w \u2192 restrict v univ \u2264 restrict w univ ** intro h s hs ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M h : v \u2264 w s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191(restrict v univ) s \u2264 \u2191(restrict w univ) s ** rw [restrict_apply _ MeasurableSet.univ hs, Set.inter_univ,\n restrict_apply _ MeasurableSet.univ hs, Set.inter_univ] ** case mpr \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 M : Type u_3 inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : AddCommMonoid M inst\u271d : PartialOrder M v w : VectorMeasure \u03b1 M h : v \u2264 w s : Set \u03b1 hs : MeasurableSet s \u22a2 \u2191v s \u2264 \u2191w s ** exact h s hs ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicTopology.DoldKan.h\u03c3'_eq' ** C : Type u_1 inst\u271d\u00b9 : Category.{u_2, u_1} C inst\u271d : Preadditive C X : SimplicialObject C q n a : \u2115 ha : n = a + q \u22a2 h\u03c3' q n (n + 1) (_ : n + 1 = n + 1) = (-1) ^ a \u2022 \u03c3 X { val := a, isLt := (_ : a < Nat.succ n) } ** rw [h\u03c3'_eq ha rfl, eqToHom_refl, comp_id] ** Qed", + "informal": "" + }, + { + "formal": "even_two ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 R : Type u_4 inst\u271d\u00b9 : Semiring \u03b1 inst\u271d : Semiring \u03b2 m n : \u03b1 \u22a2 2 = 1 + 1 ** rw[one_add_one_eq_two] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.det_one_add_col_mul_row ** l : Type u_1 m : Type u_2 n : Type u_3 \u03b1 : Type u_4 inst\u271d\u2076 : Fintype l inst\u271d\u2075 : Fintype m inst\u271d\u2074 : Fintype n inst\u271d\u00b3 : DecidableEq l inst\u271d\u00b2 : DecidableEq m inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 u v : m \u2192 \u03b1 \u22a2 det (1 + col u * row v) = 1 + v \u2b1d\u1d65 u ** rw [det_one_add_mul_comm, det_unique, Pi.add_apply, Pi.add_apply, Matrix.one_apply_eq,\n Matrix.row_mul_col_apply] ** Qed", + "informal": "" + }, + { + "formal": "Set.EqOn.piecewise_ite' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03c0 : \u03b1 \u2192 Type u_5 \u03b4 : \u03b1 \u2192 Sort u_6 s : Set \u03b1 f\u271d g\u271d : (i : \u03b1) \u2192 \u03b4 i inst\u271d : (j : \u03b1) \u2192 Decidable (j \u2208 s) f f' g : \u03b1 \u2192 \u03b2 t t' : Set \u03b1 h : EqOn f g (t \u2229 s) h' : EqOn f' g (t' \u2229 s\u1d9c) \u22a2 EqOn (piecewise s f f') g (Set.ite s t t') ** simp [eqOn_piecewise, *] ** Qed", + "informal": "" + }, + { + "formal": "sup_eq_maxDefault ** \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b2 : SemilatticeSup \u03b1 inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1 inst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1 \u22a2 (fun x x_1 => x \u2294 x_1) = maxDefault ** ext x y ** case h.h \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b2 : SemilatticeSup \u03b1 inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1 inst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1 x y : \u03b1 \u22a2 x \u2294 y = maxDefault x y ** unfold maxDefault ** case h.h \u03b1 : Type u \u03b2 : Type v inst\u271d\u00b2 : SemilatticeSup \u03b1 inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2264 x_1 inst\u271d : IsTotal \u03b1 fun x x_1 => x \u2264 x_1 x y : \u03b1 \u22a2 x \u2294 y = if x \u2264 y then y else x ** split_ifs with h' ** Qed", + "informal": "" + }, + { + "formal": "TypeVec.prod_fst_mk ** n : \u2115 \u03b1 \u03b2 : TypeVec.{u_1} n i : Fin2 n a : \u03b1 i b : \u03b2 i \u22a2 prod.fst i (prod.mk i a b) = a ** induction' i with _ _ _ i_ih ** case fz n : \u2115 \u03b1\u271d \u03b2\u271d : TypeVec.{u_1} n i : Fin2 n a\u271d : \u03b1\u271d i b\u271d : \u03b2\u271d i n\u271d : \u2115 \u03b1 \u03b2 : TypeVec.{u_1} (succ n\u271d) a : \u03b1 Fin2.fz b : \u03b2 Fin2.fz \u22a2 prod.fst Fin2.fz (prod.mk Fin2.fz a b) = a case fs n : \u2115 \u03b1\u271d \u03b2\u271d : TypeVec.{u_1} n i : Fin2 n a\u271d\u00b9 : \u03b1\u271d i b\u271d : \u03b2\u271d i n\u271d : \u2115 a\u271d : Fin2 n\u271d i_ih : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (a : \u03b1 a\u271d) (b : \u03b2 a\u271d), prod.fst a\u271d (prod.mk a\u271d a b) = a \u03b1 \u03b2 : TypeVec.{u_1} (succ n\u271d) a : \u03b1 (Fin2.fs a\u271d) b : \u03b2 (Fin2.fs a\u271d) \u22a2 prod.fst (Fin2.fs a\u271d) (prod.mk (Fin2.fs a\u271d) a b) = a ** simp_all only [prod.fst, prod.mk] ** case fs n : \u2115 \u03b1\u271d \u03b2\u271d : TypeVec.{u_1} n i : Fin2 n a\u271d\u00b9 : \u03b1\u271d i b\u271d : \u03b2\u271d i n\u271d : \u2115 a\u271d : Fin2 n\u271d i_ih : \u2200 {\u03b1 \u03b2 : TypeVec.{u_1} n\u271d} (a : \u03b1 a\u271d) (b : \u03b2 a\u271d), prod.fst a\u271d (prod.mk a\u271d a b) = a \u03b1 \u03b2 : TypeVec.{u_1} (succ n\u271d) a : \u03b1 (Fin2.fs a\u271d) b : \u03b2 (Fin2.fs a\u271d) \u22a2 prod.fst (Fin2.fs a\u271d) (prod.mk (Fin2.fs a\u271d) a b) = a ** apply i_ih ** Qed", + "informal": "" + }, + { + "formal": "Finpartition.IsEquipartition.card_part_le_average_add_one ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 P : Finpartition s hP : IsEquipartition P ht : t \u2208 P.parts \u22a2 Finset.card t \u2264 Finset.card s / Finset.card P.parts + 1 ** rw [\u2190 P.sum_card_parts] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s t : Finset \u03b1 P : Finpartition s hP : IsEquipartition P ht : t \u2208 P.parts \u22a2 Finset.card t \u2264 (Finset.sum P.parts fun i => Finset.card i) / Finset.card P.parts + 1 ** exact Finset.EquitableOn.le_add_one hP ht ** Qed", + "informal": "" + }, + { + "formal": "CliffordAlgebra.contractLeftAux_contractLeftAux ** R : Type u1 inst\u271d\u00b2 : CommRing R M : Type u2 inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M Q : QuadraticForm R M d d' : Module.Dual R M v : M x fx : CliffordAlgebra Q \u22a2 \u2191(\u2191(contractLeftAux Q d) v) (\u2191(\u03b9 Q) v * x, \u2191(\u2191(contractLeftAux Q d) v) (x, fx)) = \u2191Q v \u2022 fx ** simp only [contractLeftAux_apply_apply] ** R : Type u1 inst\u271d\u00b2 : CommRing R M : Type u2 inst\u271d\u00b9 : AddCommGroup M inst\u271d : Module R M Q : QuadraticForm R M d d' : Module.Dual R M v : M x fx : CliffordAlgebra Q \u22a2 \u2191d v \u2022 (\u2191(\u03b9 Q) v * x) - \u2191(\u03b9 Q) v * (\u2191d v \u2022 x - \u2191(\u03b9 Q) v * fx) = \u2191Q v \u2022 fx ** rw [mul_sub, \u2190 mul_assoc, \u03b9_sq_scalar, \u2190 Algebra.smul_def, \u2190 sub_add, mul_smul_comm, sub_self,\n zero_add] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.SimpleFunc.setToSimpleFunc_zero_apply ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 \ud835\udd5c : Type u_6 p : \u211d\u22650\u221e inst\u271d\u2076 : NormedAddCommGroup E inst\u271d\u2075 : NormedSpace \u211d E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : NormedAddCommGroup F' inst\u271d\u00b9 : NormedSpace \u211d F' inst\u271d : NormedAddCommGroup G m\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 m : MeasurableSpace \u03b1 T : Set \u03b1 \u2192 F \u2192L[\u211d] F' \u22a2 setToSimpleFunc T 0 = 0 ** cases isEmpty_or_nonempty \u03b1 <;> simp [setToSimpleFunc] ** Qed", + "informal": "" + }, + { + "formal": "Filter.HasAntitoneBasis.subbasis_with_rel ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 r (\u03c6 m) (\u03c6 n)) \u2227 HasAntitoneBasis f (s \u2218 \u03c6) ** suffices : \u2203 \u03c6 : \u2115 \u2192 \u2115, StrictMono \u03c6 \u2227 \u2200 m n, m < n \u2192 r (\u03c6 m) (\u03c6 n) ** case this \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n) ** have : \u2200 t : Set \u2115, t.Finite \u2192 \u2200\u1da0 n in atTop, \u2200 m \u2208 t, m < n \u2227 r m n := fun t ht =>\n (eventually_all_finite ht).2 fun m _ => (eventually_gt_atTop m).and (hr _) ** case this \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n this : \u2200 (t : Set \u2115), Set.Finite t \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (m : \u2115), m \u2208 t \u2192 m < n \u2227 r m n \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n) ** rcases seq_of_forall_finite_exists fun t ht => (this t ht).exists with \u27e8\u03c6, h\u03c6\u27e9 ** case this.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n this : \u2200 (t : Set \u2115), Set.Finite t \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (m : \u2115), m \u2208 t \u2192 m < n \u2227 r m n \u03c6 : \u2115 \u2192 \u2115 h\u03c6 : \u2200 (n m : \u2115), m \u2208 \u03c6 '' Iio n \u2192 m < \u03c6 n \u2227 r m (\u03c6 n) \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n) ** simp only [ball_image_iff, forall_and, mem_Iio] at h\u03c6 ** case this.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n this : \u2200 (t : Set \u2115), Set.Finite t \u2192 \u2200\u1da0 (n : \u2115) in atTop, \u2200 (m : \u2115), m \u2208 t \u2192 m < n \u2227 r m n \u03c6 : \u2115 \u2192 \u2115 h\u03c6 : (\u2200 (x x_1 : \u2115), x_1 < x \u2192 \u03c6 x_1 < \u03c6 x) \u2227 \u2200 (x x_1 : \u2115), x_1 < x \u2192 r (\u03c6 x_1) (\u03c6 x) \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n) ** exact \u27e8\u03c6, forall_swap.2 h\u03c6.1, forall_swap.2 h\u03c6.2\u27e9 ** \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n this : \u2203 \u03c6, StrictMono \u03c6 \u2227 \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n) \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 r (\u03c6 m) (\u03c6 n)) \u2227 HasAntitoneBasis f (s \u2218 \u03c6) ** rcases this with \u27e8\u03c6, h\u03c6, hr\u03c6\u27e9 ** case intro.intro \u03b9 : Type u_1 \u03b9' : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 \u03b3 : Type u_5 f : Filter \u03b1 s : \u2115 \u2192 Set \u03b1 hs : HasAntitoneBasis f s r : \u2115 \u2192 \u2115 \u2192 Prop hr : \u2200 (m : \u2115), \u2200\u1da0 (n : \u2115) in atTop, r m n \u03c6 : \u2115 \u2192 \u2115 h\u03c6 : StrictMono \u03c6 hr\u03c6 : \u2200 (m n : \u2115), m < n \u2192 r (\u03c6 m) (\u03c6 n) \u22a2 \u2203 \u03c6, StrictMono \u03c6 \u2227 (\u2200 \u2983m n : \u2115\u2984, m < n \u2192 r (\u03c6 m) (\u03c6 n)) \u2227 HasAntitoneBasis f (s \u2218 \u03c6) ** exact \u27e8\u03c6, h\u03c6, hr\u03c6, hs.comp_strictMono h\u03c6\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.snorm_indicator_le ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc f : \u03b1 \u2192 E s : Set \u03b1 \u22a2 snorm (Set.indicator s f) p \u03bc \u2264 snorm f p \u03bc ** refine' snorm_mono_ae (eventually_of_forall fun x => _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc f : \u03b1 \u2192 E s : Set \u03b1 x : \u03b1 \u22a2 \u2016Set.indicator s f x\u2016 \u2264 \u2016f x\u2016 ** suffices \u2016s.indicator f x\u2016\u208a \u2264 \u2016f x\u2016\u208a by exact NNReal.coe_mono this ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc f : \u03b1 \u2192 E s : Set \u03b1 x : \u03b1 \u22a2 \u2016Set.indicator s f x\u2016\u208a \u2264 \u2016f x\u2016\u208a ** rw [nnnorm_indicator_eq_indicator_nnnorm] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc f : \u03b1 \u2192 E s : Set \u03b1 x : \u03b1 \u22a2 Set.indicator s (fun a => \u2016f a\u2016\u208a) x \u2264 \u2016f x\u2016\u208a ** exact s.indicator_le_self _ x ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G c : E f\u271d : \u03b1 \u2192 E hf : AEStronglyMeasurable f\u271d \u03bc f : \u03b1 \u2192 E s : Set \u03b1 x : \u03b1 this : \u2016Set.indicator s f x\u2016\u208a \u2264 \u2016f x\u2016\u208a \u22a2 \u2016Set.indicator s f x\u2016 \u2264 \u2016f x\u2016 ** exact NNReal.coe_mono this ** Qed", + "informal": "" + }, + { + "formal": "dist_pi_const ** \u03b1 : Type u \u03b2 : Type v X : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : PseudoMetricSpace \u03b1 \u03c0 : \u03b2 \u2192 Type u_3 inst\u271d\u00b2 : Fintype \u03b2 inst\u271d\u00b9 : (b : \u03b2) \u2192 PseudoMetricSpace (\u03c0 b) inst\u271d : Nonempty \u03b2 a b : \u03b1 \u22a2 (dist (fun x => a) fun x => b) = dist a b ** simpa only [dist_edist] using congr_arg ENNReal.toReal (edist_pi_const a b) ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.lintegral_rpow_funMulInvSnorm_eq_one ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0_lt : 0 < p f : \u03b1 \u2192 \u211d\u22650\u221e hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b\u207b (c : \u03b1), funMulInvSnorm f p \u03bc c ^ p \u2202\u03bc = 1 ** simp_rw [funMulInvSnorm_rpow hp0_lt] ** \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0_lt : 0 < p f : \u03b1 \u2192 \u211d\u22650\u221e hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 \u222b\u207b (c : \u03b1), f c ^ p * (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2202\u03bc = 1 ** rw [lintegral_mul_const', ENNReal.mul_inv_cancel hf_nonzero hf_top] ** case hr \u03b1 : Type u_1 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 p : \u211d hp0_lt : 0 < p f : \u03b1 \u2192 \u211d\u22650\u221e hf_nonzero : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 0 hf_top : \u222b\u207b (a : \u03b1), f a ^ p \u2202\u03bc \u2260 \u22a4 \u22a2 (\u222b\u207b (c : \u03b1), f c ^ p \u2202\u03bc)\u207b\u00b9 \u2260 \u22a4 ** rwa [inv_ne_top] ** Qed", + "informal": "" + }, + { + "formal": "IsFiniteMeasure.lintegral_lt_top_of_bounded_to_ennreal ** \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1\u271d \u03bc\u271d \u03bd : Measure \u03b1\u271d \u03b1 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03bc_fin : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e f_bdd : \u2203 c, \u2200 (x : \u03b1), f x \u2264 \u2191c \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u22a4 ** cases' f_bdd with c hc ** case intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1\u271d \u03bc\u271d \u03bd : Measure \u03b1\u271d \u03b1 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03bc_fin : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e c : \u211d\u22650 hc : \u2200 (x : \u03b1), f x \u2264 \u2191c \u22a2 \u222b\u207b (x : \u03b1), f x \u2202\u03bc < \u22a4 ** apply lt_of_le_of_lt (@lintegral_mono _ _ \u03bc _ _ hc) ** case intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1\u271d \u03bc\u271d \u03bd : Measure \u03b1\u271d \u03b1 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03bc_fin : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e c : \u211d\u22650 hc : \u2200 (x : \u03b1), f x \u2264 \u2191c \u22a2 \u222b\u207b (a : \u03b1), \u2191c \u2202\u03bc < \u22a4 ** rw [lintegral_const] ** case intro \u03b1\u271d : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1\u271d \u03bc\u271d \u03bd : Measure \u03b1\u271d \u03b1 : Type u_5 inst\u271d : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u03bc_fin : IsFiniteMeasure \u03bc f : \u03b1 \u2192 \u211d\u22650\u221e c : \u211d\u22650 hc : \u2200 (x : \u03b1), f x \u2264 \u2191c \u22a2 \u2191c * \u2191\u2191\u03bc univ < \u22a4 ** exact ENNReal.mul_lt_top ENNReal.coe_lt_top.ne \u03bc_fin.measure_univ_lt_top.ne ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.toSignedMeasure_sub_apply ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : IsFiniteMeasure \u03bd i : Set \u03b1 hi : MeasurableSet i \u22a2 \u2191(toSignedMeasure \u03bc - toSignedMeasure \u03bd) i = ENNReal.toReal (\u2191\u2191\u03bc i) - ENNReal.toReal (\u2191\u2191\u03bd i) ** rw [VectorMeasure.sub_apply, toSignedMeasure_apply_measurable hi,\n Measure.toSignedMeasure_apply_measurable hi, sub_eq_add_neg] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_scaleRoots ** R : Type u_1 S : Type u_2 A : Type u_3 K : Type u_4 inst\u271d\u00b9 : Semiring R inst\u271d : Semiring S p : R[X] s : R i : \u2115 \u22a2 coeff (scaleRoots p s) i = coeff p i * s ^ (natDegree p - i) ** simp (config := { contextual := true }) [scaleRoots, coeff_monomial] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.ker_liftQ_eq_bot ** R : Type u_1 M : Type u_2 r : R x y : M inst\u271d\u2075 : Ring R inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : Module R M p p' : Submodule R M R\u2082 : Type u_3 M\u2082 : Type u_4 inst\u271d\u00b2 : Ring R\u2082 inst\u271d\u00b9 : AddCommGroup M\u2082 inst\u271d : Module R\u2082 M\u2082 \u03c4\u2081\u2082 : R \u2192+* R\u2082 q : Submodule R\u2082 M\u2082 f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082 h : p \u2264 ker f h' : ker f \u2264 p \u22a2 ker (liftQ p f h) = \u22a5 ** rw [ker_liftQ, le_antisymm h h', mkQ_map_self] ** Qed", + "informal": "" + }, + { + "formal": "PrimeMultiset.prod_smul ** d : \u2115 u : PrimeMultiset \u22a2 prod (d \u2022 u) = Pow.pow (prod u) d ** induction' d with n ih ** case zero u : PrimeMultiset \u22a2 prod (Nat.zero \u2022 u) = Pow.pow (prod u) Nat.zero ** rfl ** case succ u : PrimeMultiset n : \u2115 ih : prod (n \u2022 u) = Pow.pow (prod u) n \u22a2 prod (Nat.succ n \u2022 u) = Pow.pow (prod u) (Nat.succ n) ** have : \u2200 n' : \u2115, Pow.pow (prod u) n' = Monoid.npow n' (prod u) := fun _ \u21a6 rfl ** case succ u : PrimeMultiset n : \u2115 ih : prod (n \u2022 u) = Pow.pow (prod u) n this : \u2200 (n' : \u2115), Pow.pow (prod u) n' = Monoid.npow n' (prod u) \u22a2 prod (Nat.succ n \u2022 u) = Pow.pow (prod u) (Nat.succ n) ** rw [succ_nsmul, prod_add, ih, this, this, Monoid.npow_succ, mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "MvPFunctor.wPathCasesOn_eta ** n : \u2115 P : MvPFunctor.{u} (n + 1) \u03b1 : TypeVec.{u_1} n a : P.A f : PFunctor.B (last P) a \u2192 PFunctor.W (last P) h : WPath P (WType.mk a f) \u27f9 \u03b1 \u22a2 wPathCasesOn P (wPathDestLeft P h) (wPathDestRight P h) = h ** ext i x ** case a.h n : \u2115 P : MvPFunctor.{u} (n + 1) \u03b1 : TypeVec.{u_1} n a : P.A f : PFunctor.B (last P) a \u2192 PFunctor.W (last P) h : WPath P (WType.mk a f) \u27f9 \u03b1 i : Fin2 n x : WPath P (WType.mk a fun j => f j) i \u22a2 wPathCasesOn P (wPathDestLeft P h) (wPathDestRight P h) i x = h i x ** cases x <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "Submodule.mapQ_comp ** R : Type u_1 M : Type u_2 r : R x y : M inst\u271d\u2079 : Ring R inst\u271d\u2078 : AddCommGroup M inst\u271d\u2077 : Module R M p p' : Submodule R M R\u2082 : Type u_3 M\u2082 : Type u_4 inst\u271d\u2076 : Ring R\u2082 inst\u271d\u2075 : AddCommGroup M\u2082 inst\u271d\u2074 : Module R\u2082 M\u2082 \u03c4\u2081\u2082 : R \u2192+* R\u2082 q : Submodule R\u2082 M\u2082 R\u2083 : Type u_5 M\u2083 : Type u_6 inst\u271d\u00b3 : Ring R\u2083 inst\u271d\u00b2 : AddCommGroup M\u2083 inst\u271d\u00b9 : Module R\u2083 M\u2083 p\u2082 : Submodule R\u2082 M\u2082 p\u2083 : Submodule R\u2083 M\u2083 \u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083 \u03c4\u2081\u2083 : R \u2192+* R\u2083 inst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083 f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082 g : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083 hf : p \u2264 comap f p\u2082 hg : p\u2082 \u2264 comap g p\u2083 h : optParam (p \u2264 comap f (comap g p\u2083)) (_ : p \u2264 comap f (comap g p\u2083)) \u22a2 mapQ p p\u2083 (comp g f) h = comp (mapQ p\u2082 p\u2083 g hg) (mapQ p p\u2082 f hf) ** ext ** case h.h R : Type u_1 M : Type u_2 r : R x y : M inst\u271d\u2079 : Ring R inst\u271d\u2078 : AddCommGroup M inst\u271d\u2077 : Module R M p p' : Submodule R M R\u2082 : Type u_3 M\u2082 : Type u_4 inst\u271d\u2076 : Ring R\u2082 inst\u271d\u2075 : AddCommGroup M\u2082 inst\u271d\u2074 : Module R\u2082 M\u2082 \u03c4\u2081\u2082 : R \u2192+* R\u2082 q : Submodule R\u2082 M\u2082 R\u2083 : Type u_5 M\u2083 : Type u_6 inst\u271d\u00b3 : Ring R\u2083 inst\u271d\u00b2 : AddCommGroup M\u2083 inst\u271d\u00b9 : Module R\u2083 M\u2083 p\u2082 : Submodule R\u2082 M\u2082 p\u2083 : Submodule R\u2083 M\u2083 \u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083 \u03c4\u2081\u2083 : R \u2192+* R\u2083 inst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083 f : M \u2192\u209b\u2097[\u03c4\u2081\u2082] M\u2082 g : M\u2082 \u2192\u209b\u2097[\u03c4\u2082\u2083] M\u2083 hf : p \u2264 comap f p\u2082 hg : p\u2082 \u2264 comap g p\u2083 h : optParam (p \u2264 comap f (comap g p\u2083)) (_ : p \u2264 comap f (comap g p\u2083)) x\u271d : M \u22a2 \u2191(comp (mapQ p p\u2083 (comp g f) h) (mkQ p)) x\u271d = \u2191(comp (comp (mapQ p\u2082 p\u2083 g hg) (mapQ p p\u2082 f hf)) (mkQ p)) x\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "List.next_get ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x : \u03b1 x\u271d : Nodup [] i : Fin (length []) \u22a2 next [] (get [] i) (_ : get [] { val := \u2191i, isLt := (_ : \u2191i < length []) } \u2208 []) = get [] { val := (\u2191i + 1) % length [], isLt := (_ : (\u2191i + 1) % length [] < length []) } ** simpa using i.2 ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x head\u271d : \u03b1 x\u271d\u00b9 : Nodup [head\u271d] x\u271d : Fin (length [head\u271d]) \u22a2 next [head\u271d] (get [head\u271d] x\u271d) (_ : get [head\u271d] { val := \u2191x\u271d, isLt := (_ : \u2191x\u271d < length [head\u271d]) } \u2208 [head\u271d]) = get [head\u271d] { val := (\u2191x\u271d + 1) % length [head\u271d], isLt := (_ : (\u2191x\u271d + 1) % length [head\u271d] < length [head\u271d]) } ** simp ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 _h : Nodup (x :: y :: l) h0 : 0 < length (x :: y :: l) \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := 0, isLt := h0 }) (_ : get (x :: y :: l) { val := \u2191{ val := 0, isLt := h0 }, isLt := (_ : \u2191{ val := 0, isLt := h0 } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** have h\u2081 : get (x :: y :: l) { val := 0, isLt := h0 } = x := by simp ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 _h : Nodup (x :: y :: l) h0 : 0 < length (x :: y :: l) h\u2081 : get (x :: y :: l) { val := 0, isLt := h0 } = x \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := 0, isLt := h0 }) (_ : get (x :: y :: l) { val := \u2191{ val := 0, isLt := h0 }, isLt := (_ : \u2191{ val := 0, isLt := h0 } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** rw [next_cons_cons_eq' _ _ _ _ _ h\u2081] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 _h : Nodup (x :: y :: l) h0 : 0 < length (x :: y :: l) h\u2081 : get (x :: y :: l) { val := 0, isLt := h0 } = x \u22a2 y = get (x :: y :: l) { val := (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := 0, isLt := h0 } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** simp ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 _h : Nodup (x :: y :: l) h0 : 0 < length (x :: y :: l) \u22a2 get (x :: y :: l) { val := 0, isLt := h0 } = x ** simp ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi }) (_ : get (x :: y :: l) { val := \u2191{ val := i + 1, isLt := hi }, isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** have hx' : (x :: y :: l).get \u27e8i+1, hi\u27e9 \u2260 x := by\n intro H\n suffices (i + 1 : \u2115) = 0 by simpa\n rw [nodup_iff_injective_get] at hn\n refine' Fin.veq_of_eq (@hn \u27e8i + 1, hi\u27e9 \u27e80, by simp\u27e9 _)\n simpa using H ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi }) (_ : get (x :: y :: l) { val := \u2191{ val := i + 1, isLt := hi }, isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** have hi' : i \u2264 l.length := Nat.le_of_lt_succ (Nat.succ_lt_succ_iff.1 hi) ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi' : i \u2264 length l \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi }) (_ : get (x :: y :: l) { val := \u2191{ val := i + 1, isLt := hi }, isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** rcases hi'.eq_or_lt with (hi' | hi') ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x ** intro H ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) H : get (x :: y :: l) { val := i + 1, isLt := hi } = x \u22a2 False ** suffices (i + 1 : \u2115) = 0 by simpa ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) H : get (x :: y :: l) { val := i + 1, isLt := hi } = x \u22a2 i + 1 = 0 ** rw [nodup_iff_injective_get] at hn ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Function.Injective (get (x :: y :: l)) i : \u2115 hi : i + 1 < length (x :: y :: l) H : get (x :: y :: l) { val := i + 1, isLt := hi } = x \u22a2 i + 1 = 0 ** refine' Fin.veq_of_eq (@hn \u27e8i + 1, hi\u27e9 \u27e80, by simp\u27e9 _) ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Function.Injective (get (x :: y :: l)) i : \u2115 hi : i + 1 < length (x :: y :: l) H : get (x :: y :: l) { val := i + 1, isLt := hi } = x \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } = get (x :: y :: l) { val := 0, isLt := (_ : 0 < Nat.succ (length l + 1)) } ** simpa using H ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) H : get (x :: y :: l) { val := i + 1, isLt := hi } = x this : i + 1 = 0 \u22a2 False ** simpa ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Function.Injective (get (x :: y :: l)) i : \u2115 hi : i + 1 < length (x :: y :: l) H : get (x :: y :: l) { val := i + 1, isLt := hi } = x \u22a2 0 < length (x :: y :: l) ** simp ** case inl \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i = length l \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi }) (_ : get (x :: y :: l) { val := \u2191{ val := i + 1, isLt := hi }, isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** subst hi' ** case inl \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := length l + 1, isLt := hi }) (_ : get (x :: y :: l) { val := \u2191{ val := length l + 1, isLt := hi }, isLt := (_ : \u2191{ val := length l + 1, isLt := hi } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** rw [next_getLast_cons] ** case inl \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 x = get (x :: y :: l) { val := (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := length l + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** simp [hi', get] ** case inl.h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 get (x :: y :: l) { val := length l + 1, isLt := hi } \u2208 y :: l ** rw [get_cons_succ] ** case inl.h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 get (y :: l) { val := length l, isLt := (_ : length l < length (y :: l)) } \u2208 y :: l ** exact get_mem _ _ _ ** case inl.hy \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x ** exact hx' ** case inl.hx \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 get (x :: y :: l) { val := length l + 1, isLt := hi } = getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) ** simp [getLast_eq_get] ** case inl.hl \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) hi : length l + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := length l + 1, isLt := hi } \u2260 x hi' : length l \u2264 length l \u22a2 Nodup (y :: l) ** exact hn.of_cons ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 next (x :: y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi }) (_ : get (x :: y :: l) { val := \u2191{ val := i + 1, isLt := hi }, isLt := (_ : \u2191{ val := i + 1, isLt := hi } < length (x :: y :: l)) } \u2208 x :: y :: l) = get (x :: y :: l) { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } ** rw [next_ne_head_ne_getLast _ _ _ _ _ hx'] ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 next (y :: l) (get (x :: y :: l) { val := i + 1, isLt := hi }) (_ : get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l) = get (x :: y :: l) { val := (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l ** simp only [get_cons_succ] ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 next (y :: l) (get (y :: l) { val := i, isLt := (_ : i < length (y :: l)) }) (_ : get (y :: l) { val := i, isLt := (_ : i < length (y :: l)) } \u2208 y :: l) = get (x :: y :: l) { val := (i + 1 + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l ** rw [next_get (y::l), \u2190 get_cons_succ (a := x)] ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1, isLt := ?m.109878 } = get (x :: y :: l) { val := (i + 1 + 1) % length (x :: y :: l), isLt := (_ : (\u2191{ val := i + 1, isLt := hi } + 1) % length (x :: y :: l) < length (x :: y :: l)) } \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l) case inr._h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 Nodup (y :: l) case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l ** congr ** case inr.e_a.e_val \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 = (i + 1 + 1) % length (x :: y :: l) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l) case inr._h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 Nodup (y :: l) case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l ** dsimp ** case inr.e_a.e_val \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 (i + 1) % Nat.succ (length l) + 1 = (i + 1 + 1) % Nat.succ (Nat.succ (length l)) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l) case inr._h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 Nodup (y :: l) case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l ** rw [Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 hi'),\n Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 (Nat.succ_lt_succ_iff.2 hi'))] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 (\u2191{ val := i, isLt := (_ : i < length (y :: l)) } + 1) % length (y :: l) + 1 < length (x :: y :: l) ** simp [Nat.mod_eq_of_lt (Nat.succ_lt_succ_iff.2 hi'), Nat.succ_eq_add_one, hi'] ** case inr._h \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 Nodup (y :: l) ** exact hn.of_cons ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 getLast (x :: y :: l) (_ : x :: y :: l \u2260 []) ** rw [getLast_eq_get] ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) } ** intro h ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l h : get (x :: y :: l) { val := i + 1, isLt := hi } = get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) } \u22a2 False ** have := nodup_iff_injective_get.1 hn h ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l h : get (x :: y :: l) { val := i + 1, isLt := hi } = get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) } this : { val := i + 1, isLt := hi } = { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) } \u22a2 False ** simp at this ** case inr \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l h : get (x :: y :: l) { val := i + 1, isLt := hi } = get (x :: y :: l) { val := length (x :: y :: l) - 1, isLt := (_ : length (x :: y :: l) - 1 < length (x :: y :: l)) } this : i = length l \u22a2 False ** simp [this] at hi' ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (x :: y :: l) { val := i + 1, isLt := hi } \u2208 y :: l ** rw [get_cons_succ] ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 l\u271d : List \u03b1 x\u271d x y : \u03b1 l : List \u03b1 hn : Nodup (x :: y :: l) i : \u2115 hi : i + 1 < length (x :: y :: l) hx' : get (x :: y :: l) { val := i + 1, isLt := hi } \u2260 x hi'\u271d : i \u2264 length l hi' : i < length l \u22a2 get (y :: l) { val := i, isLt := (_ : i < length (y :: l)) } \u2208 y :: l ** exact get_mem _ _ _ ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.mem_iff_ofSubtype_apply_mem ** \u03b1 : Type u \u03b2 : Type v p : \u03b1 \u2192 Prop f\u271d : Perm \u03b1 inst\u271d : DecidablePred p a : \u03b1 f : Perm (Subtype p) x : \u03b1 h : p x \u22a2 p x \u2194 p (\u2191(\u2191ofSubtype f) x) ** simpa only [h, true_iff_iff, MonoidHom.coe_mk, ofSubtype_apply_of_mem f h] using (f \u27e8x, h\u27e9).2 ** \u03b1 : Type u \u03b2 : Type v p : \u03b1 \u2192 Prop f\u271d : Perm \u03b1 inst\u271d : DecidablePred p a : \u03b1 f : Perm (Subtype p) x : \u03b1 h : \u00acp x \u22a2 p x \u2194 p (\u2191(\u2191ofSubtype f) x) ** simp [h, ofSubtype_apply_of_not_mem f h] ** Qed", + "informal": "" + }, + { + "formal": "Int.sub_ediv_of_dvd ** a b c : Int hcb : c \u2223 b \u22a2 (a - b) / c = a / c - b / c ** rw [Int.sub_eq_add_neg, Int.sub_eq_add_neg, Int.add_ediv_of_dvd_right (Int.dvd_neg.2 hcb)] ** a b c : Int hcb : c \u2223 b \u22a2 a / c + -b / c = a / c + -(b / c) ** congr ** case e_a a b c : Int hcb : c \u2223 b \u22a2 -b / c = -(b / c) ** exact Int.neg_ediv_of_dvd hcb ** Qed", + "informal": "" + }, + { + "formal": "FormalMultilinearSeries.leftInv_comp ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i \u22a2 FormalMultilinearSeries.comp (leftInv p i) p = id \ud835\udd5c E ** ext (n v) ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n : \u2115 v : Fin 0 \u2192 E \u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p 0) v = \u2191(id \ud835\udd5c E 0) v ** simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne.def,\n not_false_iff, zero_ne_one, comp_coeff_zero'] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n : \u2115 v : Fin 1 \u2192 E \u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p 1) v = \u2191(id \ud835\udd5c E 1) v ** simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,\n ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFin1_symm_apply] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} \u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v ** have B :\n Disjoint ({c | Composition.length c < n + 2} : Set (Composition (n + 2))).toFinset\n {Composition.ones (n + 2)} := by\n simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} \u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v ** have C :\n ((p.leftInv i (Composition.ones (n + 2)).length)\n fun j : Fin (Composition.ones n.succ.succ).length =>\n p 1 fun _ => v ((Fin.castLE (Composition.length_le _)) j)) =\n p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j := by\n apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_\n exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j \u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v ** have D :\n (p.leftInv i (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =\n -\u2211 c : Composition (n + 2) in {c : Composition (n + 2) | c.length < n + 2}.toFinset,\n (p.leftInv i c.length) (p.applyComposition c v) := by\n simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj,\n ContinuousMultilinearMap.sum_apply]\n convert\n (sum_toFinset_eq_subtype\n (fun c : Composition (n + 2) => c.length < n + 2)\n (fun c : Composition (n + 2) =>\n (ContinuousMultilinearMap.compAlongComposition\n (p.compContinuousLinearMap (i.symm : F \u2192L[\ud835\udd5c] E)) c (p.leftInv i c.length))\n fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans\n _\n simp only [compContinuousLinearMap_applyComposition,\n ContinuousMultilinearMap.compAlongComposition_apply]\n congr\n ext c\n congr\n ext k\n simp [h, Function.comp] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j D : (\u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j) = -\u2211 c in Set.toFinset {c | Composition.length c < n + 2}, \u2191(leftInv p i (Composition.length c)) (applyComposition p c v) \u22a2 \u2191(FormalMultilinearSeries.comp (leftInv p i) p (n + 2)) v = \u2191(id \ud835\udd5c E (n + 2)) v ** simp [FormalMultilinearSeries.comp, show n + 2 \u2260 1 by norm_num, A, Finset.sum_union B,\n applyComposition_ones, C, D, -Set.toFinset_setOf] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E \u22a2 univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} ** refine' Subset.antisymm (fun c _ => _) (subset_univ _) ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E c : Composition (n + 2) x\u271d : c \u2208 univ \u22a2 c \u2208 Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} ** by_cases h : c.length < n + 2 ** case pos \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h\u271d : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E c : Composition (n + 2) x\u271d : c \u2208 univ h : Composition.length c < n + 2 \u22a2 c \u2208 Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} ** simp [h, Set.mem_toFinset (s := {c | Composition.length c < n + 2})] ** case neg \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h\u271d : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E c : Composition (n + 2) x\u271d : c \u2208 univ h : \u00acComposition.length c < n + 2 \u22a2 c \u2208 Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} ** simp [Composition.eq_ones_iff_le_length.2 (not_lt.1 h)] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} \u22a2 Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} ** simp [Set.mem_toFinset (s := {c | Composition.length c < n + 2})] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} \u22a2 (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j ** apply FormalMultilinearSeries.congr _ (Composition.ones_length _) fun j hj1 hj2 => ?_ ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} j : \u2115 hj1 : j < Composition.length (Composition.ones (n + 2)) hj2 : j < n + 2 \u22a2 (\u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) { val := j, isLt := hj1 })) = \u2191(p 1) fun x => v { val := j, isLt := hj2 } ** exact FormalMultilinearSeries.congr _ rfl fun k _ _ => by congr ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} j : \u2115 hj1 : j < Composition.length (Composition.ones (n + 2)) hj2 : j < n + 2 k : \u2115 x\u271d\u00b9 x\u271d : k < 1 \u22a2 v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) { val := j, isLt := hj1 }) = v { val := j, isLt := hj2 } ** congr ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j \u22a2 (\u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j) = -\u2211 c in Set.toFinset {c | Composition.length c < n + 2}, \u2191(leftInv p i (Composition.length c)) (applyComposition p c v) ** simp only [leftInv, ContinuousMultilinearMap.neg_apply, neg_inj,\n ContinuousMultilinearMap.sum_apply] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j \u22a2 (\u2211 x : { c // Composition.length c < n + 2 }, \u2191(ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) (\u2191x) (leftInv p i (Composition.length \u2191x))) fun j => \u2191(p 1) fun x => v j) = \u2211 x in Set.toFinset {c | Composition.length c < n + 2}, \u2191(leftInv p i (Composition.length x)) (applyComposition p x v) ** convert\n (sum_toFinset_eq_subtype\n (fun c : Composition (n + 2) => c.length < n + 2)\n (fun c : Composition (n + 2) =>\n (ContinuousMultilinearMap.compAlongComposition\n (p.compContinuousLinearMap (i.symm : F \u2192L[\ud835\udd5c] E)) c (p.leftInv i c.length))\n fun j : Fin (n + 2) => p 1 fun _ : Fin 1 => v j)).symm.trans\n _ ** case convert_2 \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j \u22a2 (\u2211 a in Set.toFinset {x | Composition.length x < n + 2}, \u2191(ContinuousMultilinearMap.compAlongComposition (compContinuousLinearMap p \u2191(ContinuousLinearEquiv.symm i)) a (leftInv p i (Composition.length a))) fun j => \u2191(p 1) fun x => v j) = \u2211 x in Set.toFinset {c | Composition.length c < n + 2}, \u2191(leftInv p i (Composition.length x)) (applyComposition p x v) ** simp only [compContinuousLinearMap_applyComposition,\n ContinuousMultilinearMap.compAlongComposition_apply] ** case convert_2 \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j \u22a2 \u2211 x in Set.toFinset {x | Composition.length x < n + 2}, \u2191(leftInv p i (Composition.length x)) (applyComposition p x (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j)) = \u2211 x in Set.toFinset {x | Composition.length x < n + 2}, \u2191(leftInv p i (Composition.length x)) (applyComposition p x v) ** congr ** case convert_2.e_f \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j \u22a2 (fun x => \u2191(leftInv p i (Composition.length x)) (applyComposition p x (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j))) = fun x => \u2191(leftInv p i (Composition.length x)) (applyComposition p x v) ** ext c ** case convert_2.e_f.h \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j c : Composition (n + 2) \u22a2 \u2191(leftInv p i (Composition.length c)) (applyComposition p c (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j)) = \u2191(leftInv p i (Composition.length c)) (applyComposition p c v) ** congr ** case convert_2.e_f.h.h.e_6.h \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j c : Composition (n + 2) \u22a2 applyComposition p c (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j) = applyComposition p c v ** ext k ** case convert_2.e_f.h.h.e_6.h.h \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j c : Composition (n + 2) k : Fin (Composition.length c) \u22a2 applyComposition p c (\u2191\u2191(ContinuousLinearEquiv.symm i) \u2218 fun j => \u2191(p 1) fun x => v j) k = applyComposition p c v k ** simp [h, Function.comp] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i n\u271d n : \u2115 v : Fin (n + 2) \u2192 E A : univ = Set.toFinset {c | Composition.length c < n + 2} \u222a {Composition.ones (n + 2)} B : Disjoint (Set.toFinset {c | Composition.length c < n + 2}) {Composition.ones (n + 2)} C : (\u2191(leftInv p i (Composition.length (Composition.ones (n + 2)))) fun j => \u2191(p 1) fun x => v (Fin.castLE (_ : Composition.length (Composition.ones (Nat.succ (Nat.succ n))) \u2264 n + 2) j)) = \u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j D : (\u2191(leftInv p i (n + 2)) fun j => \u2191(p 1) fun x => v j) = -\u2211 c in Set.toFinset {c | Composition.length c < n + 2}, \u2191(leftInv p i (Composition.length c)) (applyComposition p c v) \u22a2 n + 2 \u2260 1 ** norm_num ** Qed", + "informal": "" + }, + { + "formal": "List.perm_merge ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r \u22a2 merge r [] [] ~ [] ++ [] ** simp [merge] ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r b : \u03b1 l' : List \u03b1 \u22a2 merge r [] (b :: l') ~ [] ++ b :: l' ** simp [merge] ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a : \u03b1 l : List \u03b1 \u22a2 merge r (a :: l) [] ~ a :: l ++ [] ** simp [merge] ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a : \u03b1 l : List \u03b1 b : \u03b1 l' : List \u03b1 \u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l' ** by_cases h : a \u227c b ** case pos \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a : \u03b1 l : List \u03b1 b : \u03b1 l' : List \u03b1 h : r a b \u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l' ** simpa [merge, h] using perm_merge _ _ ** case neg \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a : \u03b1 l : List \u03b1 b : \u03b1 l' : List \u03b1 h : \u00acr a b \u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l' ** suffices b :: merge r (a :: l) l' ~ a :: (l ++ b :: l') by simpa [merge, h] ** case neg \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a : \u03b1 l : List \u03b1 b : \u03b1 l' : List \u03b1 h : \u00acr a b \u22a2 b :: merge r (a :: l) l' ~ a :: (l ++ b :: l') ** exact ((perm_merge _ _).cons _).trans ((swap _ _ _).trans (perm_middle.symm.cons _)) ** \u03b1 : Type uu r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : DecidableRel r a : \u03b1 l : List \u03b1 b : \u03b1 l' : List \u03b1 h : \u00acr a b this : b :: merge r (a :: l) l' ~ a :: (l ++ b :: l') \u22a2 merge r (a :: l) (b :: l') ~ a :: l ++ b :: l' ** simpa [merge, h] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.det_smul_inv_vecMul_eq_cramer_transpose ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq n inst\u271d : CommRing \u03b1 A\u271d B A : Matrix n n \u03b1 b : n \u2192 \u03b1 h : IsUnit (det A) \u22a2 det A \u2022 vecMul b A\u207b\u00b9 = \u2191(cramer A\u1d40) b ** rw [\u2190 A\u207b\u00b9.transpose_transpose, vecMul_transpose, transpose_nonsing_inv, \u2190 det_transpose,\n A\u1d40.det_smul_inv_mulVec_eq_cramer _ (isUnit_det_transpose A h)] ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.pi_mem_of_mulSingle_mem ** G : Type u_1 inst\u271d\u2074 : Group G A : Type u_2 inst\u271d\u00b3 : AddGroup A \u03b7 : Type u_3 f : \u03b7 \u2192 Type u_4 inst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i) inst\u271d\u00b9 : Finite \u03b7 inst\u271d : DecidableEq \u03b7 H : Subgroup ((i : \u03b7) \u2192 f i) x : (i : \u03b7) \u2192 f i h : \u2200 (i : \u03b7), Pi.mulSingle i (x i) \u2208 H \u22a2 x \u2208 H ** cases nonempty_fintype \u03b7 ** case intro G : Type u_1 inst\u271d\u2074 : Group G A : Type u_2 inst\u271d\u00b3 : AddGroup A \u03b7 : Type u_3 f : \u03b7 \u2192 Type u_4 inst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i) inst\u271d\u00b9 : Finite \u03b7 inst\u271d : DecidableEq \u03b7 H : Subgroup ((i : \u03b7) \u2192 f i) x : (i : \u03b7) \u2192 f i h : \u2200 (i : \u03b7), Pi.mulSingle i (x i) \u2208 H val\u271d : Fintype \u03b7 \u22a2 x \u2208 H ** exact pi_mem_of_mulSingle_mem_aux Finset.univ x (by simp) fun i _ => h i ** G : Type u_1 inst\u271d\u2074 : Group G A : Type u_2 inst\u271d\u00b3 : AddGroup A \u03b7 : Type u_3 f : \u03b7 \u2192 Type u_4 inst\u271d\u00b2 : (i : \u03b7) \u2192 Group (f i) inst\u271d\u00b9 : Finite \u03b7 inst\u271d : DecidableEq \u03b7 H : Subgroup ((i : \u03b7) \u2192 f i) x : (i : \u03b7) \u2192 f i h : \u2200 (i : \u03b7), Pi.mulSingle i (x i) \u2208 H val\u271d : Fintype \u03b7 \u22a2 \u2200 (i : \u03b7), \u00aci \u2208 Finset.univ \u2192 x i = 1 ** simp ** Qed", + "informal": "" + }, + { + "formal": "writtenInExtChartAt_chartAt_symm ** \ud835\udd5c : Type u_1 E : Type u_2 M : Type u_3 H : Type u_4 E' : Type u_5 M' : Type u_6 H' : Type u_7 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : TopologicalSpace H inst\u271d\u2076 : TopologicalSpace M f f' : LocalHomeomorph M H I : ModelWithCorners \ud835\udd5c E H inst\u271d\u2075 : NormedAddCommGroup E' inst\u271d\u2074 : NormedSpace \ud835\udd5c E' inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' I' : ModelWithCorners \ud835\udd5c E' H' x\u271d : M s t : Set M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : ChartedSpace H' M' x : M y : E h : y \u2208 (extChartAt I x).target \u22a2 writtenInExtChartAt I I (\u2191(chartAt H x) x) (\u2191(LocalHomeomorph.symm (chartAt H x))) y = y ** simp_all only [mfld_simps] ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.sub_le_integral_of_hasDeriv_right_of_le ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** obtain rfl | a_lt_b := hab.eq_or_lt ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** set s := {t | g b - g t \u2264 \u222b u in t..b, \u03c6 u} \u2229 Icc a b ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** have s_closed : IsClosed s := by\n have : ContinuousOn (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (Icc a b) := by\n rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2\n exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int)\n simp only [inter_comm]\n exact this.preimage_closed_of_closed isClosed_Icc isClosed_le_prod ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** have A : closure (Ioc a b) \u2286 s := by\n apply s_closed.closure_subset_iff.2\n intro t ht\n refine' \u27e8_, \u27e8ht.1.le, ht.2\u27e9\u27e9\n exact\n sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n (fun x hx => hderiv x \u27e8ht.1.trans_le hx.1, hx.2\u27e9)\n (\u03c6int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => h\u03c6g x \u27e8ht.1.trans_le hx.1, hx.2\u27e9 ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s A : closure (Ioc a b) \u2286 s \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** rw [closure_Ioc a_lt_b.ne] at A ** case inr \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A\u271d : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s A : Icc a b \u2286 s \u22a2 g b - g a \u2264 \u222b (y : \u211d) in a..b, \u03c6 y ** exact (A (left_mem_Icc.2 hab)).1 ** case inl \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a : \u211d hab : a \u2264 a hcont : ContinuousOn g (Icc a a) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a a) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a a \u2192 g' x \u2264 \u03c6 x \u22a2 g a - g a \u2264 \u222b (y : \u211d) in a..a, \u03c6 y ** simp ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 IsClosed s ** have : ContinuousOn (fun t => (g b - g t, \u222b u in t..b, \u03c6 u)) (Icc a b) := by\n rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2\n exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b this : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b) \u22a2 IsClosed s ** simp only [inter_comm] ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b this : ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b) \u22a2 IsClosed (Icc a b \u2229 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u}) ** exact this.preimage_closed_of_closed isClosed_Icc isClosed_le_prod ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) (Icc a b) ** rw [\u2190 uIcc_of_le hab] at hcont \u03c6int \u22a2 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g [[a, b]] hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 [[a, b]] h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b \u22a2 ContinuousOn (fun t => (g b - g t, \u222b (u : \u211d) in t..b, \u03c6 u)) [[a, b]] ** exact (continuousOn_const.sub hcont).prod (continuousOn_primitive_interval_left \u03c6int) ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s \u22a2 closure (Ioc a b) \u2286 s ** apply s_closed.closure_subset_iff.2 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s \u22a2 Ioc a b \u2286 s ** intro t ht ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 Ioc a b \u22a2 t \u2208 s ** refine' \u27e8_, \u27e8ht.1.le, ht.2\u27e9\u27e9 ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f : \u211d \u2192 E g' g \u03c6 : \u211d \u2192 \u211d a b : \u211d hab : a \u2264 b hcont : ContinuousOn g (Icc a b) hderiv : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 HasDerivWithinAt g (g' x) (Ioi x) x \u03c6int : IntegrableOn \u03c6 (Icc a b) h\u03c6g : \u2200 (x : \u211d), x \u2208 Ioo a b \u2192 g' x \u2264 \u03c6 x a_lt_b : a < b s : Set \u211d := {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} \u2229 Icc a b s_closed : IsClosed s t : \u211d ht : t \u2208 Ioc a b \u22a2 t \u2208 {t | g b - g t \u2264 \u222b (u : \u211d) in t..b, \u03c6 u} ** exact\n sub_le_integral_of_hasDeriv_right_of_le_Ico ht.2 (hcont.mono (Icc_subset_Icc ht.1.le le_rfl))\n (fun x hx => hderiv x \u27e8ht.1.trans_le hx.1, hx.2\u27e9)\n (\u03c6int.mono_set (Icc_subset_Icc ht.1.le le_rfl)) fun x hx => h\u03c6g x \u27e8ht.1.trans_le hx.1, hx.2\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.setLintegral_eq_top_of_measure_eq_top_ne_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : AEMeasurable f h\u03bcf : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x = \u22a4} \u2260 0 \u22a2 \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x = \u22a4} \u2264 \u2191\u2191(Measure.restrict \u03bc s) {x | f x = \u22a4} ** rw [\u2190setOf_inter_eq_sep] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e s : Set \u03b1 hf : AEMeasurable f h\u03bcf : \u2191\u2191\u03bc {x | x \u2208 s \u2227 f x = \u22a4} \u2260 0 \u22a2 \u2191\u2191\u03bc ({a | f a = \u22a4} \u2229 s) \u2264 \u2191\u2191(Measure.restrict \u03bc s) {x | f x = \u22a4} ** exact Measure.le_restrict_apply _ _ ** Qed", + "informal": "" + }, + { + "formal": "Multiset.range_add_eq_union ** a b : \u2115 \u22a2 range (a + b) = range a \u222a map (fun x => a + x) (range b) ** rw [range_add, add_eq_union_iff_disjoint] ** a b : \u2115 \u22a2 Disjoint (range a) (map (fun x => a + x) (range b)) ** apply range_disjoint_map_add ** Qed", + "informal": "" + }, + { + "formal": "QuadraticForm.nonneg_pi_iff ** \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) \u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2191(pi Q) x) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x ** simp_rw [pi, sum_apply, comp_apply, LinearMap.proj_apply] ** \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) \u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2194 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x ** constructor ** case mp \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) \u22a2 (\u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1)) \u2192 \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x ** intro h i x ** case mp \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) h : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1) i : \u03b9 x : M\u1d62 i \u22a2 0 \u2264 \u2191(Q i) x ** classical\nconvert h (Pi.single i x) using 1\nrw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same]\nrw [Pi.single_eq_of_ne hji, map_zero] ** case mp \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) h : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1) i : \u03b9 x : M\u1d62 i \u22a2 0 \u2264 \u2191(Q i) x ** convert h (Pi.single i x) using 1 ** case h.e'_4 \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) h : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1) i : \u03b9 x : M\u1d62 i \u22a2 \u2191(Q i) x = \u2211 x_1 : \u03b9, \u2191(Q x_1) (Pi.single i x x_1) ** rw [Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hji => ?_, Pi.single_eq_same] ** \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) h : \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1) i : \u03b9 x : M\u1d62 i j : \u03b9 x\u271d : j \u2208 Finset.univ hji : j \u2260 i \u22a2 \u2191(Q j) (Pi.single i x j) = 0 ** rw [Pi.single_eq_of_ne hji, map_zero] ** case mpr \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) \u22a2 (\u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x) \u2192 \u2200 (x : (i : \u03b9) \u2192 M\u1d62 i), 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1) ** rintro h x ** case mpr \u03b9 : Type u_1 R\u271d : Type u_2 M\u2081 : Type u_3 M\u2082 : Type u_4 N\u2081 : Type u_5 N\u2082 : Type u_6 M\u1d62 : \u03b9 \u2192 Type u_7 N\u1d62 : \u03b9 \u2192 Type u_8 inst\u271d\u2077 : CommSemiring R\u271d inst\u271d\u2076 : (i : \u03b9) \u2192 AddCommMonoid (M\u1d62 i) inst\u271d\u2075 : (i : \u03b9) \u2192 AddCommMonoid (N\u1d62 i) inst\u271d\u2074 : (i : \u03b9) \u2192 Module R\u271d (M\u1d62 i) inst\u271d\u00b3 : (i : \u03b9) \u2192 Module R\u271d (N\u1d62 i) inst\u271d\u00b2 : Fintype \u03b9 R : Type u_9 inst\u271d\u00b9 : OrderedCommRing R inst\u271d : (i : \u03b9) \u2192 Module R (M\u1d62 i) Q : (i : \u03b9) \u2192 QuadraticForm R (M\u1d62 i) h : \u2200 (i : \u03b9) (x : M\u1d62 i), 0 \u2264 \u2191(Q i) x x : (i : \u03b9) \u2192 M\u1d62 i \u22a2 0 \u2264 \u2211 x_1 : \u03b9, \u2191(Q x_1) (x x_1) ** exact Finset.sum_nonneg fun i _ => h i (x i) ** Qed", + "informal": "" + }, + { + "formal": "iSup_ne_bot_subtype ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 \u22a2 \u2a06 i, f \u2191i = \u2a06 i, f i ** by_cases htriv : \u2200 i, f i = \u22a5 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5 \u22a2 \u2a06 i, f \u2191i = \u2a06 i, f i ** refine' (iSup_comp_le f _).antisymm (iSup_mono' fun i => _) ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5 i : \u03b9 \u22a2 \u2203 i', f i \u2264 f \u2191i' ** by_cases hi : f i = \u22a5 ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u2200 (i : \u03b9), f i = \u22a5 \u22a2 \u2a06 i, f \u2191i = \u2a06 i, f i ** simp only [iSup_bot, (funext htriv : f = _)] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5 i : \u03b9 hi : f i = \u22a5 \u22a2 \u2203 i', f i \u2264 f \u2191i' ** rw [hi] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5 i : \u03b9 hi : f i = \u22a5 \u22a2 \u2203 i', \u22a5 \u2264 f \u2191i' ** obtain \u27e8i\u2080, hi\u2080\u27e9 := not_forall.mp htriv ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5 i : \u03b9 hi : f i = \u22a5 i\u2080 : \u03b9 hi\u2080 : \u00acf i\u2080 = \u22a5 \u22a2 \u2203 i', \u22a5 \u2264 f \u2191i' ** exact \u27e8\u27e8i\u2080, hi\u2080\u27e9, bot_le\u27e9 ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b2\u2082 : Type u_3 \u03b3 : Type u_4 \u03b9 : Sort u_5 \u03b9' : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba' : \u03b9' \u2192 Sort u_8 inst\u271d : CompleteLattice \u03b1 f\u271d g s t : \u03b9 \u2192 \u03b1 a b : \u03b1 f : \u03b9 \u2192 \u03b1 htriv : \u00ac\u2200 (i : \u03b9), f i = \u22a5 i : \u03b9 hi : \u00acf i = \u22a5 \u22a2 \u2203 i', f i \u2264 f \u2191i' ** exact \u27e8\u27e8i, hi\u27e9, rfl.le\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "smul_eq_zero_iff_eq ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b2 : Group \u03b1 inst\u271d\u00b9 : AddMonoid \u03b2 inst\u271d : DistribMulAction \u03b1 \u03b2 a : \u03b1 x : \u03b2 h : a \u2022 x = 0 \u22a2 x = 0 ** rw [\u2190 inv_smul_smul a x, h, smul_zero] ** Qed", + "informal": "" + }, + { + "formal": "Sylow.smul_eq_of_normal ** p : \u2115 G : Type u_1 inst\u271d : Group G g : G P : Sylow p G h : Normal \u2191P \u22a2 g \u2022 P = P ** simp only [Sylow.smul_eq_iff_mem_normalizer, normalizer_eq_top.mpr h, mem_top] ** Qed", + "informal": "" + }, + { + "formal": "tangentCone_mono_nhds ** \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x \u22a2 tangentConeAt \ud835\udd5c s x \u2286 tangentConeAt \ud835\udd5c t x ** rintro y \u27e8c, d, ds, ctop, clim\u27e9 ** case intro.intro.intro.intro \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y\u271d : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x y : E c : \u2115 \u2192 \ud835\udd5c d : \u2115 \u2192 E ds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s ctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop clim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y) \u22a2 y \u2208 tangentConeAt \ud835\udd5c t x ** refine' \u27e8c, d, _, ctop, clim\u27e9 ** case intro.intro.intro.intro \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y\u271d : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x y : E c : \u2115 \u2192 \ud835\udd5c d : \u2115 \u2192 E ds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s ctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop clim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y) \u22a2 \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 t ** suffices : Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x) ** case intro.intro.intro.intro \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y\u271d : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x y : E c : \u2115 \u2192 \ud835\udd5c d : \u2115 \u2192 E ds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s ctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop clim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y) this : Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x) \u22a2 \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 t case this \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y\u271d : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x y : E c : \u2115 \u2192 \ud835\udd5c d : \u2115 \u2192 E ds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s ctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop clim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y) \u22a2 Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x) ** exact tendsto_principal.1 (tendsto_inf.1 this).2 ** case this \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y\u271d : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x y : E c : \u2115 \u2192 \ud835\udd5c d : \u2115 \u2192 E ds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s ctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop clim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y) \u22a2 Tendsto (fun n => x + d n) atTop (\ud835\udcdd[t] x) ** refine' (tendsto_inf.2 \u27e8_, tendsto_principal.2 ds\u27e9).mono_right h ** case this \ud835\udd5c : Type u_1 inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup F inst\u271d\u00b2 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup G inst\u271d : NormedSpace \u211d G x y\u271d : E s t : Set E h : \ud835\udcdd[s] x \u2264 \ud835\udcdd[t] x y : E c : \u2115 \u2192 \ud835\udd5c d : \u2115 \u2192 E ds : \u2200\u1da0 (n : \u2115) in atTop, x + d n \u2208 s ctop : Tendsto (fun n => \u2016c n\u2016) atTop atTop clim : Tendsto (fun n => c n \u2022 d n) atTop (\ud835\udcdd y) \u22a2 Tendsto (fun a => x + d a) atTop (\ud835\udcdd x) ** simpa only [add_zero] using tendsto_const_nhds.add (tangentConeAt.lim_zero atTop ctop clim) ** Qed", + "informal": "" + }, + { + "formal": "iInter_halfspaces_eq ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2075 : TopologicalSpace E inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : TopologicalAddGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : ContinuousSMul \u211d E s t : Set E x y : E inst\u271d : LocallyConvexSpace \u211d E hs\u2081 : Convex \u211d s hs\u2082 : IsClosed s \u22a2 \u22c2 l, {x | \u2203 y, y \u2208 s \u2227 \u2191l x \u2264 \u2191l y} = s ** rw [Set.iInter_setOf] ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2075 : TopologicalSpace E inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : TopologicalAddGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : ContinuousSMul \u211d E s t : Set E x y : E inst\u271d : LocallyConvexSpace \u211d E hs\u2081 : Convex \u211d s hs\u2082 : IsClosed s \u22a2 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s \u2227 \u2191i x \u2264 \u2191i y} = s ** refine' Set.Subset.antisymm (fun x hx => _) fun x hx l => \u27e8x, hx, le_rfl\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2075 : TopologicalSpace E inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : TopologicalAddGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : ContinuousSMul \u211d E s t : Set E x\u271d y : E inst\u271d : LocallyConvexSpace \u211d E hs\u2081 : Convex \u211d s hs\u2082 : IsClosed s x : E hx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s \u2227 \u2191i x \u2264 \u2191i y} \u22a2 x \u2208 s ** by_contra h ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2075 : TopologicalSpace E inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : TopologicalAddGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : ContinuousSMul \u211d E s t : Set E x\u271d y : E inst\u271d : LocallyConvexSpace \u211d E hs\u2081 : Convex \u211d s hs\u2082 : IsClosed s x : E hx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s \u2227 \u2191i x \u2264 \u2191i y} h : \u00acx \u2208 s \u22a2 False ** obtain \u27e8l, s, hlA, hl\u27e9 := geometric_hahn_banach_closed_point hs\u2081 hs\u2082 h ** case intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2075 : TopologicalSpace E inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : TopologicalAddGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : ContinuousSMul \u211d E s\u271d t : Set E x\u271d y : E inst\u271d : LocallyConvexSpace \u211d E hs\u2081 : Convex \u211d s\u271d hs\u2082 : IsClosed s\u271d x : E hx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s\u271d \u2227 \u2191i x \u2264 \u2191i y} h : \u00acx \u2208 s\u271d l : E \u2192L[\u211d] \u211d s : \u211d hlA : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191l a < s hl : s < \u2191l x \u22a2 False ** obtain \u27e8y, hy, hxy\u27e9 := hx l ** case intro.intro.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2075 : TopologicalSpace E inst\u271d\u2074 : AddCommGroup E inst\u271d\u00b3 : TopologicalAddGroup E inst\u271d\u00b2 : Module \u211d E inst\u271d\u00b9 : ContinuousSMul \u211d E s\u271d t : Set E x\u271d y\u271d : E inst\u271d : LocallyConvexSpace \u211d E hs\u2081 : Convex \u211d s\u271d hs\u2082 : IsClosed s\u271d x : E hx : x \u2208 {x | \u2200 (i : E \u2192L[\u211d] \u211d), \u2203 y, y \u2208 s\u271d \u2227 \u2191i x \u2264 \u2191i y} h : \u00acx \u2208 s\u271d l : E \u2192L[\u211d] \u211d s : \u211d hlA : \u2200 (a : E), a \u2208 s\u271d \u2192 \u2191l a < s hl : s < \u2191l x y : E hy : y \u2208 s\u271d hxy : \u2191l x \u2264 \u2191l y \u22a2 False ** exact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl ** Qed", + "informal": "" + }, + { + "formal": "Function.Surjective.lieModule_lcs_map_eq ** R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 \u22a2 Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) ** induction' k with k ih ** case zero R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 \u22a2 Submodule.map g \u2191(lowerCentralSeries R L M Nat.zero) = \u2191(lowerCentralSeries R L\u2082 M\u2082 Nat.zero) ** simpa [LinearMap.range_eq_top] ** case succ R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) \u22a2 Submodule.map g \u2191(lowerCentralSeries R L M (Nat.succ k)) = \u2191(lowerCentralSeries R L\u2082 M\u2082 (Nat.succ k)) ** suffices\n g '' {m | \u2203 (x : L) (n : _), n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} =\n {m | \u2203 (x : L\u2082) (n : _), n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, g n\u2046 = m} by\n simp only [\u2190 LieSubmodule.mem_coeSubmodule] at this\n simp_rw [lowerCentralSeries_succ, LieSubmodule.lieIdeal_oper_eq_linear_span',\n Submodule.map_span, LieSubmodule.mem_top, true_and, \u2190 LieSubmodule.mem_coeSubmodule, this,\n \u2190 ih, Submodule.mem_map, exists_exists_and_eq_and] ** case succ R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) \u22a2 \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} = {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, \u2191g n\u2046 = m} ** ext m\u2082 ** case succ.h R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) m\u2082 : M\u2082 \u22a2 m\u2082 \u2208 \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} \u2194 m\u2082 \u2208 {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, \u2191g n\u2046 = m} ** constructor ** R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) this : \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} = {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, \u2191g n\u2046 = m} \u22a2 Submodule.map g \u2191(lowerCentralSeries R L M (Nat.succ k)) = \u2191(lowerCentralSeries R L\u2082 M\u2082 (Nat.succ k)) ** simp only [\u2190 LieSubmodule.mem_coeSubmodule] at this ** R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) this : (fun a => \u2191g a) '' {m | \u2203 x n, n \u2208 \u2191(lowerCentralSeries R L M k) \u2227 \u2045x, n\u2046 = m} = {m | \u2203 x n, n \u2208 \u2191(lowerCentralSeries R L M k) \u2227 \u2045x, \u2191g n\u2046 = m} \u22a2 Submodule.map g \u2191(lowerCentralSeries R L M (Nat.succ k)) = \u2191(lowerCentralSeries R L\u2082 M\u2082 (Nat.succ k)) ** simp_rw [lowerCentralSeries_succ, LieSubmodule.lieIdeal_oper_eq_linear_span',\n Submodule.map_span, LieSubmodule.mem_top, true_and, \u2190 LieSubmodule.mem_coeSubmodule, this,\n \u2190 ih, Submodule.mem_map, exists_exists_and_eq_and] ** case succ.h.mp R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) m\u2082 : M\u2082 \u22a2 m\u2082 \u2208 \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} \u2192 m\u2082 \u2208 {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, \u2191g n\u2046 = m} ** rintro \u27e8m, \u27e8x, n, hn, rfl\u27e9, rfl\u27e9 ** case succ.h.mp.intro.intro.intro.intro.intro R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) x : L n : M hn : n \u2208 lowerCentralSeries R L M k \u22a2 \u2191g \u2045x, n\u2046 \u2208 {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, \u2191g n\u2046 = m} ** exact \u27e8f x, n, hn, hfg x n\u27e9 ** case succ.h.mpr R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) m\u2082 : M\u2082 \u22a2 m\u2082 \u2208 {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, \u2191g n\u2046 = m} \u2192 m\u2082 \u2208 \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} ** rintro \u27e8x, n, hn, rfl\u27e9 ** case succ.h.mpr.intro.intro.intro R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) x : L\u2082 n : M hn : n \u2208 lowerCentralSeries R L M k \u22a2 \u2045x, \u2191g n\u2046 \u2208 \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} ** obtain \u27e8y, rfl\u27e9 := hf x ** case succ.h.mpr.intro.intro.intro.intro R : Type u L : Type v M : Type w inst\u271d\u00b9\u00b2 : CommRing R inst\u271d\u00b9\u00b9 : LieRing L inst\u271d\u00b9\u2070 : LieAlgebra R L inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : Module R M inst\u271d\u2077 : LieRingModule L M inst\u271d\u2076 : LieModule R L M k\u271d : \u2115 N : LieSubmodule R L M L\u2082 : Type u_1 M\u2082 : Type u_2 inst\u271d\u2075 : LieRing L\u2082 inst\u271d\u2074 : LieAlgebra R L\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L\u2082 M\u2082 inst\u271d : LieModule R L\u2082 M\u2082 f : L \u2192\u2097\u2045R\u2046 L\u2082 g : M \u2192\u2097[R] M\u2082 hf : Surjective \u2191f hg : Surjective \u2191g hfg : \u2200 (x : L) (m : M), \u2045\u2191f x, \u2191g m\u2046 = \u2191g \u2045x, m\u2046 k : \u2115 ih : Submodule.map g \u2191(lowerCentralSeries R L M k) = \u2191(lowerCentralSeries R L\u2082 M\u2082 k) n : M hn : n \u2208 lowerCentralSeries R L M k y : L \u22a2 \u2045\u2191f y, \u2191g n\u2046 \u2208 \u2191g '' {m | \u2203 x n, n \u2208 lowerCentralSeries R L M k \u2227 \u2045x, n\u2046 = m} ** exact \u27e8\u2045y, n\u2046, \u27e8y, n, hn, rfl\u27e9, (hfg y n).symm\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Set.iUnion_prod_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 s : \u03b9 \u2192 Set \u03b1 t : Set \u03b2 \u22a2 (\u22c3 i, s i) \u00d7\u02e2 t = \u22c3 i, s i \u00d7\u02e2 t ** ext ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 s : \u03b9 \u2192 Set \u03b1 t : Set \u03b2 x\u271d : \u03b1 \u00d7 \u03b2 \u22a2 x\u271d \u2208 (\u22c3 i, s i) \u00d7\u02e2 t \u2194 x\u271d \u2208 \u22c3 i, s i \u00d7\u02e2 t ** simp ** Qed", + "informal": "" + }, + { + "formal": "Equiv.refl_trans ** \u03b1 : Sort u \u03b2 : Sort v \u03b3 : Sort w e : \u03b1 \u2243 \u03b2 \u22a2 (Equiv.refl \u03b1).trans e = e ** cases e ** case mk \u03b1 : Sort u \u03b2 : Sort v \u03b3 : Sort w toFun\u271d : \u03b1 \u2192 \u03b2 invFun\u271d : \u03b2 \u2192 \u03b1 left_inv\u271d : LeftInverse invFun\u271d toFun\u271d right_inv\u271d : Function.RightInverse invFun\u271d toFun\u271d \u22a2 (Equiv.refl \u03b1).trans { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } = { toFun := toFun\u271d, invFun := invFun\u271d, left_inv := left_inv\u271d, right_inv := right_inv\u271d } ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.integral_eq_lintegral_of_nonneg_ae ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** by_cases hfi : Integrable f \u03bc ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** rw [integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi] ** case pos \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f h_min : \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = 0 \u22a2 ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) - ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc) = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** rw [h_min, zero_toReal, _root_.sub_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (-f a) \u2202\u03bc = 0 ** rw [lintegral_eq_zero_iff'] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f \u22a2 (fun a => ENNReal.ofReal (-f a)) =\u1d50[\u03bc] 0 ** refine' hf.mono _ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f \u22a2 \u2200 (x : \u03b1), OfNat.ofNat 0 x \u2264 f x \u2192 (fun a => ENNReal.ofReal (-f a)) x = OfNat.ofNat 0 x ** simp only [Pi.zero_apply] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f \u22a2 \u2200 (x : \u03b1), 0 \u2264 f x \u2192 ENNReal.ofReal (-f x) = 0 ** intro a h ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f a : \u03b1 h : 0 \u2264 f a \u22a2 ENNReal.ofReal (-f a) = 0 ** simp only [h, neg_nonpos, ofReal_eq_zero] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : Integrable f \u22a2 AEMeasurable fun a => ENNReal.ofReal (-f a) ** exact measurable_ofReal.comp_aemeasurable hfm.aemeasurable.neg ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u00acIntegrable f \u22a2 \u222b (a : \u03b1), f a \u2202\u03bc = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** rw [integral_undef hfi] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u00acIntegrable f \u22a2 0 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** simp_rw [Integrable, hfm, hasFiniteIntegral_iff_norm, lt_top_iff_ne_top, Ne.def, true_and_iff,\n Classical.not_not] at hfi ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc = \u22a4 \u22a2 0 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** have : \u222b\u207b a : \u03b1, ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b a, ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc := by\n refine' lintegral_congr_ae (hf.mono fun a h => _)\n dsimp only\n rw [Real.norm_eq_abs, abs_of_nonneg h] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc = \u22a4 this : \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc \u22a2 0 = ENNReal.toReal (\u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc) ** rw [this, hfi] ** case neg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc = \u22a4 this : \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc \u22a2 0 = ENNReal.toReal \u22a4 ** rfl ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc = \u22a4 \u22a2 \u222b\u207b (a : \u03b1), ENNReal.ofReal (f a) \u2202\u03bc = \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc ** refine' lintegral_congr_ae (hf.mono fun a h => _) ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc = \u22a4 a : \u03b1 h : OfNat.ofNat 0 a \u2264 f a \u22a2 (fun a => ENNReal.ofReal (f a)) a = (fun a => ENNReal.ofReal \u2016f a\u2016) a ** dsimp only ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \u211d E hE : CompleteSpace E inst\u271d\u2079 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : SMulCommClass \u211d \ud835\udd5c E inst\u271d\u2076 : NormedAddCommGroup F inst\u271d\u2075 : NormedSpace \u211d F inst\u271d\u2074 : CompleteSpace F G : Type u_5 inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \u211d G f\u271d g : \u03b1 \u2192 E m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 X : Type u_6 inst\u271d\u00b9 : TopologicalSpace X inst\u271d : FirstCountableTopology X f : \u03b1 \u2192 \u211d hf : 0 \u2264\u1d50[\u03bc] f hfm : AEStronglyMeasurable f \u03bc hfi : \u222b\u207b (a : \u03b1), ENNReal.ofReal \u2016f a\u2016 \u2202\u03bc = \u22a4 a : \u03b1 h : OfNat.ofNat 0 a \u2264 f a \u22a2 ENNReal.ofReal (f a) = ENNReal.ofReal \u2016f a\u2016 ** rw [Real.norm_eq_abs, abs_of_nonneg h] ** Qed", + "informal": "" + }, + { + "formal": "eq_orthogonalProjection_of_mem_orthogonal' ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2076 : IsROrC \ud835\udd5c inst\u271d\u2075 : NormedAddCommGroup E inst\u271d\u2074 : NormedAddCommGroup F inst\u271d\u00b3 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b2 : InnerProductSpace \u211d F K : Submodule \ud835\udd5c E inst\u271d\u00b9 inst\u271d : HasOrthogonalProjection K u v z : E hv : v \u2208 K hz : z \u2208 K\u15ee hu : u = v + z \u22a2 u - v \u2208 K\u15ee ** simpa [hu] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Subgraph.iSup_adj ** \u03b9 : Sort u_1 V : Type u W : Type v G : SimpleGraph V G\u2081 G\u2082 : Subgraph G a b : V f : \u03b9 \u2192 Subgraph G \u22a2 Adj (\u2a06 i, f i) a b \u2194 \u2203 i, Adj (f i) a b ** simp [iSup] ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.IsThreeCycle.alternating_normalClosure ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f \u22a2 \u22a4 \u2264 normalClosure {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} ** have hi : Function.Injective (alternatingGroup \u03b1).subtype := Subtype.coe_injective ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) \u22a2 \u22a4 \u2264 normalClosure {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} ** refine' eq_top_iff.1 (map_injective hi (le_antisymm (map_mono le_top) _)) ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) \u22a2 map (Subgroup.subtype (alternatingGroup \u03b1)) \u22a4 \u2264 map (Subgroup.subtype (alternatingGroup \u03b1)) (normalClosure {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}) ** rw [\u2190 MonoidHom.range_eq_map, subtype_range, normalClosure, MonoidHom.map_closure] ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) \u22a2 alternatingGroup \u03b1 \u2264 closure (\u2191(Subgroup.subtype (alternatingGroup \u03b1)) '' Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }}) ** refine' (le_of_eq closure_three_cycles_eq_alternating.symm).trans (closure_mono _) ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) \u22a2 {\u03c3 | IsThreeCycle \u03c3} \u2286 \u2191(Subgroup.subtype (alternatingGroup \u03b1)) '' Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} ** intro g h ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) g : Perm \u03b1 h : g \u2208 {\u03c3 | IsThreeCycle \u03c3} \u22a2 g \u2208 \u2191(Subgroup.subtype (alternatingGroup \u03b1)) '' Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} ** obtain \u27e8c, rfl\u27e9 := isConj_iff.1 (isConj_iff_cycleType_eq.2 (hf.trans h.symm)) ** case intro \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) c : Perm \u03b1 h : c * f * c\u207b\u00b9 \u2208 {\u03c3 | IsThreeCycle \u03c3} \u22a2 c * f * c\u207b\u00b9 \u2208 \u2191(Subgroup.subtype (alternatingGroup \u03b1)) '' Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} ** refine' \u27e8\u27e8c * f * c\u207b\u00b9, h.mem_alternatingGroup\u27e9, _, rfl\u27e9 ** case intro \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) c : Perm \u03b1 h : c * f * c\u207b\u00b9 \u2208 {\u03c3 | IsThreeCycle \u03c3} \u22a2 { val := c * f * c\u207b\u00b9, property := (_ : c * f * c\u207b\u00b9 \u2208 alternatingGroup \u03b1) } \u2208 Group.conjugatesOfSet {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} ** rw [Group.mem_conjugatesOfSet_iff] ** case intro \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 inst\u271d : DecidableEq \u03b1 h5 : 5 \u2264 card \u03b1 f : Perm \u03b1 hf : IsThreeCycle f hi : Function.Injective \u2191(Subgroup.subtype (alternatingGroup \u03b1)) c : Perm \u03b1 h : c * f * c\u207b\u00b9 \u2208 {\u03c3 | IsThreeCycle \u03c3} \u22a2 \u2203 a, a \u2208 {{ val := f, property := (_ : f \u2208 alternatingGroup \u03b1) }} \u2227 IsConj a { val := c * f * c\u207b\u00b9, property := (_ : c * f * c\u207b\u00b9 \u2208 alternatingGroup \u03b1) } ** exact \u27e8\u27e8f, hf.mem_alternatingGroup\u27e9, Set.mem_singleton _, isThreeCycle_isConj h5 hf h\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Set.sigmaToiUnion_injective ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 t : \u03b1 \u2192 Set \u03b2 h : \u2200 (i j : \u03b1), i \u2260 j \u2192 Disjoint (t i) (t j) a\u2081 : \u03b1 b\u2081 : \u03b2 h\u2081 : b\u2081 \u2208 t a\u2081 a\u2082 : \u03b1 b\u2082 : \u03b2 h\u2082 : b\u2082 \u2208 t a\u2082 eq : sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } } = sigmaToiUnion t { fst := a\u2082, snd := { val := b\u2082, property := h\u2082 } } b_eq : b\u2081 = b\u2082 a_eq : a\u2081 = a\u2082 \u22a2 \u2191(Eq.recOn a_eq { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } }.snd) = \u2191{ fst := a\u2082, snd := { val := b\u2082, property := h\u2082 } }.snd ** subst b_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 t : \u03b1 \u2192 Set \u03b2 h : \u2200 (i j : \u03b1), i \u2260 j \u2192 Disjoint (t i) (t j) a\u2081 : \u03b1 b\u2081 : \u03b2 h\u2081 : b\u2081 \u2208 t a\u2081 a\u2082 : \u03b1 a_eq : a\u2081 = a\u2082 h\u2082 : b\u2081 \u2208 t a\u2082 eq : sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } } = sigmaToiUnion t { fst := a\u2082, snd := { val := b\u2081, property := h\u2082 } } \u22a2 \u2191(Eq.recOn a_eq { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } }.snd) = \u2191{ fst := a\u2082, snd := { val := b\u2081, property := h\u2082 } }.snd ** subst a_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 t : \u03b1 \u2192 Set \u03b2 h : \u2200 (i j : \u03b1), i \u2260 j \u2192 Disjoint (t i) (t j) a\u2081 : \u03b1 b\u2081 : \u03b2 h\u2081 h\u2082 : b\u2081 \u2208 t a\u2081 eq : sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } } = sigmaToiUnion t { fst := a\u2081, snd := { val := b\u2081, property := h\u2082 } } \u22a2 \u2191(Eq.recOn (_ : a\u2081 = a\u2081) { fst := a\u2081, snd := { val := b\u2081, property := h\u2081 } }.snd) = \u2191{ fst := a\u2081, snd := { val := b\u2081, property := h\u2082 } }.snd ** rfl ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.map_smul_inv ** R : Type u_1 R\u2081 : Type u_2 R\u2082 : Type u_3 R\u2083 : Type u_4 k : Type u_5 S : Type u_6 S\u2083 : Type u_7 T : Type u_8 M : Type u_9 M\u2081 : Type u_10 M\u2082 : Type u_11 M\u2083 : Type u_12 N\u2081 : Type u_13 N\u2082 : Type u_14 N\u2083 : Type u_15 \u03b9 : Type u_16 inst\u271d\u00b9\u00b2 : Semiring R inst\u271d\u00b9\u00b9 : Semiring S inst\u271d\u00b9\u2070 : AddCommMonoid M inst\u271d\u2079 : AddCommMonoid M\u2081 inst\u271d\u2078 : AddCommMonoid M\u2082 inst\u271d\u2077 : AddCommMonoid M\u2083 inst\u271d\u2076 : AddCommMonoid N\u2081 inst\u271d\u2075 : AddCommMonoid N\u2082 inst\u271d\u2074 : AddCommMonoid N\u2083 inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : Module S M\u2083 \u03c3 : R \u2192+* S f\u2097 g\u2097 : M \u2192\u2097[R] M\u2082 f g : M \u2192\u209b\u2097[\u03c3] M\u2083 \u03c3' : S \u2192+* R inst\u271d : RingHomInvPair \u03c3 \u03c3' c : S x : M \u22a2 c \u2022 \u2191f x = \u2191f (\u2191\u03c3' c \u2022 x) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.derivFamily_eq_enumOrd ** \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) \u22a2 derivFamily f = enumOrd (\u22c2 i, fixedPoints (f i)) ** rw [\u2190 eq_enumOrd _ (fp_family_unbounded.{u, v} H)] ** \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) \u22a2 StrictMono (derivFamily f) \u2227 Set.range (derivFamily f) = \u22c2 i, fixedPoints (f i) ** use (derivFamily_isNormal f).strictMono ** case right \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) \u22a2 Set.range (derivFamily f) = \u22c2 i, fixedPoints (f i) ** rw [Set.range_eq_iff] ** case right \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) \u22a2 (\u2200 (a : Ordinal.{max u v}), derivFamily f a \u2208 \u22c2 i, fixedPoints (f i)) \u2227 \u2200 (b : Ordinal.{max u v}), b \u2208 \u22c2 i, fixedPoints (f i) \u2192 \u2203 a, derivFamily f a = b ** refine' \u27e8_, fun a ha => _\u27e9 ** case right.refine'_2 \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) a : Ordinal.{max u v} ha : a \u2208 \u22c2 i, fixedPoints (f i) \u22a2 \u2203 a_1, derivFamily f a_1 = a ** rw [Set.mem_iInter] at ha ** case right.refine'_2 \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) a : Ordinal.{max u v} ha : \u2200 (i : \u03b9), a \u2208 fixedPoints (f i) \u22a2 \u2203 a_1, derivFamily f a_1 = a ** rwa [\u2190 fp_iff_derivFamily H] ** case right.refine'_1 \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) \u22a2 \u2200 (a : Ordinal.{max u v}), derivFamily f a \u2208 \u22c2 i, fixedPoints (f i) ** rintro a S \u27e8i, hi\u27e9 ** case right.refine'_1.intro \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) a : Ordinal.{max u v} S : Set Ordinal.{max u v} i : \u03b9 hi : (fun i => fixedPoints (f i)) i = S \u22a2 derivFamily f a \u2208 S ** rw [\u2190 hi] ** case right.refine'_1.intro \u03b9 : Type u f : \u03b9 \u2192 Ordinal.{max u v} \u2192 Ordinal.{max u v} H : \u2200 (i : \u03b9), IsNormal (f i) a : Ordinal.{max u v} S : Set Ordinal.{max u v} i : \u03b9 hi : (fun i => fixedPoints (f i)) i = S \u22a2 derivFamily f a \u2208 (fun i => fixedPoints (f i)) i ** exact derivFamily_fp (H i) a ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_nnnorm_eq_lintegral_edist ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b4 inst\u271d\u00b9 : NormedAddCommGroup \u03b2 inst\u271d : NormedAddCommGroup \u03b3 f : \u03b1 \u2192 \u03b2 \u22a2 \u222b\u207b (a : \u03b1), \u2191\u2016f a\u2016\u208a \u2202\u03bc = \u222b\u207b (a : \u03b1), edist (f a) 0 \u2202\u03bc ** simp only [edist_eq_coe_nnnorm] ** Qed", + "informal": "" + }, + { + "formal": "PowerBasis.leftMulMatrix ** R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S \u22a2 \u2191(Algebra.leftMulMatrix pb.basis) pb.gen = \u2191Matrix.of fun i j => if \u2191j + 1 = pb.dim then -coeff (minpolyGen pb) \u2191i else if \u2191i = \u2191j + 1 then 1 else 0 ** cases subsingleton_or_nontrivial A ** case inr R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A \u22a2 \u2191(Algebra.leftMulMatrix pb.basis) pb.gen = \u2191Matrix.of fun i j => if \u2191j + 1 = pb.dim then -coeff (minpolyGen pb) \u2191i else if \u2191i = \u2191j + 1 then 1 else 0 ** rw [Algebra.leftMulMatrix_apply, \u2190 LinearEquiv.eq_symm_apply, LinearMap.toMatrix_symm] ** case inr R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A \u22a2 \u2191(Algebra.lmul A S) pb.gen = \u2191(Matrix.toLin pb.basis pb.basis) (\u2191Matrix.of fun i j => if \u2191j + 1 = pb.dim then -coeff (minpolyGen pb) \u2191i else if \u2191i = \u2191j + 1 then 1 else 0) ** refine' pb.basis.ext fun k => _ ** case inr R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim \u22a2 \u2191(\u2191(Algebra.lmul A S) pb.gen) (\u2191pb.basis k) = \u2191(\u2191(Matrix.toLin pb.basis pb.basis) (\u2191Matrix.of fun i j => if \u2191j + 1 = pb.dim then -coeff (minpolyGen pb) \u2191i else if \u2191i = \u2191j + 1 then 1 else 0)) (\u2191pb.basis k) ** simp_rw [Matrix.toLin_self, Matrix.of_apply, pb.basis_eq_pow] ** case inr R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim \u22a2 \u2191(\u2191(Algebra.lmul A S) pb.gen) (pb.gen ^ \u2191k) = \u2211 x : Fin pb.dim, (if \u2191k + 1 = pb.dim then -coeff (minpolyGen pb) \u2191x else if \u2191x = \u2191k + 1 then 1 else 0) \u2022 pb.gen ^ \u2191x ** apply (pow_succ _ _).symm.trans ** case inr R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim \u22a2 pb.gen ^ (\u2191k + 1) = \u2211 x : Fin pb.dim, (if \u2191k + 1 = pb.dim then -coeff (minpolyGen pb) \u2191x else if \u2191x = \u2191k + 1 then 1 else 0) \u2022 pb.gen ^ \u2191x ** split_ifs with h ** case inl R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Subsingleton A \u22a2 \u2191(Algebra.leftMulMatrix pb.basis) pb.gen = \u2191Matrix.of fun i j => if \u2191j + 1 = pb.dim then -coeff (minpolyGen pb) \u2191i else if \u2191i = \u2191j + 1 then 1 else 0 ** apply Subsingleton.elim ** case pos R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u2191k + 1 = pb.dim \u22a2 pb.gen ^ (\u2191k + 1) = \u2211 x : Fin pb.dim, -coeff (minpolyGen pb) \u2191x \u2022 pb.gen ^ \u2191x ** simp_rw [h, neg_smul, Finset.sum_neg_distrib, eq_neg_iff_add_eq_zero] ** case pos R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u2191k + 1 = pb.dim \u22a2 pb.gen ^ pb.dim + \u2211 x : Fin pb.dim, coeff (minpolyGen pb) \u2191x \u2022 pb.gen ^ \u2191x = 0 ** convert pb.aeval_minpolyGen ** case h.e'_2 R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u2191k + 1 = pb.dim \u22a2 pb.gen ^ pb.dim + \u2211 x : Fin pb.dim, coeff (minpolyGen pb) \u2191x \u2022 pb.gen ^ \u2191x = \u2191(aeval pb.gen) (minpolyGen pb) ** rw [add_comm, aeval_eq_sum_range, Finset.sum_range_succ, \u2190 leadingCoeff,\n pb.minpolyGen_monic.leadingCoeff, one_smul, natDegree_minpolyGen, Finset.sum_range] ** case neg R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u00ac\u2191k + 1 = pb.dim \u22a2 pb.gen ^ (\u2191k + 1) = \u2211 x : Fin pb.dim, (if \u2191x = \u2191k + 1 then 1 else 0) \u2022 pb.gen ^ \u2191x ** rw [Fintype.sum_eq_single (\u27e8(k : \u2115) + 1, lt_of_le_of_ne k.2 h\u27e9 : Fin pb.dim), if_pos, one_smul] ** case neg R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u00ac\u2191k + 1 = pb.dim \u22a2 \u2200 (x : Fin pb.dim), x \u2260 { val := \u2191k + 1, isLt := (_ : \u2191k + 1 < pb.dim) } \u2192 (if \u2191x = \u2191k + 1 then 1 else 0) \u2022 pb.gen ^ \u2191x = 0 ** intro x hx ** case neg R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u00ac\u2191k + 1 = pb.dim x : Fin pb.dim hx : x \u2260 { val := \u2191k + 1, isLt := (_ : \u2191k + 1 < pb.dim) } \u22a2 (if \u2191x = \u2191k + 1 then 1 else 0) \u2022 pb.gen ^ \u2191x = 0 ** rw [if_neg, zero_smul] ** case neg.hnc R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u00ac\u2191k + 1 = pb.dim x : Fin pb.dim hx : x \u2260 { val := \u2191k + 1, isLt := (_ : \u2191k + 1 < pb.dim) } \u22a2 \u00ac\u2191x = \u2191k + 1 ** apply mt Fin.ext hx ** case neg.hc R : Type u_1 S : Type u_2 T : Type u_3 inst\u271d\u2078 : CommRing R inst\u271d\u2077 : Ring S inst\u271d\u2076 : Algebra R S A : Type u_4 B : Type u_5 inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing B inst\u271d\u00b3 : IsDomain B inst\u271d\u00b2 : Algebra A B K : Type u_6 inst\u271d\u00b9 : Field K inst\u271d : Algebra A S pb : PowerBasis A S h\u271d : Nontrivial A k : Fin pb.dim h : \u00ac\u2191k + 1 = pb.dim \u22a2 \u2191{ val := \u2191k + 1, isLt := (_ : \u2191k + 1 < pb.dim) } = \u2191k + 1 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "NumberField.isUnit_iff_norm ** K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K x : { x // x \u2208 \ud835\udcde K } \u22a2 IsUnit x \u2194 |\u2191(\u2191(RingOfIntegers.norm \u211a) x)| = 1 ** convert (RingOfIntegers.isUnit_norm \u211a (F := K)).symm ** case h.e'_2.a K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K x : { x // x \u2208 \ud835\udcde K } \u22a2 |\u2191(\u2191(RingOfIntegers.norm \u211a) x)| = 1 \u2194 IsUnit (\u2191(RingOfIntegers.norm \u211a) x) ** rw [\u2190 abs_one, abs_eq_abs, \u2190 Rat.RingOfIntegers.isUnit_iff] ** Qed", + "informal": "" + }, + { + "formal": "Int.neg_clog_inv_eq_log ** R : Type u_1 inst\u271d\u00b9 : LinearOrderedSemifield R inst\u271d : FloorSemiring R b : \u2115 r : R \u22a2 -clog b r\u207b\u00b9 = log b r ** rw [clog_inv, neg_neg] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.finrank_le_one_iff_isPrincipal ** K : Type u V : Type v inst\u271d\u00b3 : DivisionRing K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V W : Submodule K V inst\u271d : FiniteDimensional K { x // x \u2208 W } \u22a2 finrank K { x // x \u2208 W } \u2264 1 \u2194 IsPrincipal W ** rw [\u2190 W.rank_le_one_iff_isPrincipal, \u2190 finrank_eq_rank, \u2190 Cardinal.natCast_le, Nat.cast_one] ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.integral_mono_ae ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f g : \u211d \u2192 \u211d a b : \u211d \u03bc : Measure \u211d hab : a \u2264 b hf : IntervalIntegrable f \u03bc a b hg : IntervalIntegrable g \u03bc a b h : f \u2264\u1d50[\u03bc] g \u22a2 \u222b (u : \u211d) in a..b, f u \u2202\u03bc \u2264 \u222b (u : \u211d) in a..b, g u \u2202\u03bc ** simpa only [integral_of_le hab] using set_integral_mono_ae hf.1 hg.1 h ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearEquiv.coord_norm ** \ud835\udd5c : Type u_1 \ud835\udd5c\u2082 : Type u_2 \ud835\udd5c\u2083 : Type u_3 E : Type u_4 E\u2097 : Type u_5 F : Type u_6 F\u2097 : Type u_7 G : Type u_8 G\u2097 : Type u_9 \ud835\udcd5 : Type u_10 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup F\u2097 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2082 inst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u2083 inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2082 F \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 \u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c inst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081 inst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082 inst\u271d : RingHomIsometric \u03c3\u2082\u2081 x : E h : x \u2260 0 \u22a2 \u2016coord \ud835\udd5c x h\u2016 = \u2016x\u2016\u207b\u00b9 ** have hx : 0 < \u2016x\u2016 := norm_pos_iff.mpr h ** \ud835\udd5c : Type u_1 \ud835\udd5c\u2082 : Type u_2 \ud835\udd5c\u2083 : Type u_3 E : Type u_4 E\u2097 : Type u_5 F : Type u_6 F\u2097 : Type u_7 G : Type u_8 G\u2097 : Type u_9 \ud835\udcd5 : Type u_10 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup F\u2097 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2082 inst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u2083 inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2082 F \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 \u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c inst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081 inst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082 inst\u271d : RingHomIsometric \u03c3\u2082\u2081 x : E h : x \u2260 0 hx : 0 < \u2016x\u2016 \u22a2 \u2016coord \ud835\udd5c x h\u2016 = \u2016x\u2016\u207b\u00b9 ** haveI : Nontrivial (\ud835\udd5c \u2219 x) := Submodule.nontrivial_span_singleton h ** \ud835\udd5c : Type u_1 \ud835\udd5c\u2082 : Type u_2 \ud835\udd5c\u2083 : Type u_3 E : Type u_4 E\u2097 : Type u_5 F : Type u_6 F\u2097 : Type u_7 G : Type u_8 G\u2097 : Type u_9 \ud835\udcd5 : Type u_10 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedAddCommGroup G inst\u271d\u2078 : NormedAddCommGroup F\u2097 inst\u271d\u2077 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2076 : NontriviallyNormedField \ud835\udd5c\u2082 inst\u271d\u2075 : NontriviallyNormedField \ud835\udd5c\u2083 inst\u271d\u2074 : NormedSpace \ud835\udd5c E inst\u271d\u00b3 : NormedSpace \ud835\udd5c\u2082 F \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 \u03c3\u2082\u2081 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c inst\u271d\u00b2 : RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081 inst\u271d\u00b9 : RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082 inst\u271d : RingHomIsometric \u03c3\u2082\u2081 x : E h : x \u2260 0 hx : 0 < \u2016x\u2016 this : Nontrivial { x_1 // x_1 \u2208 Submodule.span \ud835\udd5c {x} } \u22a2 \u2016coord \ud835\udd5c x h\u2016 = \u2016x\u2016\u207b\u00b9 ** exact ContinuousLinearMap.homothety_norm _ fun y =>\n homothety_inverse _ hx _ (toSpanNonzeroSingleton_homothety \ud835\udd5c x h) _ ** Qed", + "informal": "" + }, + { + "formal": "Holor.mul_scalar_mul ** \u03b1 : Type d : \u2115 ds ds\u2081 ds\u2082 ds\u2083 : List \u2115 inst\u271d : Monoid \u03b1 x : Holor \u03b1 [] y : Holor \u03b1 ds \u22a2 x \u2297 y = x { val := [], property := (_ : Forall\u2082 (fun x x_1 => x < x_1) [] []) } \u2022 y ** simp [mul, SMul.smul, HolorIndex.take, HolorIndex.drop, HSMul.hSMul] ** Qed", + "informal": "" + }, + { + "formal": "Nat.succ_ascFactorial ** n : \u2115 \u22a2 (n + 1) * ascFactorial (succ n) 0 = (n + 0 + 1) * ascFactorial n 0 ** rw [add_zero, ascFactorial_zero, ascFactorial_zero] ** n k : \u2115 \u22a2 (n + 1) * ascFactorial (succ n) (k + 1) = (n + (k + 1) + 1) * ascFactorial n (k + 1) ** rw [ascFactorial, mul_left_comm, succ_ascFactorial n k, ascFactorial,\n succ_add, \u2190 add_assoc, succ_eq_add_one] ** Qed", + "informal": "" + }, + { + "formal": "eVariationOn.lowerSemicontinuous_uniformOn ** \u03b1 : Type u_1 inst\u271d\u00b9 : LinearOrder \u03b1 E : Type u_2 inst\u271d : PseudoEMetricSpace E s : Set \u03b1 f : \u03b1 \u2192\u1d64[{s}] E \u22a2 LowerSemicontinuousAt (fun f => eVariationOn f s) f ** apply @lowerSemicontinuous_aux _ _ _ _ (UniformOnFun \u03b1 E {s}) id (\ud835\udcdd f) f s _ ** \u03b1 : Type u_1 inst\u271d\u00b9 : LinearOrder \u03b1 E : Type u_2 inst\u271d : PseudoEMetricSpace E s : Set \u03b1 f : \u03b1 \u2192\u1d64[{s}] E \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x)) ** have := @tendsto_id _ (\ud835\udcdd f) ** \u03b1 : Type u_1 inst\u271d\u00b9 : LinearOrder \u03b1 E : Type u_2 inst\u271d : PseudoEMetricSpace E s : Set \u03b1 f : \u03b1 \u2192\u1d64[{s}] E this : Tendsto id (\ud835\udcdd f) (\ud835\udcdd f) \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x)) ** rw [UniformOnFun.tendsto_iff_tendstoUniformlyOn] at this ** \u03b1 : Type u_1 inst\u271d\u00b9 : LinearOrder \u03b1 E : Type u_2 inst\u271d : PseudoEMetricSpace E s : Set \u03b1 f : \u03b1 \u2192\u1d64[{s}] E this : \u2200 (s_1 : Set \u03b1), s_1 \u2208 {s} \u2192 TendstoUniformlyOn id f (\ud835\udcdd f) s_1 \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 Tendsto (fun i => id i x) (\ud835\udcdd f) (\ud835\udcdd (f x)) ** simp_rw [\u2190 tendstoUniformlyOn_singleton_iff_tendsto] ** \u03b1 : Type u_1 inst\u271d\u00b9 : LinearOrder \u03b1 E : Type u_2 inst\u271d : PseudoEMetricSpace E s : Set \u03b1 f : \u03b1 \u2192\u1d64[{s}] E this : \u2200 (s_1 : Set \u03b1), s_1 \u2208 {s} \u2192 TendstoUniformlyOn id f (\ud835\udcdd f) s_1 \u22a2 \u2200 (x : \u03b1), x \u2208 s \u2192 TendstoUniformlyOn (fun n => id n) f (\ud835\udcdd f) {x} ** exact fun x xs => (this s rfl).mono (singleton_subset_iff.mpr xs) ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicIndependent.mono ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 K : Type u_4 A : Type u_5 A' : Type u_6 A'' : Type u_7 V : Type u V' : Type u_8 x : \u03b9 \u2192 A inst\u271d\u2076 : CommRing R inst\u271d\u2075 : CommRing A inst\u271d\u2074 : CommRing A' inst\u271d\u00b3 : CommRing A'' inst\u271d\u00b2 : Algebra R A inst\u271d\u00b9 : Algebra R A' inst\u271d : Algebra R A'' a b : R t s : Set A h : t \u2286 s hx : AlgebraicIndependent R Subtype.val \u22a2 AlgebraicIndependent R Subtype.val ** simpa [Function.comp] using hx.comp (inclusion h) (inclusion_injective h) ** Qed", + "informal": "" + }, + { + "formal": "DomMulAct.smul_Lp_sub ** M : Type u_1 N : Type u_2 \u03b1 : Type u_3 E : Type u_4 inst\u271d\u2076 : MeasurableSpace M inst\u271d\u2075 : MeasurableSpace N inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : SMul M \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc inst\u271d : MeasurableSMul M \u03b1 c : M\u1d48\u1d50\u1d43 \u22a2 \u2200 (f g : { x // x \u2208 Lp E p }), c \u2022 (f - g) = c \u2022 f - c \u2022 g ** rintro \u27e8\u27e8\u27e9, _\u27e9 \u27e8\u27e8\u27e9, _\u27e9 ** case mk.mk.mk.mk M : Type u_1 N : Type u_2 \u03b1 : Type u_3 E : Type u_4 inst\u271d\u2076 : MeasurableSpace M inst\u271d\u2075 : MeasurableSpace N inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E \u03bc : Measure \u03b1 p : \u211d\u22650\u221e inst\u271d\u00b2 : SMul M \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure M \u03b1 \u03bc inst\u271d : MeasurableSMul M \u03b1 c : M\u1d48\u1d50\u1d43 val\u271d\u00b9 : \u03b1 \u2192\u2098[\u03bc] E a\u271d\u00b9 : { f // AEStronglyMeasurable f \u03bc } property\u271d\u00b9 : Quot.mk Setoid.r a\u271d\u00b9 \u2208 Lp E p val\u271d : \u03b1 \u2192\u2098[\u03bc] E a\u271d : { f // AEStronglyMeasurable f \u03bc } property\u271d : Quot.mk Setoid.r a\u271d \u2208 Lp E p \u22a2 c \u2022 ({ val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } - { val := Quot.mk Setoid.r a\u271d, property := property\u271d }) = c \u2022 { val := Quot.mk Setoid.r a\u271d\u00b9, property := property\u271d\u00b9 } - c \u2022 { val := Quot.mk Setoid.r a\u271d, property := property\u271d } ** rfl ** Qed", + "informal": "" + }, + { + "formal": "nhdsWithin_Icc_eq_nhdsWithin_Iic ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w inst\u271d\u00b3 : TopologicalSpace \u03b1 inst\u271d\u00b2 : LinearOrder \u03b1 inst\u271d\u00b9 : OrderClosedTopology \u03b1 a\u271d b\u271d : \u03b1 inst\u271d : TopologicalSpace \u03b3 a b : \u03b1 h : a < b \u22a2 \ud835\udcdd[Icc a b] b = \ud835\udcdd[Iic b] b ** simpa only [dual_Icc] using nhdsWithin_Icc_eq_nhdsWithin_Ici h.dual ** Qed", + "informal": "" + }, + { + "formal": "Even.neg_one_pow ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 R : Type u_4 inst\u271d\u00b9 : Monoid \u03b1 n : \u2115 a : \u03b1 inst\u271d : HasDistribNeg \u03b1 h : Even n \u22a2 (-1) ^ n = 1 ** rw [h.neg_pow, one_pow] ** Qed", + "informal": "" + }, + { + "formal": "derivedSeries_normal ** G : Type u_1 G' : Type u_2 inst\u271d\u00b9 : Group G inst\u271d : Group G' f : G \u2192* G' n : \u2115 \u22a2 Normal (derivedSeries G n) ** induction' n with n ih ** case zero G : Type u_1 G' : Type u_2 inst\u271d\u00b9 : Group G inst\u271d : Group G' f : G \u2192* G' \u22a2 Normal (derivedSeries G Nat.zero) ** exact (\u22a4 : Subgroup G).normal_of_characteristic ** case succ G : Type u_1 G' : Type u_2 inst\u271d\u00b9 : Group G inst\u271d : Group G' f : G \u2192* G' n : \u2115 ih : Normal (derivedSeries G n) \u22a2 Normal (derivedSeries G (Nat.succ n)) ** exact @Subgroup.commutator_normal G _ (derivedSeries G n) (derivedSeries G n) ih ih ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Lp.tendsto_Lp_iff_tendsto_\u2112p'' ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b9 : Type u_5 fi : Filter \u03b9 inst\u271d : Fact (1 \u2264 p) f : \u03b9 \u2192 \u03b1 \u2192 E f_\u2112p : \u2200 (n : \u03b9), Mem\u2112p (f n) p f_lim : \u03b1 \u2192 E f_lim_\u2112p : Mem\u2112p f_lim p \u22a2 Tendsto (fun n => Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) fi (\ud835\udcdd (Mem\u2112p.toLp f_lim f_lim_\u2112p)) \u2194 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) fi (\ud835\udcdd 0) ** rw [Lp.tendsto_Lp_iff_tendsto_\u2112p' (fun n => (f_\u2112p n).toLp (f n)) (f_lim_\u2112p.toLp f_lim)] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b9 : Type u_5 fi : Filter \u03b9 inst\u271d : Fact (1 \u2264 p) f : \u03b9 \u2192 \u03b1 \u2192 E f_\u2112p : \u2200 (n : \u03b9), Mem\u2112p (f n) p f_lim : \u03b1 \u2192 E f_lim_\u2112p : Mem\u2112p f_lim p \u22a2 Tendsto (fun n => snorm (\u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) - \u2191\u2191(Mem\u2112p.toLp f_lim f_lim_\u2112p)) p \u03bc) fi (\ud835\udcdd 0) \u2194 Tendsto (fun n => snorm (f n - f_lim) p \u03bc) fi (\ud835\udcdd 0) ** refine Filter.tendsto_congr fun n => ?_ ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b9 : Type u_5 fi : Filter \u03b9 inst\u271d : Fact (1 \u2264 p) f : \u03b9 \u2192 \u03b1 \u2192 E f_\u2112p : \u2200 (n : \u03b9), Mem\u2112p (f n) p f_lim : \u03b1 \u2192 E f_lim_\u2112p : Mem\u2112p f_lim p n : \u03b9 \u22a2 snorm (\u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) - \u2191\u2191(Mem\u2112p.toLp f_lim f_lim_\u2112p)) p \u03bc = snorm (f n - f_lim) p \u03bc ** apply snorm_congr_ae ** case hfg \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b9 : Type u_5 fi : Filter \u03b9 inst\u271d : Fact (1 \u2264 p) f : \u03b9 \u2192 \u03b1 \u2192 E f_\u2112p : \u2200 (n : \u03b9), Mem\u2112p (f n) p f_lim : \u03b1 \u2192 E f_lim_\u2112p : Mem\u2112p f_lim p n : \u03b9 \u22a2 \u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) - \u2191\u2191(Mem\u2112p.toLp f_lim f_lim_\u2112p) =\u1d50[\u03bc] f n - f_lim ** filter_upwards [((f_\u2112p n).sub f_lim_\u2112p).coeFn_toLp,\n Lp.coeFn_sub ((f_\u2112p n).toLp (f n)) (f_lim_\u2112p.toLp f_lim)] with _ hx\u2081 hx\u2082 ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b9 : Type u_5 fi : Filter \u03b9 inst\u271d : Fact (1 \u2264 p) f : \u03b9 \u2192 \u03b1 \u2192 E f_\u2112p : \u2200 (n : \u03b9), Mem\u2112p (f n) p f_lim : \u03b1 \u2192 E f_lim_\u2112p : Mem\u2112p f_lim p n : \u03b9 a\u271d : \u03b1 hx\u2081 : \u2191\u2191(Mem\u2112p.toLp (f n - f_lim) (_ : Mem\u2112p (f n - f_lim) p)) a\u271d = (f n - f_lim) a\u271d hx\u2082 : \u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p) - Mem\u2112p.toLp f_lim f_lim_\u2112p) a\u271d = (\u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) - \u2191\u2191(Mem\u2112p.toLp f_lim f_lim_\u2112p)) a\u271d \u22a2 (\u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) - \u2191\u2191(Mem\u2112p.toLp f_lim f_lim_\u2112p)) a\u271d = (f n - f_lim) a\u271d ** rw [\u2190 hx\u2082] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b9 : Type u_5 fi : Filter \u03b9 inst\u271d : Fact (1 \u2264 p) f : \u03b9 \u2192 \u03b1 \u2192 E f_\u2112p : \u2200 (n : \u03b9), Mem\u2112p (f n) p f_lim : \u03b1 \u2192 E f_lim_\u2112p : Mem\u2112p f_lim p n : \u03b9 a\u271d : \u03b1 hx\u2081 : \u2191\u2191(Mem\u2112p.toLp (f n - f_lim) (_ : Mem\u2112p (f n - f_lim) p)) a\u271d = (f n - f_lim) a\u271d hx\u2082 : \u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p) - Mem\u2112p.toLp f_lim f_lim_\u2112p) a\u271d = (\u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p)) - \u2191\u2191(Mem\u2112p.toLp f_lim f_lim_\u2112p)) a\u271d \u22a2 \u2191\u2191(Mem\u2112p.toLp (f n) (_ : Mem\u2112p (f n) p) - Mem\u2112p.toLp f_lim f_lim_\u2112p) a\u271d = (f n - f_lim) a\u271d ** exact hx\u2081 ** Qed", + "informal": "" + }, + { + "formal": "Rack.self_invAct_act_eq ** R : Type u_1 inst\u271d : Rack R x y : R \u22a2 (x \u25c3\u207b\u00b9 x) \u25c3 y = x \u25c3 y ** have h := @self_act_invAct_eq _ _ (op x) (op y) ** R : Type u_1 inst\u271d : Rack R x y : R h : (op x \u25c3 op x) \u25c3\u207b\u00b9 op y = op x \u25c3\u207b\u00b9 op y \u22a2 (x \u25c3\u207b\u00b9 x) \u25c3 y = x \u25c3 y ** simpa using h ** Qed", + "informal": "" + }, + { + "formal": "HasFDerivWithinAt.smul_const ** \ud835\udd5c : Type u_1 inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b9\u00b9 : NormedAddCommGroup E inst\u271d\u00b9\u2070 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u2079 : NormedAddCommGroup F inst\u271d\u2078 : NormedSpace \ud835\udd5c F G : Type u_4 inst\u271d\u2077 : NormedAddCommGroup G inst\u271d\u2076 : NormedSpace \ud835\udd5c G G' : Type u_5 inst\u271d\u2075 : NormedAddCommGroup G' inst\u271d\u2074 : NormedSpace \ud835\udd5c G' f\u271d f\u2080 f\u2081 g : E \u2192 F f' f\u2080' f\u2081' g' e : E \u2192L[\ud835\udd5c] F x : E s t : Set E L L\u2081 L\u2082 : Filter E \ud835\udd5c' : Type u_6 inst\u271d\u00b3 : NontriviallyNormedField \ud835\udd5c' inst\u271d\u00b2 : NormedAlgebra \ud835\udd5c \ud835\udd5c' inst\u271d\u00b9 : NormedSpace \ud835\udd5c' F inst\u271d : IsScalarTower \ud835\udd5c \ud835\udd5c' F c : E \u2192 \ud835\udd5c' c' : E \u2192L[\ud835\udd5c] \ud835\udd5c' hc : HasFDerivWithinAt c c' s x f : F \u22a2 HasFDerivWithinAt (fun y => c y \u2022 f) (smulRight c' f) s x ** simpa only [smul_zero, zero_add] using hc.smul (hasFDerivWithinAt_const f x s) ** Qed", + "informal": "" + }, + { + "formal": "OreLocalization.mul_inv ** R : Type u_1 inst\u271d\u00b9 : Monoid R S : Submonoid R inst\u271d : OreSet S s s' : { x // x \u2208 S } \u22a2 \u2191s /\u2092 s' * (\u2191s' /\u2092 s) = 1 ** simp [oreDiv_mul_char (s : R) s' s' s 1 1 (by simp)] ** R : Type u_1 inst\u271d\u00b9 : Monoid R S : Submonoid R inst\u271d : OreSet S s s' : { x // x \u2208 S } \u22a2 \u2191s' * \u21911 = \u2191s' * 1 ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Lp.mem_Lp_iff_mem\u2112p ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f : \u03b1 \u2192\u2098[\u03bc] E \u22a2 f \u2208 Lp E p \u2194 Mem\u2112p (\u2191f) p ** simp [mem_Lp_iff_snorm_lt_top, Mem\u2112p, f.stronglyMeasurable.aestronglyMeasurable] ** Qed", + "informal": "" + }, + { + "formal": "Cycle.mem_lists_iff_coe_eq ** \u03b1 : Type u_1 s : Cycle \u03b1 l\u271d l : List \u03b1 \u22a2 l\u271d \u2208 lists (Quotient.mk'' l) \u2194 \u2191l\u271d = Quotient.mk'' l ** rw [lists, Quotient.liftOn'_mk''] ** \u03b1 : Type u_1 s : Cycle \u03b1 l\u271d l : List \u03b1 \u22a2 l\u271d \u2208 \u2191(cyclicPermutations l) \u2194 \u2191l\u271d = Quotient.mk'' l ** simp ** Qed", + "informal": "" + }, + { + "formal": "WithTop.coe_sInf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 inst\u271d : ConditionallyCompleteLinearOrderBot \u03b1 s : Set \u03b1 hs : Set.Nonempty s \u22a2 \u2191(sInf s) = \u2a05 a \u2208 s, \u2191a ** rw [coe_sInf' hs, sInf_image] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.exists_primitive_lcm_of_isPrimitive ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q \u22a2 \u2203 r, IsPrimitive r \u2227 \u2200 (s : R[X]), p \u2223 s \u2227 q \u2223 s \u2194 r \u2223 s ** have h : \u2203 (n : \u2115) (r : R[X]), r.natDegree = n \u2227 r.IsPrimitive \u2227 p \u2223 r \u2227 q \u2223 r :=\n \u27e8(p * q).natDegree, p * q, rfl, hp.mul hq, dvd_mul_right _ _, dvd_mul_left _ _\u27e9 ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r \u22a2 \u2203 r, IsPrimitive r \u2227 \u2200 (s : R[X]), p \u2223 s \u2227 q \u2223 s \u2194 r \u2223 s ** rcases Nat.find_spec h with \u27e8r, rdeg, rprim, pr, qr\u27e9 ** case intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r \u22a2 \u2203 r, IsPrimitive r \u2227 \u2200 (s : R[X]), p \u2223 s \u2227 q \u2223 s \u2194 r \u2223 s ** refine' \u27e8r, rprim, fun s => \u27e8_, fun rs => \u27e8pr.trans rs, qr.trans rs\u27e9\u27e9\u27e9 ** case intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r s : R[X] \u22a2 p \u2223 s \u2227 q \u2223 s \u2192 r \u2223 s ** suffices hs : \u2200 (n : \u2115) (s : R[X]), s.natDegree = n \u2192 p \u2223 s \u2227 q \u2223 s \u2192 r \u2223 s ** case hs R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r s : R[X] \u22a2 \u2200 (n : \u2115) (s : R[X]), natDegree s = n \u2192 p \u2223 s \u2227 q \u2223 s \u2192 r \u2223 s ** clear s ** case hs R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r \u22a2 \u2200 (n : \u2115) (s : R[X]), natDegree s = n \u2192 p \u2223 s \u2227 q \u2223 s \u2192 r \u2223 s ** by_contra' con ** case hs R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s \u22a2 False ** rcases Nat.find_spec con with \u27e8s, sdeg, \u27e8ps, qs\u27e9, rs\u27e9 ** case hs.intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s \u22a2 False ** have s0 : s \u2260 0 := by\n contrapose! rs\n simp [rs] ** case hs.intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 \u22a2 False ** have hs :=\n Nat.find_min' h\n \u27e8_, s.natDegree_primPart, s.isPrimitive_primPart, (hp.dvd_primPart_iff_dvd s0).2 ps,\n (hq.dvd_primPart_iff_dvd s0).2 qs\u27e9 ** case hs.intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : Nat.find h \u2264 natDegree s \u22a2 False ** rw [\u2190 rdeg] at hs ** case hs.intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s \u22a2 False ** by_cases sC : s.natDegree \u2264 0 ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 \u22a2 False ** have hcancel := natDegree_cancelLeads_lt_of_natDegree_le_natDegree hs (lt_of_not_ge sC) ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < natDegree s \u22a2 False ** rw [sdeg] at hcancel ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con \u22a2 False ** apply Nat.find_min con hcancel ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con \u22a2 \u2203 s_1, natDegree s_1 = natDegree (cancelLeads r s) \u2227 (p \u2223 s_1 \u2227 q \u2223 s_1) \u2227 \u00acr \u2223 s_1 ** refine'\n \u27e8_, rfl, \u27e8dvd_cancelLeads_of_dvd_of_dvd pr ps, dvd_cancelLeads_of_dvd_of_dvd qr qs\u27e9,\n fun rcs => rs _\u27e9 ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 cancelLeads r s \u22a2 r \u2223 s ** rw [\u2190 rprim.dvd_primPart_iff_dvd s0] ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 cancelLeads r s \u22a2 r \u2223 primPart s ** rw [cancelLeads, tsub_eq_zero_iff_le.mpr hs, pow_zero, mul_one] at rcs ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r \u22a2 r \u2223 primPart s ** have h :=\n dvd_add rcs (Dvd.intro_left (C (leadingCoeff s) * X ^ (natDegree s - natDegree r)) rfl) ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h\u271d : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h\u271d rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r h : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r + \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r \u22a2 r \u2223 primPart s ** have hC0 := rprim.ne_zero ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h\u271d : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h\u271d rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r h : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r + \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r hC0 : r \u2260 0 \u22a2 r \u2223 primPart s ** rw [Ne.def, \u2190 leadingCoeff_eq_zero, \u2190 C_eq_zero] at hC0 ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h\u271d : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h\u271d rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r h : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r + \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r hC0 : \u00ac\u2191C (leadingCoeff r) = 0 \u22a2 r \u2223 primPart s ** rw [sub_add_cancel, \u2190 rprim.dvd_primPart_iff_dvd (mul_ne_zero hC0 s0)] at h ** case neg R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h\u271d : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h\u271d rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r h : r \u2223 primPart (\u2191C (leadingCoeff r) * s) hC0 : \u00ac\u2191C (leadingCoeff r) = 0 \u22a2 r \u2223 primPart s ** rcases isUnit_primPart_C r.leadingCoeff with \u27e8u, hu\u27e9 ** case neg.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h\u271d : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h\u271d rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r h : r \u2223 primPart (\u2191C (leadingCoeff r) * s) hC0 : \u00ac\u2191C (leadingCoeff r) = 0 u : R[X]\u02e3 hu : \u2191u = primPart (\u2191C (leadingCoeff r)) \u22a2 r \u2223 primPart s ** apply h.trans (Associated.symm \u27e8u, _\u27e9).dvd ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h\u271d : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h\u271d rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : \u00acnatDegree s \u2264 0 hcancel : natDegree (cancelLeads r s) < Nat.find con rcs : r \u2223 \u2191C (leadingCoeff r) * s - \u2191C (leadingCoeff s) * X ^ (natDegree s - natDegree r) * r h : r \u2223 primPart (\u2191C (leadingCoeff r) * s) hC0 : \u00ac\u2191C (leadingCoeff r) = 0 u : R[X]\u02e3 hu : \u2191u = primPart (\u2191C (leadingCoeff r)) \u22a2 primPart s * \u2191u = primPart (\u2191C (leadingCoeff r) * s) ** rw [primPart_mul (mul_ne_zero hC0 s0), hu, mul_comm] ** case intro.intro.intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r s : R[X] hs : \u2200 (n : \u2115) (s : R[X]), natDegree s = n \u2192 p \u2223 s \u2227 q \u2223 s \u2192 r \u2223 s \u22a2 p \u2223 s \u2227 q \u2223 s \u2192 r \u2223 s ** apply hs s.natDegree s rfl ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s \u22a2 s \u2260 0 ** contrapose! rs ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con ps : p \u2223 s qs : q \u2223 s rs : s = 0 \u22a2 r \u2223 s ** simp [rs] ** case pos R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim : IsPrimitive r pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : natDegree s \u2264 0 \u22a2 False ** rw [eq_C_of_natDegree_le_zero (le_trans hs sC), isPrimitive_iff_content_eq_one, content_C,\n normalize_eq_one] at rprim ** case pos R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim\u271d : IsPrimitive (\u2191C (coeff r 0)) rprim : IsUnit (coeff r 0) pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs : \u00acr \u2223 s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : natDegree s \u2264 0 \u22a2 False ** rw [eq_C_of_natDegree_le_zero (le_trans hs sC), \u2190 dvd_content_iff_C_dvd] at rs ** case pos R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : NormalizedGCDMonoid R p q : R[X] hp : IsPrimitive p hq : IsPrimitive q h : \u2203 n r, natDegree r = n \u2227 IsPrimitive r \u2227 p \u2223 r \u2227 q \u2223 r r : R[X] rdeg : natDegree r = Nat.find h rprim\u271d : IsPrimitive (\u2191C (coeff r 0)) rprim : IsUnit (coeff r 0) pr : p \u2223 r qr : q \u2223 r con : \u2203 n s, natDegree s = n \u2227 (p \u2223 s \u2227 q \u2223 s) \u2227 \u00acr \u2223 s s : R[X] sdeg : natDegree s = Nat.find con rs\u271d : \u00ac\u2191C (coeff r 0) \u2223 s rs : \u00accoeff r 0 \u2223 content s ps : p \u2223 s qs : q \u2223 s s0 : s \u2260 0 hs : natDegree r \u2264 natDegree s sC : natDegree s \u2264 0 \u22a2 False ** apply rs rprim.dvd ** Qed", + "informal": "" + }, + { + "formal": "Finset.sup'_union ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b9 : SemilatticeSup \u03b1 s : Finset \u03b2 H : Finset.Nonempty s f\u271d : \u03b2 \u2192 \u03b1 inst\u271d : DecidableEq \u03b2 s\u2081 s\u2082 : Finset \u03b2 h\u2081 : Finset.Nonempty s\u2081 h\u2082 : Finset.Nonempty s\u2082 f : \u03b2 \u2192 \u03b1 a : \u03b1 \u22a2 sup' (s\u2081 \u222a s\u2082) (_ : Finset.Nonempty (s\u2081 \u222a s\u2082)) f \u2264 a \u2194 sup' s\u2081 h\u2081 f \u2294 sup' s\u2082 h\u2082 f \u2264 a ** simp [or_imp, forall_and] ** Qed", + "informal": "" + }, + { + "formal": "toIxxMod_total' ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : (fun x x_1 => x + x_1) (toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - a)) (toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - c)) = (fun x x_1 => x + x_1) p p \u22a2 toIcoMod hp b a \u2264 toIocMod hp b c \u2228 toIcoMod hp b c \u2264 toIocMod hp b a ** simp only [add_add_add_comm] at this ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : toIcoMod hp 0 (a - b) + toIcoMod hp 0 (c - b) + (toIocMod hp 0 (b - a) + toIocMod hp 0 (b - c)) = p + p \u22a2 toIcoMod hp b a \u2264 toIocMod hp b c \u2228 toIcoMod hp b c \u2264 toIocMod hp b a ** rw [_root_.add_comm (toIocMod _ _ _), add_add_add_comm, \u2190 two_nsmul] at this ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - c) + (toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - a)) = 2 \u2022 p \u22a2 toIcoMod hp b a \u2264 toIocMod hp b c \u2228 toIcoMod hp b c \u2264 toIocMod hp b a ** replace := min_le_of_add_le_two_nsmul this.le ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : min (toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - c)) (toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - a)) \u2264 p \u22a2 toIcoMod hp b a \u2264 toIocMod hp b c \u2228 toIcoMod hp b c \u2264 toIocMod hp b a ** rw [min_le_iff] at this ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - c) \u2264 p \u2228 toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - a) \u2264 p \u22a2 toIcoMod hp b a \u2264 toIocMod hp b c \u2228 toIcoMod hp b c \u2264 toIocMod hp b a ** rw [toIxxMod_iff, toIxxMod_iff] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - c) \u2264 p \u2228 toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - a) \u2264 p \u22a2 toIcoMod hp 0 (a - b) + toIcoMod hp 0 (b - c) \u2264 p \u2228 toIcoMod hp 0 (c - b) + toIcoMod hp 0 (b - a) \u2264 p ** refine' this.imp (le_trans <| add_le_add_left _ _) (le_trans <| add_le_add_left _ _) ** case refine'_1 \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - c) \u2264 p \u2228 toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - a) \u2264 p \u22a2 toIcoMod hp 0 (b - c) \u2264 toIocMod hp 0 (b - c) ** apply toIcoMod_le_toIocMod ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c\u271d : \u03b1 n : \u2124 a b c : \u03b1 this : toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - c) \u2264 p \u2228 toIcoMod hp 0 (c - b) + toIocMod hp 0 (b - a) \u2264 p \u22a2 toIcoMod hp 0 (b - a) \u2264 toIocMod hp 0 (b - a) ** apply toIcoMod_le_toIocMod ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.IntegrableAtFilter.add ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d g\u271d : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 f g : \u03b1 \u2192 E hf : IntegrableAtFilter f l hg : IntegrableAtFilter g l \u22a2 IntegrableAtFilter (f + g) l ** rcases hf with \u27e8s, sl, hs\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d g\u271d : \u03b1 \u2192 E s\u271d t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 f g : \u03b1 \u2192 E hg : IntegrableAtFilter g l s : Set \u03b1 sl : s \u2208 l hs : IntegrableOn f s \u22a2 IntegrableAtFilter (f + g) l ** rcases hg with \u27e8t, tl, ht\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d g\u271d : \u03b1 \u2192 E s\u271d t\u271d : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 f g : \u03b1 \u2192 E s : Set \u03b1 sl : s \u2208 l hs : IntegrableOn f s t : Set \u03b1 tl : t \u2208 l ht : IntegrableOn g t \u22a2 IntegrableAtFilter (f + g) l ** refine \u27e8s \u2229 t, inter_mem sl tl, ?_\u27e9 ** case intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f\u271d g\u271d : \u03b1 \u2192 E s\u271d t\u271d : Set \u03b1 \u03bc \u03bd : Measure \u03b1 l l' : Filter \u03b1 f g : \u03b1 \u2192 E s : Set \u03b1 sl : s \u2208 l hs : IntegrableOn f s t : Set \u03b1 tl : t \u2208 l ht : IntegrableOn g t \u22a2 IntegrableOn (f + g) (s \u2229 t) ** exact (hs.mono_set (inter_subset_left _ _)).add (ht.mono_set (inter_subset_right _ _)) ** Qed", + "informal": "" + }, + { + "formal": "Multiset.map_surjective_of_surjective ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Surjective f \u22a2 Surjective (map f) ** intro s ** \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Surjective f s : Multiset \u03b2 \u22a2 \u2203 a, map f a = s ** induction' s using Multiset.induction_on with x s ih ** case empty \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Surjective f \u22a2 \u2203 a, map f a = 0 ** exact \u27e80, map_zero _\u27e9 ** case cons \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Surjective f x : \u03b2 s : Multiset \u03b2 ih : \u2203 a, map f a = s \u22a2 \u2203 a, map f a = x ::\u2098 s ** obtain \u27e8y, rfl\u27e9 := hf x ** case cons.intro \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Surjective f s : Multiset \u03b2 ih : \u2203 a, map f a = s y : \u03b1 \u22a2 \u2203 a, map f a = f y ::\u2098 s ** obtain \u27e8t, rfl\u27e9 := ih ** case cons.intro.intro \u03b1 : Type u_1 \u03b2 : Type v \u03b3 : Type u_2 f : \u03b1 \u2192 \u03b2 hf : Surjective f y : \u03b1 t : Multiset \u03b1 \u22a2 \u2203 a, map f a = f y ::\u2098 map f t ** exact \u27e8y ::\u2098 t, map_cons _ _ _\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "IsCyclotomicExtension.adjoin_roots_cyclotomic_eq_adjoin_nth_roots ** n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n \u22a2 adjoin A (rootSet (cyclotomic (\u2191n) A) B) = adjoin A {b | \u2203 a, a \u2208 {n} \u2227 b ^ \u2191a = 1} ** simp only [mem_singleton_iff, exists_eq_left, map_cyclotomic] ** n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n \u22a2 adjoin A (rootSet (cyclotomic (\u2191n) A) B) = adjoin A {b | b ^ \u2191n = 1} ** refine' le_antisymm (adjoin_mono fun x hx => _) (adjoin_le fun x hx => _) ** case refine'_1 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : x \u2208 rootSet (cyclotomic (\u2191n) A) B \u22a2 x \u2208 {b | b ^ \u2191n = 1} ** rw [mem_rootSet'] at hx ** case refine'_1 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0 \u22a2 x \u2208 {b | b ^ \u2191n = 1} ** simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] ** case refine'_1 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0 \u22a2 x ^ \u2191n = 1 ** rw [isRoot_of_unity_iff n.pos] ** case refine'_1 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0 \u22a2 \u2203 i, i \u2208 Nat.divisors \u2191n \u2227 IsRoot (cyclotomic i B) x ** refine' \u27e8n, Nat.mem_divisors_self n n.ne_zero, _\u27e9 ** case refine'_1 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0 \u22a2 IsRoot (cyclotomic (\u2191n) B) x ** rw [IsRoot.def, \u2190 map_cyclotomic n (algebraMap A B), eval_map, \u2190 aeval_def] ** case refine'_1 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : map (algebraMap A B) (cyclotomic (\u2191n) A) \u2260 0 \u2227 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0 \u22a2 \u2191(aeval x) (cyclotomic (\u2191n) A) = 0 ** exact hx.2 ** case refine'_2 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : x \u2208 {b | b ^ \u2191n = 1} \u22a2 x \u2208 \u2191(adjoin A (rootSet (cyclotomic (\u2191n) A) B)) ** simp only [mem_singleton_iff, exists_eq_left, mem_setOf_eq] at hx ** case refine'_2 n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n x : B hx : x ^ \u2191n = 1 \u22a2 x \u2208 \u2191(adjoin A (rootSet (cyclotomic (\u2191n) A) B)) ** obtain \u27e8i, _, rfl\u27e9 := h\u03b6.eq_pow_of_pow_eq_one hx n.pos ** case refine'_2.intro.intro n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n i : \u2115 left\u271d : i < \u2191n hx : (\u03b6 ^ i) ^ \u2191n = 1 \u22a2 \u03b6 ^ i \u2208 \u2191(adjoin A (rootSet (cyclotomic (\u2191n) A) B)) ** refine' SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin _) _) ** case refine'_2.intro.intro n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n i : \u2115 left\u271d : i < \u2191n hx : (\u03b6 ^ i) ^ \u2191n = 1 \u22a2 \u03b6 \u2208 rootSet (cyclotomic (\u2191n) A) B ** rw [mem_rootSet', map_cyclotomic, aeval_def, \u2190 eval_map, map_cyclotomic, \u2190 IsRoot] ** case refine'_2.intro.intro n\u271d : \u2115+ S T : Set \u2115+ A : Type u B : Type v K : Type w L : Type z inst\u271d\u2076 : CommRing A inst\u271d\u2075 : CommRing B inst\u271d\u2074 : Algebra A B inst\u271d\u00b3 : Field K inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Algebra K L inst\u271d : IsDomain B \u03b6 : B n : \u2115+ h\u03b6 : IsPrimitiveRoot \u03b6 \u2191n i : \u2115 left\u271d : i < \u2191n hx : (\u03b6 ^ i) ^ \u2191n = 1 \u22a2 cyclotomic (\u2191n) B \u2260 0 \u2227 IsRoot (cyclotomic (\u2191n) B) \u03b6 ** refine' \u27e8cyclotomic_ne_zero n B, h\u03b6.isRoot_cyclotomic n.pos\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "TensorProduct.ext_threefold ** R : Type u_1 inst\u271d\u00b9\u2074 : CommSemiring R R' : Type u_2 inst\u271d\u00b9\u00b3 : Monoid R' R'' : Type u_3 inst\u271d\u00b9\u00b2 : Semiring R'' M : Type u_4 N : Type u_5 P : Type u_6 Q : Type u_7 S : Type u_8 inst\u271d\u00b9\u00b9 : AddCommMonoid M inst\u271d\u00b9\u2070 : AddCommMonoid N inst\u271d\u2079 : AddCommMonoid P inst\u271d\u2078 : AddCommMonoid Q inst\u271d\u2077 : AddCommMonoid S inst\u271d\u2076 : Module R M inst\u271d\u2075 : Module R N inst\u271d\u2074 : Module R P inst\u271d\u00b3 : Module R Q inst\u271d\u00b2 : Module R S inst\u271d\u00b9 : DistribMulAction R' M inst\u271d : Module R'' M f : M \u2192\u2097[R] N \u2192\u2097[R] P g h : (M \u2297[R] N) \u2297[R] P \u2192\u2097[R] Q H : \u2200 (x : M) (y : N) (z : P), \u2191g ((x \u2297\u209c[R] y) \u2297\u209c[R] z) = \u2191h ((x \u2297\u209c[R] y) \u2297\u209c[R] z) \u22a2 g = h ** ext x y z ** case H.H.h.h.h R : Type u_1 inst\u271d\u00b9\u2074 : CommSemiring R R' : Type u_2 inst\u271d\u00b9\u00b3 : Monoid R' R'' : Type u_3 inst\u271d\u00b9\u00b2 : Semiring R'' M : Type u_4 N : Type u_5 P : Type u_6 Q : Type u_7 S : Type u_8 inst\u271d\u00b9\u00b9 : AddCommMonoid M inst\u271d\u00b9\u2070 : AddCommMonoid N inst\u271d\u2079 : AddCommMonoid P inst\u271d\u2078 : AddCommMonoid Q inst\u271d\u2077 : AddCommMonoid S inst\u271d\u2076 : Module R M inst\u271d\u2075 : Module R N inst\u271d\u2074 : Module R P inst\u271d\u00b3 : Module R Q inst\u271d\u00b2 : Module R S inst\u271d\u00b9 : DistribMulAction R' M inst\u271d : Module R'' M f : M \u2192\u2097[R] N \u2192\u2097[R] P g h : (M \u2297[R] N) \u2297[R] P \u2192\u2097[R] Q H : \u2200 (x : M) (y : N) (z : P), \u2191g ((x \u2297\u209c[R] y) \u2297\u209c[R] z) = \u2191h ((x \u2297\u209c[R] y) \u2297\u209c[R] z) x : M y : N z : P \u22a2 \u2191(\u2191(\u2191(LinearMap.compr\u2082 (mk R M N) (LinearMap.compr\u2082 (mk R (M \u2297[R] N) P) g)) x) y) z = \u2191(\u2191(\u2191(LinearMap.compr\u2082 (mk R M N) (LinearMap.compr\u2082 (mk R (M \u2297[R] N) P) h)) x) y) z ** exact H x y z ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.edge_other_ne ** \u03b9 : Sort u_1 \ud835\udd5c : Type u_2 V : Type u W : Type v X : Type w G : SimpleGraph V G' : SimpleGraph W a b c u v\u271d w : V e\u271d : Sym2 V G\u2081 G\u2082 : SimpleGraph V e : Sym2 V he : e \u2208 edgeSet G v : V h : v \u2208 e \u22a2 Sym2.Mem.other h \u2260 v ** erw [\u2190 Sym2.other_spec h, Sym2.eq_swap] at he ** \u03b9 : Sort u_1 \ud835\udd5c : Type u_2 V : Type u W : Type v X : Type w G : SimpleGraph V G' : SimpleGraph W a b c u v\u271d w : V e\u271d : Sym2 V G\u2081 G\u2082 : SimpleGraph V e : Sym2 V v : V h : v \u2208 e he : Quotient.mk (Sym2.Rel.setoid V) (Sym2.Mem.other h, v) \u2208 edgeSet G \u22a2 Sym2.Mem.other h \u2260 v ** exact G.ne_of_adj he ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.count_map_roots_of_injective ** R : Type u S : Type v T : Type w a b\u271d : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : DecidableEq B p : A[X] f : A \u2192+* B hf : Function.Injective \u2191f b : B \u22a2 Multiset.count b (Multiset.map (\u2191f) (roots p)) \u2264 rootMultiplicity b (map f p) ** by_cases hp0 : p = 0 ** case pos R : Type u S : Type v T : Type w a b\u271d : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : DecidableEq B p : A[X] f : A \u2192+* B hf : Function.Injective \u2191f b : B hp0 : p = 0 \u22a2 Multiset.count b (Multiset.map (\u2191f) (roots p)) \u2264 rootMultiplicity b (map f p) ** simp only [hp0, roots_zero, Multiset.map_zero, Multiset.count_zero, Polynomial.map_zero,\n rootMultiplicity_zero] ** case neg R : Type u S : Type v T : Type w a b\u271d : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : DecidableEq B p : A[X] f : A \u2192+* B hf : Function.Injective \u2191f b : B hp0 : \u00acp = 0 \u22a2 Multiset.count b (Multiset.map (\u2191f) (roots p)) \u2264 rootMultiplicity b (map f p) ** exact count_map_roots ((Polynomial.map_ne_zero_iff hf).mpr hp0) b ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.ProjectiveResolution.lift_commutes ** C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroObject C inst\u271d\u00b2 : HasZeroMorphisms C inst\u271d\u00b9 : HasEqualizers C inst\u271d : HasImages C Y Z : C f : Y \u27f6 Z P : ProjectiveResolution Y Q : ProjectiveResolution Z \u22a2 lift f P Q \u226b Q.\u03c0 = P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f ** ext ** case h C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroObject C inst\u271d\u00b2 : HasZeroMorphisms C inst\u271d\u00b9 : HasEqualizers C inst\u271d : HasImages C Y Z : C f : Y \u27f6 Z P : ProjectiveResolution Y Q : ProjectiveResolution Z \u22a2 HomologicalComplex.Hom.f (lift f P Q \u226b Q.\u03c0) 0 = HomologicalComplex.Hom.f (P.\u03c0 \u226b (ChainComplex.single\u2080 C).map f) 0 ** simp [lift, liftZero] ** Qed", + "informal": "" + }, + { + "formal": "IsPrimitiveRoot.pow_eq_one_iff_dvd ** M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u00b2 : CommMonoid M inst\u271d\u00b9 : CommMonoid N inst\u271d : DivisionCommMonoid G k l\u271d : \u2115 \u03b6 : M f : F h : IsPrimitiveRoot \u03b6 k l : \u2115 \u22a2 k \u2223 l \u2192 \u03b6 ^ l = 1 ** rintro \u27e8i, rfl\u27e9 ** case intro M : Type u_1 N : Type u_2 G : Type u_3 R : Type u_4 S : Type u_5 F : Type u_6 inst\u271d\u00b2 : CommMonoid M inst\u271d\u00b9 : CommMonoid N inst\u271d : DivisionCommMonoid G k l : \u2115 \u03b6 : M f : F h : IsPrimitiveRoot \u03b6 k i : \u2115 \u22a2 \u03b6 ^ (k * i) = 1 ** simp only [pow_mul, h.pow_eq_one, one_pow, PNat.mul_coe] ** Qed", + "informal": "" + }, + { + "formal": "Nat.one_le_cast ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u2074 : AddCommMonoidWithOne \u03b1 inst\u271d\u00b3 : PartialOrder \u03b1 inst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 inst\u271d\u00b9 : ZeroLEOneClass \u03b1 inst\u271d : CharZero \u03b1 m n : \u2115 \u22a2 1 \u2264 \u2191n \u2194 1 \u2264 n ** rw [\u2190 cast_one, cast_le] ** Qed", + "informal": "" + }, + { + "formal": "LinearIndependent.comp ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f \u22a2 LinearIndependent R (v \u2218 f) ** rw [linearIndependent_iff, Finsupp.total_comp] ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f \u22a2 \u2200 (l : \u03b9' \u2192\u2080 R), \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 \u2192 l = 0 ** intro l hl ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f l : \u03b9' \u2192\u2080 R hl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 \u22a2 l = 0 ** have h_map_domain : \u2200 x, (Finsupp.mapDomain f l) (f x) = 0 := by\n rw [linearIndependent_iff.1 h (Finsupp.mapDomain f l) hl]; simp ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f l : \u03b9' \u2192\u2080 R hl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 h_map_domain : \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0 \u22a2 l = 0 ** ext x ** case h \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x\u271d y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f l : \u03b9' \u2192\u2080 R hl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 h_map_domain : \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0 x : \u03b9' \u22a2 \u2191l x = \u21910 x ** convert h_map_domain x ** case h.e'_2 \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x\u271d y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f l : \u03b9' \u2192\u2080 R hl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 h_map_domain : \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0 x : \u03b9' \u22a2 \u2191l x = \u2191(Finsupp.mapDomain f l) (f x) ** rw [Finsupp.mapDomain_apply hf] ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f l : \u03b9' \u2192\u2080 R hl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 \u22a2 \u2200 (x : \u03b9'), \u2191(Finsupp.mapDomain f l) (f x) = 0 ** rw [linearIndependent_iff.1 h (Finsupp.mapDomain f l) hl] ** \u03b9 : Type u' \u03b9' : Type u_1 R : Type u_2 K : Type u_3 M : Type u_4 M' : Type u_5 M'' : Type u_6 V : Type u V' : Type u_7 v : \u03b9 \u2192 M inst\u271d\u2076 : Semiring R inst\u271d\u2075 : AddCommMonoid M inst\u271d\u2074 : AddCommMonoid M' inst\u271d\u00b3 : AddCommMonoid M'' inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : Module R M' inst\u271d : Module R M'' a b : R x y : M h : LinearIndependent R v f : \u03b9' \u2192 \u03b9 hf : Injective f l : \u03b9' \u2192\u2080 R hl : \u2191(LinearMap.comp (Finsupp.total \u03b9 M R v) (Finsupp.lmapDomain R R f)) l = 0 \u22a2 \u2200 (x : \u03b9'), \u21910 (f x) = 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "union_support_maximal_linearIndependent_eq_range_basis ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i \u22a2 \u22c3 k, \u2191(\u2191b.repr (v k)).support = Set.univ ** by_contra h ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i h : \u00ac\u22c3 k, \u2191(\u2191b.repr (v k)).support = Set.univ \u22a2 False ** simp only [\u2190 Ne.def, ne_univ_iff_exists_not_mem, mem_iUnion, not_exists_not,\n Finsupp.mem_support_iff, Finset.mem_coe] at h ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i h : \u2203 a, \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) a = 0 \u22a2 False ** obtain \u27e8b', w\u27e9 := h ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 \u22a2 False ** let v' : Option \u03ba \u2192 M := fun o => o.elim (b b') v ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v \u22a2 False ** have r : range v \u2286 range v' := by\n rintro - \u27e8k, rfl\u27e9\n use some k\n rfl ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' \u22a2 False ** have r' : b b' \u2209 range v := by\n rintro \u27e8k, p\u27e9\n simpa [w] using congr_arg (fun m => (b.repr m) b') p ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v \u22a2 False ** have r'' : range v \u2260 range v' := by\n intro e\n have p : b b' \u2208 range v' := by\n use none\n rfl\n rw [\u2190 e] at p\n exact r' p ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' i' : LinearIndependent R Subtype.val \u22a2 False ** dsimp [LinearIndependent.Maximal] at m ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : \u2200 (s : Set M), LinearIndependent R Subtype.val \u2192 range v \u2286 s \u2192 range v = s b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' i' : LinearIndependent R Subtype.val \u22a2 False ** specialize m (range v') i' r ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' i' : LinearIndependent R Subtype.val m : range v = range v' \u22a2 False ** exact r'' m ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v \u22a2 range v \u2286 range v' ** rintro - \u27e8k, rfl\u27e9 ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v k : \u03ba \u22a2 v k \u2208 range v' ** use some k ** case h K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v k : \u03ba \u22a2 v' (some k) = v k ** rfl ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' \u22a2 \u00ac\u2191b b' \u2208 range v ** rintro \u27e8k, p\u27e9 ** case intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' k : \u03ba p : v k = \u2191b b' \u22a2 False ** simpa [w] using congr_arg (fun m => (b.repr m) b') p ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v \u22a2 range v \u2260 range v' ** intro e ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v e : range v = range v' \u22a2 False ** have p : b b' \u2208 range v' := by\n use none\n rfl ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v e : range v = range v' p : \u2191b b' \u2208 range v' \u22a2 False ** rw [\u2190 e] at p ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v e : range v = range v' p : \u2191b b' \u2208 range v \u22a2 False ** exact r' p ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v e : range v = range v' \u22a2 \u2191b b' \u2208 range v' ** use none ** case h K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v e : range v = range v' \u22a2 v' none = \u2191b b' ** rfl ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' \u22a2 Injective v' ** rintro (_ | k) (_ | k) z ** case none.none K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' z : v' none = v' none \u22a2 none = none ** rfl ** case none.some K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' k : \u03ba z : v' none = v' (some k) \u22a2 none = some k ** exfalso ** case none.some.h K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' k : \u03ba z : v' none = v' (some k) \u22a2 False ** exact r' \u27e8k, z.symm\u27e9 ** case some.none K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' k : \u03ba z : v' (some k) = v' none \u22a2 some k = none ** exfalso ** case some.none.h K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' k : \u03ba z : v' (some k) = v' none \u22a2 False ** exact r' \u27e8k, z\u27e9 ** case some.some K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' k\u271d k : \u03ba z : v' (some k\u271d) = v' (some k) \u22a2 some k\u271d = some k ** congr ** case some.some.e_val K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' k\u271d k : \u03ba z : v' (some k\u271d) = v' (some k) \u22a2 k\u271d = k ** exact i.injective z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' \u22a2 LinearIndependent R Subtype.val ** rw [linearIndependent_subtype_range inj', linearIndependent_iff] ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' \u22a2 \u2200 (l : Option \u03ba \u2192\u2080 R), \u2191(Finsupp.total (Option \u03ba) M R v') l = 0 \u2192 l = 0 ** intro l z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191(Finsupp.total (Option \u03ba) M R v') l = 0 \u22a2 l = 0 ** rw [Finsupp.total_option] at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 v' none + \u2191(Finsupp.total \u03ba M R (v' \u2218 some)) (Finsupp.some l) = 0 \u22a2 l = 0 ** simp only [Option.elim'] at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R ((fun o => Option.elim o (\u2191b b') v) \u2218 some)) (Finsupp.some l) = 0 \u22a2 l = 0 ** change _ + Finsupp.total \u03ba M R v l.some = 0 at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 \u22a2 l = 0 ** have l\u2080 : l none = 0 := by\n rw [\u2190 eq_neg_iff_add_eq_zero] at z\n replace z := neg_eq_iff_eq_neg.mpr z\n apply_fun fun x => b.repr x b' at z\n simp only [repr_self, LinearEquiv.map_smul, mul_one, Finsupp.single_eq_same, Pi.neg_apply,\n Finsupp.smul_single', LinearEquiv.map_neg, Finsupp.coe_neg] at z\n erw [FunLike.congr_fun (Finsupp.apply_total R (b.repr : M \u2192\u2097[R] \u03b9 \u2192\u2080 R) v l.some) b'] at z\n simpa [Finsupp.total_apply, w] using z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 \u22a2 l = 0 ** have l\u2081 : l.some = 0 := by\n rw [l\u2080, zero_smul, zero_add] at z\n exact linearIndependent_iff.mp i _ z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 l\u2081 : Finsupp.some l = 0 \u22a2 l = 0 ** ext (_ | a) ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 \u22a2 \u2191l none = 0 ** rw [\u2190 eq_neg_iff_add_eq_zero] at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v = -\u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) \u22a2 \u2191l none = 0 ** replace z := neg_eq_iff_eq_neg.mpr z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : -(\u2191l none \u2022 Option.elim none (\u2191b b') v) = \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) \u22a2 \u2191l none = 0 ** apply_fun fun x => b.repr x b' at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191(\u2191b.repr (-(\u2191l none \u2022 Option.elim none (\u2191b b') v))) b' = \u2191(\u2191b.repr (\u2191(Finsupp.total \u03ba M R v) (Finsupp.some l))) b' \u22a2 \u2191l none = 0 ** simp only [repr_self, LinearEquiv.map_smul, mul_one, Finsupp.single_eq_same, Pi.neg_apply,\n Finsupp.smul_single', LinearEquiv.map_neg, Finsupp.coe_neg] at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : -\u2191(\u2191l none \u2022 \u2191b.repr (Option.elim none (\u2191b b') v)) b' = \u2191(\u2191b.repr (\u2191(Finsupp.total \u03ba M R v) (Finsupp.some l))) b' \u22a2 \u2191l none = 0 ** erw [FunLike.congr_fun (Finsupp.apply_total R (b.repr : M \u2192\u2097[R] \u03b9 \u2192\u2080 R) v l.some) b'] at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : -\u2191(\u2191l none \u2022 \u2191b.repr (Option.elim none (\u2191b b') v)) b' = \u2191(\u2191(Finsupp.total \u03ba (\u03b9 \u2192\u2080 R) R (\u2191\u2191b.repr \u2218 v)) (Finsupp.some l)) b' \u22a2 \u2191l none = 0 ** simpa [Finsupp.total_apply, w] using z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 \u22a2 Finsupp.some l = 0 ** rw [l\u2080, zero_smul, zero_add] at z ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 \u22a2 Finsupp.some l = 0 ** exact linearIndependent_iff.mp i _ z ** case h.none K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 l\u2081 : Finsupp.some l = 0 \u22a2 \u2191l none = \u21910 none ** simp only [l\u2080, Finsupp.coe_zero, Pi.zero_apply] ** case h.some K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 l\u2081 : Finsupp.some l = 0 a : \u03ba \u22a2 \u2191l (some a) = \u21910 (some a) ** erw [FunLike.congr_fun l\u2081 a] ** case h.some K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9\u271d : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u03b9 : Type w b : Basis \u03b9 R M \u03ba : Type w' v : \u03ba \u2192 M i : LinearIndependent R v m : LinearIndependent.Maximal i b' : \u03b9 w : \u2200 (x : \u03ba), \u2191(\u2191b.repr (v x)) b' = 0 v' : Option \u03ba \u2192 M := fun o => Option.elim o (\u2191b b') v r : range v \u2286 range v' r' : \u00ac\u2191b b' \u2208 range v r'' : range v \u2260 range v' inj' : Injective v' l : Option \u03ba \u2192\u2080 R z : \u2191l none \u2022 Option.elim none (\u2191b b') v + \u2191(Finsupp.total \u03ba M R v) (Finsupp.some l) = 0 l\u2080 : \u2191l none = 0 l\u2081 : Finsupp.some l = 0 a : \u03ba \u22a2 \u21910 a = \u21910 (some a) ** simp only [Finsupp.coe_zero, Pi.zero_apply] ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.degrees_add_of_disjoint ** R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees (p + q) = degrees p \u222a degrees q ** apply le_antisymm ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees (p + q) \u2264 degrees p \u222a degrees q ** apply degrees_add ** case a R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees p \u222a degrees q \u2264 degrees (p + q) ** apply Multiset.union_le ** case a.h\u2081 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees p \u2264 degrees (p + q) ** apply le_degrees_add h ** case a.h\u2082 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees q \u2264 degrees (p + q) ** rw [add_comm] ** case a.h\u2082 R : Type u S : Type v \u03c3 : Type u_1 \u03c4 : Type u_2 r : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : MvPolynomial \u03c3 R inst\u271d : DecidableEq \u03c3 p q : MvPolynomial \u03c3 R h : Multiset.Disjoint (degrees p) (degrees q) \u22a2 degrees q \u2264 degrees (q + p) ** apply le_degrees_add h.symm ** Qed", + "informal": "" + }, + { + "formal": "Mathlib.Meta.NormNum.isNat_natDiv ** n\u271d\u00b9 n\u271d : \u2115 \u22a2 \u2191n\u271d\u00b9 / \u2191n\u271d = \u2191(Nat.div n\u271d\u00b9 n\u271d) ** aesop ** Qed", + "informal": "" + }, + { + "formal": "NNReal.mul_lt_of_lt_div ** a b r : \u211d\u22650 h : a < b / r hr : r = 0 \u22a2 False ** simp [hr] at h ** Qed", + "informal": "" + }, + { + "formal": "Matrix.isHermitian_transpose_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : Type u_3 n : Type u_4 A\u271d : Matrix n n \u03b1 inst\u271d\u00b9 : Star \u03b1 inst\u271d : Star \u03b2 A : Matrix n n \u03b1 \u22a2 IsHermitian A\u1d40 \u2192 IsHermitian A ** intro h ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : Type u_3 n : Type u_4 A\u271d : Matrix n n \u03b1 inst\u271d\u00b9 : Star \u03b1 inst\u271d : Star \u03b2 A : Matrix n n \u03b1 h : IsHermitian A\u1d40 \u22a2 IsHermitian A ** rw [\u2190 transpose_transpose A] ** \u03b1 : Type u_1 \u03b2 : Type u_2 m : Type u_3 n : Type u_4 A\u271d : Matrix n n \u03b1 inst\u271d\u00b9 : Star \u03b1 inst\u271d : Star \u03b2 A : Matrix n n \u03b1 h : IsHermitian A\u1d40 \u22a2 IsHermitian A\u1d40\u1d40 ** exact IsHermitian.transpose h ** Qed", + "informal": "" + }, + { + "formal": "sum_Ioc_inv_sq_le_sub ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n : \u2115 hk : k \u2260 0 h : k \u2264 n \u22a2 \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 ** refine' Nat.le_induction _ _ n h ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n : \u2115 hk : k \u2260 0 h : k \u2264 n \u22a2 \u2200 (n : \u2115), k \u2264 n \u2192 \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u2192 \u2211 i in Ioc k (n + 1), (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9 ** intro n hn IH ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u22a2 \u2211 i in Ioc k (n + 1), (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9 ** rw [sum_Ioc_succ_top hn] ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u22a2 \u2211 k in Ioc k n, (\u2191k ^ 2)\u207b\u00b9 + (\u2191(n + 1) ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9 ** apply (add_le_add IH le_rfl).trans ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u22a2 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 + (\u2191(n + 1) ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191(n + 1))\u207b\u00b9 ** simp only [sub_eq_add_neg, add_assoc, Nat.cast_add, Nat.cast_one, le_add_neg_iff_add_le,\n add_le_iff_nonpos_right, neg_add_le_iff_le_add, add_zero] ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u22a2 ((\u2191n + 1) ^ 2)\u207b\u00b9 + (\u2191n + 1)\u207b\u00b9 \u2264 (\u2191n)\u207b\u00b9 ** have A : 0 < (n : \u03b1) := by simpa using hk.bot_lt.trans_le hn ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n \u22a2 ((\u2191n + 1) ^ 2)\u207b\u00b9 + (\u2191n + 1)\u207b\u00b9 \u2264 (\u2191n)\u207b\u00b9 ** have B : 0 < (n : \u03b1) + 1 := by linarith ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n B : 0 < \u2191n + 1 \u22a2 ((\u2191n + 1) ^ 2)\u207b\u00b9 + (\u2191n + 1)\u207b\u00b9 \u2264 (\u2191n)\u207b\u00b9 ** field_simp ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n B : 0 < \u2191n + 1 \u22a2 (\u2191n + 1 + (\u2191n + 1) ^ 2) / ((\u2191n + 1) ^ 2 * (\u2191n + 1)) \u2264 1 / \u2191n ** rw [div_le_div_iff _ A, \u2190 sub_nonneg] ** case refine'_1 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n : \u2115 hk : k \u2260 0 h : k \u2264 n \u22a2 \u2211 i in Ioc k k, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191k)\u207b\u00b9 ** simp only [Ioc_self, sum_empty, sub_self, le_refl] ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 \u22a2 0 < \u2191n ** simpa using hk.bot_lt.trans_le hn ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n \u22a2 0 < \u2191n + 1 ** linarith ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n B : 0 < \u2191n + 1 \u22a2 0 \u2264 1 * ((\u2191n + 1) ^ 2 * (\u2191n + 1)) - (\u2191n + 1 + (\u2191n + 1) ^ 2) * \u2191n ** ring_nf ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n B : 0 < \u2191n + 1 \u22a2 0 \u2264 1 + \u2191n ** rw [add_comm] ** case refine'_2 \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n B : 0 < \u2191n + 1 \u22a2 0 \u2264 \u2191n + 1 ** exact B.le ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 k n\u271d : \u2115 hk : k \u2260 0 h : k \u2264 n\u271d n : \u2115 hn : k \u2264 n IH : \u2211 i in Ioc k n, (\u2191i ^ 2)\u207b\u00b9 \u2264 (\u2191k)\u207b\u00b9 - (\u2191n)\u207b\u00b9 A : 0 < \u2191n B : 0 < \u2191n + 1 \u22a2 0 < (\u2191n + 1) ^ 2 * (\u2191n + 1) ** positivity ** Qed", + "informal": "" + }, + { + "formal": "Nat.pow_dvd_pow_iff_le_right ** x k l : \u2115 w : 1 < x \u22a2 x ^ k \u2223 x ^ l \u2194 k \u2264 l ** rw [pow_dvd_pow_iff_pow_le_pow (lt_of_succ_lt w), pow_le_iff_le_right w] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_nat_cast_ite ** R : Type u a b : R m n : \u2115 inst\u271d : Semiring R p q : R[X] \u22a2 coeff (\u2191m) n = \u2191(if n = 0 then m else 0) ** simp only [\u2190 C_eq_nat_cast, coeff_C, Nat.cast_ite, Nat.cast_zero] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.cgf_undef ** \u03a9 : Type u_1 \u03b9 : Type u_2 m : MeasurableSpace \u03a9 X : \u03a9 \u2192 \u211d p : \u2115 \u03bc : Measure \u03a9 t : \u211d hX : \u00acIntegrable fun \u03c9 => rexp (t * X \u03c9) \u22a2 cgf X \u03bc t = 0 ** simp only [cgf, mgf_undef hX, log_zero] ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.support_indicator_subset ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 i : \u03b9 \u22a2 \u2191(indicator s f).support \u2286 \u2191s ** intro i hi ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 i\u271d i : \u03b9 hi : i \u2208 \u2191(indicator s f).support \u22a2 i \u2208 \u2191s ** rw [mem_coe, mem_support_iff] at hi ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 i\u271d i : \u03b9 hi : \u2191(indicator s f) i \u2260 0 \u22a2 i \u2208 \u2191s ** by_contra h ** \u03b9 : Type u_1 \u03b1 : Type u_2 inst\u271d : Zero \u03b1 s : Finset \u03b9 f : (i : \u03b9) \u2192 i \u2208 s \u2192 \u03b1 i\u271d i : \u03b9 hi : \u2191(indicator s f) i \u2260 0 h : \u00aci \u2208 \u2191s \u22a2 False ** exact hi (indicator_of_not_mem h _) ** Qed", + "informal": "" + }, + { + "formal": "Set.chainHeight_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 \u22a2 chainHeight (f '' s) = chainHeight s ** apply le_antisymm <;> rw [chainHeight_le_chainHeight_iff] ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 \u22a2 \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 length l = length l' ** suffices \u2200 l \u2208 (f '' s).subchain, \u2203 l' \u2208 s.subchain, map f l' = l by\n intro l hl\n obtain \u27e8l', h\u2081, rfl\u27e9 := this l hl\n exact \u27e8l', h\u2081, length_map _ _\u27e9 ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 \u22a2 \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l ** intro l ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b2 \u22a2 l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l ** induction' l with x xs hx ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 this : \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l \u22a2 \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 length l = length l' ** intro l hl ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 this : \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l l : List \u03b2 hl : l \u2208 subchain (f '' s) \u22a2 \u2203 l', l' \u2208 subchain s \u2227 length l = length l' ** obtain \u27e8l', h\u2081, rfl\u27e9 := this l hl ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 this : \u2200 (l : List \u03b2), l \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = l l' : List \u03b1 h\u2081 : l' \u2208 subchain s hl : map f l' \u2208 subchain (f '' s) \u22a2 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 length (map f l') = length l'_1 ** exact \u27e8l', h\u2081, length_map _ _\u27e9 ** case a.nil \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 \u22a2 [] \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = [] ** exact fun _ \u21a6 \u27e8nil, \u27e8trivial, fun x h \u21a6 (not_mem_nil x h).elim\u27e9, rfl\u27e9 ** case a.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b2 xs : List \u03b2 hx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs \u22a2 x :: xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = x :: xs ** intro h ** case a.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b2 xs : List \u03b2 hx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs h : x :: xs \u2208 subchain (f '' s) \u22a2 \u2203 l', l' \u2208 subchain s \u2227 map f l' = x :: xs ** rw [cons_mem_subchain_iff] at h ** case a.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b2 xs : List \u03b2 hx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs h : x \u2208 f '' s \u2227 xs \u2208 subchain (f '' s) \u2227 \u2200 (b : \u03b2), b \u2208 head? xs \u2192 x < b \u22a2 \u2203 l', l' \u2208 subchain s \u2227 map f l' = x :: xs ** obtain \u27e8\u27e8x, hx', rfl\u27e9, h\u2081, h\u2082\u27e9 := h ** case a.cons.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 xs : List \u03b2 hx : xs \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = xs x : \u03b1 hx' : x \u2208 s h\u2081 : xs \u2208 subchain (f '' s) h\u2082 : \u2200 (b : \u03b2), b \u2208 head? xs \u2192 f x < b \u22a2 \u2203 l', l' \u2208 subchain s \u2227 map f l' = f x :: xs ** obtain \u27e8l', h\u2083, rfl\u27e9 := hx h\u2081 ** case a.cons.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b1 hx' : x \u2208 s l' : List \u03b1 h\u2083 : l' \u2208 subchain s hx : map f l' \u2208 subchain (f '' s) \u2192 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 map f l'_1 = map f l' h\u2081 : map f l' \u2208 subchain (f '' s) h\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f l') \u2192 f x < b \u22a2 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 map f l'_1 = f x :: map f l' ** refine' \u27e8x::l', Set.cons_mem_subchain_iff.mpr \u27e8hx', h\u2083, _\u27e9, rfl\u27e9 ** case a.cons.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b1 hx' : x \u2208 s l' : List \u03b1 h\u2083 : l' \u2208 subchain s hx : map f l' \u2208 subchain (f '' s) \u2192 \u2203 l'_1, l'_1 \u2208 subchain s \u2227 map f l'_1 = map f l' h\u2081 : map f l' \u2208 subchain (f '' s) h\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f l') \u2192 f x < b \u22a2 \u2200 (b : \u03b1), b \u2208 head? l' \u2192 x < b ** cases l' ** case a.cons.intro.intro.intro.intro.intro.intro.nil \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b1 hx' : x \u2208 s h\u2083 : [] \u2208 subchain s hx : map f [] \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = map f [] h\u2081 : map f [] \u2208 subchain (f '' s) h\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f []) \u2192 f x < b \u22a2 \u2200 (b : \u03b1), b \u2208 head? [] \u2192 x < b ** simp ** case a.cons.intro.intro.intro.intro.intro.intro.cons \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 x : \u03b1 hx' : x \u2208 s head\u271d : \u03b1 tail\u271d : List \u03b1 h\u2083 : head\u271d :: tail\u271d \u2208 subchain s hx : map f (head\u271d :: tail\u271d) \u2208 subchain (f '' s) \u2192 \u2203 l', l' \u2208 subchain s \u2227 map f l' = map f (head\u271d :: tail\u271d) h\u2081 : map f (head\u271d :: tail\u271d) \u2208 subchain (f '' s) h\u2082 : \u2200 (b : \u03b2), b \u2208 head? (map f (head\u271d :: tail\u271d)) \u2192 f x < b \u22a2 \u2200 (b : \u03b1), b \u2208 head? (head\u271d :: tail\u271d) \u2192 x < b ** simpa [\u2190 hf] using h\u2082 ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 \u22a2 \u2200 (l : List \u03b1), l \u2208 subchain s \u2192 \u2203 l', l' \u2208 subchain (f '' s) \u2227 length l = length l' ** intro l hl ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s \u22a2 \u2203 l', l' \u2208 subchain (f '' s) \u2227 length l = length l' ** refine' \u27e8l.map f, \u27e8_, _\u27e9, _\u27e9 ** case a.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s \u22a2 Chain' (fun x x_1 => x < x_1) (map f l) ** simp_rw [chain'_map, \u2190 hf] ** case a.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s \u22a2 Chain' (fun a b => a < b) l ** exact hl.1 ** case a.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s \u22a2 \u2200 (i : \u03b2), i \u2208 map f l \u2192 i \u2208 f '' s ** intro _ e ** case a.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s i\u271d : \u03b2 e : i\u271d \u2208 map f l \u22a2 i\u271d \u2208 f '' s ** obtain \u27e8a, ha, rfl\u27e9 := mem_map.mp e ** case a.refine'_2.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a\u271d : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s a : \u03b1 ha : a \u2208 l e : f a \u2208 map f l \u22a2 f a \u2208 f '' s ** exact Set.mem_image_of_mem _ (hl.2 _ ha) ** case a.refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LT \u03b1 inst\u271d : LT \u03b2 s\u271d t : Set \u03b1 l\u271d : List \u03b1 a : \u03b1 f : \u03b1 \u2192 \u03b2 hf : \u2200 {x y : \u03b1}, x < y \u2194 f x < f y s : Set \u03b1 l : List \u03b1 hl : l \u2208 subchain s \u22a2 length l = length (map f l) ** rw [length_map] ** Qed", + "informal": "" + }, + { + "formal": "Finset.eq_univ_of_card ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Fintype \u03b1 s : Finset \u03b1 hs : card s = Fintype.card \u03b1 \u22a2 card univ \u2264 card s ** rw [hs, Finset.card_univ] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_erase_lt_of_one_lt ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3\u271d : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : CommMonoid \u03b2 \u03b3 : Type u_2 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : OrderedCommMonoid \u03b3 inst\u271d : CovariantClass \u03b3 \u03b3 (fun x x_1 => x * x_1) fun x x_1 => x < x_1 s : Finset \u03b1 d : \u03b1 hd : d \u2208 s f : \u03b1 \u2192 \u03b3 hdf : 1 < f d \u22a2 \u220f m in erase s d, f m < \u220f m in s, f m ** conv in \u220f m in s, f m => rw [\u2190 Finset.insert_erase hd] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3\u271d : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : CommMonoid \u03b2 \u03b3 : Type u_2 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : OrderedCommMonoid \u03b3 inst\u271d : CovariantClass \u03b3 \u03b3 (fun x x_1 => x * x_1) fun x x_1 => x < x_1 s : Finset \u03b1 d : \u03b1 hd : d \u2208 s f : \u03b1 \u2192 \u03b3 hdf : 1 < f d \u22a2 \u220f m in erase s d, f m < \u220f m in insert d (erase s d), f m ** rw [Finset.prod_insert (Finset.not_mem_erase d s)] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3\u271d : Type w s\u271d s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g : \u03b1 \u2192 \u03b2 inst\u271d\u00b3 : CommMonoid \u03b2 \u03b3 : Type u_2 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : OrderedCommMonoid \u03b3 inst\u271d : CovariantClass \u03b3 \u03b3 (fun x x_1 => x * x_1) fun x x_1 => x < x_1 s : Finset \u03b1 d : \u03b1 hd : d \u2208 s f : \u03b1 \u2192 \u03b3 hdf : 1 < f d \u22a2 \u220f m in erase s d, f m < f d * \u220f x in erase s d, f x ** exact lt_mul_of_one_lt_left' _ hdf ** Qed", + "informal": "" + }, + { + "formal": "pow_eq_mod_orderOf ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Monoid G inst\u271d : AddMonoid A x y : G a b : A n\u271d m n : \u2115 \u22a2 x ^ n = x ^ (n % orderOf x + orderOf x * (n / orderOf x)) ** rw [Nat.mod_add_div] ** G : Type u_1 H : Type u_2 A : Type u_3 \u03b1 : Type u_4 \u03b2 : Type u_5 inst\u271d\u00b9 : Monoid G inst\u271d : AddMonoid A x y : G a b : A n\u271d m n : \u2115 \u22a2 x ^ (n % orderOf x + orderOf x * (n / orderOf x)) = x ^ (n % orderOf x) ** simp [pow_add, pow_mul, pow_orderOf_eq_one] ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.Finset.prod ** \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03b2 : Type u_2 p : \u03b1 hp : Prime p s : Finset \u03b2 f : \u03b2 \u2192 \u03b1 \u22a2 multiplicity p (\u220f x in s, f x) = \u2211 x in s, multiplicity p (f x) ** induction' s using Finset.induction with a s has ih h ** case empty \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03b2 : Type u_2 p : \u03b1 hp : Prime p f : \u03b2 \u2192 \u03b1 \u22a2 multiplicity p (\u220f x in \u2205, f x) = \u2211 x in \u2205, multiplicity p (f x) ** simp only [Finset.sum_empty, Finset.prod_empty] ** case empty \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03b2 : Type u_2 p : \u03b1 hp : Prime p f : \u03b2 \u2192 \u03b1 \u22a2 multiplicity p 1 = 0 ** convert one_right hp.not_unit ** case insert \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03b2 : Type u_2 p : \u03b1 hp : Prime p f : \u03b2 \u2192 \u03b1 a : \u03b2 s : Finset \u03b2 has : \u00aca \u2208 s ih : multiplicity p (\u220f x in s, f x) = \u2211 x in s, multiplicity p (f x) \u22a2 multiplicity p (\u220f x in insert a s, f x) = \u2211 x in insert a s, multiplicity p (f x) ** simp [has, \u2190 ih] ** case insert \u03b1 : Type u_1 inst\u271d\u00b9 : CancelCommMonoidWithZero \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 \u03b2 : Type u_2 p : \u03b1 hp : Prime p f : \u03b2 \u2192 \u03b1 a : \u03b2 s : Finset \u03b2 has : \u00aca \u2208 s ih : multiplicity p (\u220f x in s, f x) = \u2211 x in s, multiplicity p (f x) \u22a2 multiplicity p (f a * \u220f x in s, f x) = multiplicity p (f a) + multiplicity p (\u220f x in s, f x) ** convert multiplicity.mul hp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.IsFundamentalDomain.measure_eq ** G : Type u_1 H : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 E : Type u_5 inst\u271d\u00b9\u2070 : Group G inst\u271d\u2079 : Group H inst\u271d\u2078 : MulAction G \u03b1 inst\u271d\u2077 : MeasurableSpace \u03b1 inst\u271d\u2076 : MulAction H \u03b2 inst\u271d\u2075 : MeasurableSpace \u03b2 inst\u271d\u2074 : NormedAddCommGroup E s t : Set \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b3 : MeasurableSpace G inst\u271d\u00b2 : MeasurableSMul G \u03b1 inst\u271d\u00b9 : SMulInvariantMeasure G \u03b1 \u03bc inst\u271d : Countable G \u03bd : Measure \u03b1 hs : IsFundamentalDomain G s ht : IsFundamentalDomain G t \u22a2 \u2191\u2191\u03bc s = \u2191\u2191\u03bc t ** simpa only [set_lintegral_one] using hs.set_lintegral_eq ht (fun _ => 1) fun _ _ => rfl ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.deleteEdges_empty_eq ** \u03b9 : Sort u_1 \ud835\udd5c : Type u_2 V : Type u W : Type v X : Type w G : SimpleGraph V G' : SimpleGraph W a b c u v w : V e : Sym2 V \u22a2 deleteEdges G \u2205 = G ** ext ** case Adj.h.h.a \u03b9 : Sort u_1 \ud835\udd5c : Type u_2 V : Type u W : Type v X : Type w G : SimpleGraph V G' : SimpleGraph W a b c u v w : V e : Sym2 V x\u271d\u00b9 x\u271d : V \u22a2 Adj (deleteEdges G \u2205) x\u271d\u00b9 x\u271d \u2194 Adj G x\u271d\u00b9 x\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "Set.Ioo_inter_Ioc_of_right_lt ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b9 : LinearOrder \u03b1 inst\u271d : LinearOrder \u03b2 f : \u03b1 \u2192 \u03b2 a a\u2081 a\u2082 b b\u2081 b\u2082 c d : \u03b1 h : b\u2082 < b\u2081 \u22a2 Ioo a\u2081 b\u2081 \u2229 Ioc a\u2082 b\u2082 = Ioc (max a\u2081 a\u2082) b\u2082 ** rw [inter_comm, Ioc_inter_Ioo_of_left_lt h, max_comm] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.isCokernelEpiComp_desc ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasZeroMorphisms C X Y : C f : X \u27f6 Y c : CokernelCofork f i : IsColimit c W : C g : W \u27f6 X hg : Epi g h : W \u27f6 Y hh : h = g \u226b f s : CokernelCofork h \u22a2 f \u226b Cofork.\u03c0 s = 0 \u226b Cofork.\u03c0 s ** rw [\u2190 cancel_epi g, \u2190 Category.assoc, \u2190 hh] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasZeroMorphisms C X Y : C f : X \u27f6 Y c : CokernelCofork f i : IsColimit c W : C g : W \u27f6 X hg : Epi g h : W \u27f6 Y hh : h = g \u226b f s : CokernelCofork h \u22a2 h \u226b Cofork.\u03c0 s = g \u226b 0 \u226b Cofork.\u03c0 s ** simp ** Qed", + "informal": "" + }, + { + "formal": "Orientation.rightAngleRotation_symm ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u22a2 LinearIsometryEquiv.symm (rightAngleRotation o) = LinearIsometryEquiv.trans (rightAngleRotation o) (LinearIsometryEquiv.neg \u211d) ** rw [rightAngleRotation] ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) \u22a2 LinearIsometryEquiv.symm (LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o) (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id) (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id)) = LinearIsometryEquiv.trans (LinearIsometryEquiv.ofLinearIsometry (rightAngleRotationAux\u2082 o) (-rightAngleRotationAux\u2081 o) (_ : LinearMap.comp (rightAngleRotationAux\u2082 o).toLinearMap (-rightAngleRotationAux\u2081 o) = LinearMap.id) (_ : LinearMap.comp (-rightAngleRotationAux\u2081 o) (rightAngleRotationAux\u2082 o).toLinearMap = LinearMap.id)) (LinearIsometryEquiv.neg \u211d) ** exact LinearIsometryEquiv.toLinearIsometry_injective rfl ** Qed", + "informal": "" + }, + { + "formal": "Orientation.norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V hd2 : Fact (finrank \u211d V = 2) o : Orientation \u211d V (Fin 2) x y : V h : oangle o x y = \u2191(\u03c0 / 2) \u22a2 \u2016y\u2016 / Real.Angle.sin (oangle o (x - y) x) = \u2016x - y\u2016 ** rw [\u2190 neg_inj, oangle_rev, \u2190 oangle_neg_orientation_eq_neg, neg_inj] at h \u22a2 ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V hd2 : Fact (finrank \u211d V = 2) o : Orientation \u211d V (Fin 2) x y : V h\u271d : oangle o x y = \u2191(\u03c0 / 2) h : oangle (-o) y x = \u2191(\u03c0 / 2) \u22a2 \u2016y\u2016 / Real.Angle.sin (oangle (-o) x (x - y)) = \u2016x - y\u2016 ** exact (-o).norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two h ** Qed", + "informal": "" + }, + { + "formal": "Multiset.support_sum_eq ** \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s \u22a2 (sum s).support = sup (map Finsupp.support s) ** induction' s using Quot.inductionOn with a ** case h \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs\u271d : Pairwise (_root_.Disjoint on Finsupp.support) s a : List (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) (Quot.mk Setoid.r a) \u22a2 (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a)) ** obtain \u27e8l, hl, hd\u27e9 := hs ** case h.intro.intro \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hl : Quot.mk Setoid.r a = \u2191l hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l \u22a2 (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a)) ** suffices : a.Pairwise (_root_.Disjoint on Finsupp.support) ** case h.intro.intro \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hl : Quot.mk Setoid.r a = \u2191l hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l this : List.Pairwise (_root_.Disjoint on Finsupp.support) a \u22a2 (sum (Quot.mk Setoid.r a)).support = sup (map Finsupp.support (Quot.mk Setoid.r a)) ** convert List.support_sum_eq a this ** case h.e'_2.h.e'_4 \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hl : Quot.mk Setoid.r a = \u2191l hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l this : List.Pairwise (_root_.Disjoint on Finsupp.support) a \u22a2 sum (Quot.mk Setoid.r a) = List.sum a ** simp only [Multiset.quot_mk_to_coe'', Multiset.coe_sum] ** case h.e'_3 \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hl : Quot.mk Setoid.r a = \u2191l hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l this : List.Pairwise (_root_.Disjoint on Finsupp.support) a \u22a2 sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr ((fun x x_1 => x \u2294 x_1) \u2218 Finsupp.support) \u2205 a ** dsimp only [Function.comp] ** case h.e'_3 \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hl : Quot.mk Setoid.r a = \u2191l hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l this : List.Pairwise (_root_.Disjoint on Finsupp.support) a \u22a2 sup (map Finsupp.support (Quot.mk Setoid.r a)) = List.foldr (fun x x_1 => x.support \u2294 x_1) \u2205 a ** simp only [quot_mk_to_coe'', coe_map, sup_coe, ge_iff_le, Finset.le_eq_subset,\n Finset.sup_eq_union, Finset.bot_eq_empty, List.foldr_map] ** case this \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hl : Quot.mk Setoid.r a = \u2191l hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l \u22a2 List.Pairwise (_root_.Disjoint on Finsupp.support) a ** simp only [Multiset.quot_mk_to_coe'', Multiset.coe_map, Multiset.coe_eq_coe] at hl ** case this \u03b9 : Type u_1 M : Type u_2 inst\u271d\u00b9 : DecidableEq \u03b9 inst\u271d : AddCommMonoid M s : Multiset (\u03b9 \u2192\u2080 M) hs : Pairwise (_root_.Disjoint on Finsupp.support) s a l : List (\u03b9 \u2192\u2080 M) hd : List.Pairwise (_root_.Disjoint on Finsupp.support) l hl : a ~ l \u22a2 List.Pairwise (_root_.Disjoint on Finsupp.support) a ** exact hl.symm.pairwise hd fun _ _ h \u21a6 _root_.Disjoint.symm h ** Qed", + "informal": "" + }, + { + "formal": "Nat.size_pow ** n : \u2115 \u22a2 1 < 2 ** decide ** Qed", + "informal": "" + }, + { + "formal": "List.insertNth_eq_insertNthTR ** \u22a2 @insertNth = @insertNthTR ** funext \u03b1 f n l ** case h.h.h.h \u03b1 : Type u_1 f : Nat n : \u03b1 l : List \u03b1 \u22a2 insertNth f n l = insertNthTR f n l ** simp [insertNthTR, insertNthTR_go_eq] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.condCdfRat_nonneg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 0 \u2264 condCdfRat \u03c1 a r ** unfold condCdfRat ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a \u22a2 0 \u2264 ite (a \u2208 condCdfSet \u03c1) (fun r => ENNReal.toReal (preCdf \u03c1 r a)) (fun r => if r < 0 then 0 else 1) r ** split_ifs ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h\u271d : \u00aca \u2208 condCdfSet \u03c1 \u22a2 0 \u2264 (fun r => if r < 0 then 0 else 1) r ** dsimp only ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h\u271d : \u00aca \u2208 condCdfSet \u03c1 \u22a2 0 \u2264 if r < 0 then 0 else 1 ** split_ifs ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h\u271d\u00b9 : \u00aca \u2208 condCdfSet \u03c1 h\u271d : r < 0 \u22a2 0 \u2264 0 case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h\u271d\u00b9 : \u00aca \u2208 condCdfSet \u03c1 h\u271d : \u00acr < 0 \u22a2 0 \u2264 1 ** exacts [le_rfl, zero_le_one] ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 \u03c1 : Measure (\u03b1 \u00d7 \u211d) a : \u03b1 r : \u211a h\u271d : a \u2208 condCdfSet \u03c1 \u22a2 0 \u2264 (fun r => ENNReal.toReal (preCdf \u03c1 r a)) r ** exact ENNReal.toReal_nonneg ** Qed", + "informal": "" + }, + { + "formal": "Module.End.hasEigenvalue_iff_mem_spectrum ** K R : Type v V M : Type w inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Field K inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module K V inst\u271d : FiniteDimensional K V f : End K V \u03bc : K h : \u03bc \u2208 spectrum K f \u22a2 HasEigenvalue f \u03bc ** rwa [spectrum.mem_iff, IsUnit.sub_iff, LinearMap.isUnit_iff_ker_eq_bot] at h ** Qed", + "informal": "" + }, + { + "formal": "TopologicalSpace.NoetherianSpace.exists_finset_irreducible ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d\u00b2 : TopologicalSpace \u03b1 inst\u271d\u00b9 : TopologicalSpace \u03b2 inst\u271d : NoetherianSpace \u03b1 s : Closeds \u03b1 \u22a2 \u2203 S, (\u2200 (k : { x // x \u2208 S }), IsIrreducible \u2191\u2191k) \u2227 s = Finset.sup S id ** simpa [Set.exists_finite_iff_finset, Finset.sup_id_eq_sSup]\n using NoetherianSpace.exists_finite_set_closeds_irreducible s ** Qed", + "informal": "" + }, + { + "formal": "StateT.goto_mkLabel ** m : Type u \u2192 Type v inst\u271d : Monad m \u03b1 \u03b2 \u03c3 : Type u x : Label (\u03b1 \u00d7 \u03c3) m (\u03b2 \u00d7 \u03c3) i : \u03b1 \u22a2 goto (mkLabel x) i = StateT.mk fun s => goto x (i, s) ** cases x ** case mk m : Type u \u2192 Type v inst\u271d : Monad m \u03b1 \u03b2 \u03c3 : Type u i : \u03b1 apply\u271d : \u03b1 \u00d7 \u03c3 \u2192 m (\u03b2 \u00d7 \u03c3) \u22a2 goto (mkLabel { apply := apply\u271d }) i = StateT.mk fun s => goto { apply := apply\u271d } (i, s) ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Real.not_summable_nat_cast_inv ** \u22a2 \u00acSummable fun n => (\u2191n)\u207b\u00b9 ** have : \u00acSummable (fun n => ((n : \u211d) ^ 1)\u207b\u00b9 : \u2115 \u2192 \u211d) :=\n mt (Real.summable_nat_pow_inv (p := 1)).1 (lt_irrefl 1) ** this : \u00acSummable fun n => (\u2191n ^ 1)\u207b\u00b9 \u22a2 \u00acSummable fun n => (\u2191n)\u207b\u00b9 ** simpa ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.support_zipWith ** \u03b9 : Type u \u03b3 : Type w \u03b2 : \u03b9 \u2192 Type v \u03b2\u2081 : \u03b9 \u2192 Type v\u2081 \u03b2\u2082 : \u03b9 \u2192 Type v\u2082 dec : DecidableEq \u03b9 inst\u271d\u2075 : (i : \u03b9) \u2192 Zero (\u03b2 i) inst\u271d\u2074 : (i : \u03b9) \u2192 (x : \u03b2 i) \u2192 Decidable (x \u2260 0) inst\u271d\u00b3 : (i : \u03b9) \u2192 Zero (\u03b2\u2081 i) inst\u271d\u00b2 : (i : \u03b9) \u2192 Zero (\u03b2\u2082 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 (x : \u03b2\u2081 i) \u2192 Decidable (x \u2260 0) inst\u271d : (i : \u03b9) \u2192 (x : \u03b2\u2082 i) \u2192 Decidable (x \u2260 0) f : (i : \u03b9) \u2192 \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i hf : \u2200 (i : \u03b9), f i 0 0 = 0 g\u2081 : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i g\u2082 : \u03a0\u2080 (i : \u03b9), \u03b2\u2082 i \u22a2 support (zipWith f hf g\u2081 g\u2082) \u2286 support g\u2081 \u222a support g\u2082 ** simp [zipWith_def] ** Qed", + "informal": "" + }, + { + "formal": "Zspan.vadd_mem_fundamentalDomain ** E : Type u_1 \u03b9 : Type u_2 K : Type u_3 inst\u271d\u2074 : NormedLinearOrderedField K inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace K E b : Basis \u03b9 K E inst\u271d\u00b9 : FloorRing K inst\u271d : Fintype \u03b9 y : { x // x \u2208 span \u2124 (Set.range \u2191b) } x : E \u22a2 y +\u1d65 x \u2208 fundamentalDomain b \u2194 y = -floor b x ** rw [Subtype.ext_iff, \u2190 add_right_inj x, AddSubgroupClass.coe_neg, \u2190 sub_eq_add_neg, \u2190 fract_apply,\n \u2190 fract_zspan_add b _ (Subtype.mem y), add_comm, \u2190 vadd_eq_add, \u2190 vadd_def, eq_comm, \u2190\n fract_eq_self] ** Qed", + "informal": "" + }, + { + "formal": "List.mem_keys_kunion ** \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v l l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2) inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2081 l\u2082 : List (Sigma \u03b2) \u22a2 a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082 ** induction l\u2081 generalizing l\u2082 with\n| nil => simp\n| cons s l\u2081 ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]] ** case nil \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v l l\u2081 l\u2082\u271d : List (Sigma \u03b2) inst\u271d : DecidableEq \u03b1 a : \u03b1 l\u2082 : List (Sigma \u03b2) \u22a2 a \u2208 keys (kunion [] l\u2082) \u2194 a \u2208 keys [] \u2228 a \u2208 keys l\u2082 ** simp ** case cons \u03b1 : Type u \u03b2 : \u03b1 \u2192 Type v l l\u2081\u271d l\u2082\u271d : List (Sigma \u03b2) inst\u271d : DecidableEq \u03b1 a : \u03b1 s : Sigma \u03b2 l\u2081 : List (Sigma \u03b2) ih : \u2200 {l\u2082 : List (Sigma \u03b2)}, a \u2208 keys (kunion l\u2081 l\u2082) \u2194 a \u2208 keys l\u2081 \u2228 a \u2208 keys l\u2082 l\u2082 : List (Sigma \u03b2) \u22a2 a \u2208 keys (kunion (s :: l\u2081) l\u2082) \u2194 a \u2208 keys (s :: l\u2081) \u2228 a \u2208 keys l\u2082 ** by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]] ** Qed", + "informal": "" + }, + { + "formal": "iSup_unpair ** \u03b1 : Type u_1 inst\u271d : CompleteLattice \u03b1 f : \u2115 \u2192 \u2115 \u2192 \u03b1 \u22a2 \u2a06 n, f (unpair n).1 (unpair n).2 = \u2a06 i, \u2a06 j, f i j ** rw [\u2190 (iSup_prod : \u2a06 i : \u2115 \u00d7 \u2115, f i.1 i.2 = _), \u2190 Nat.surjective_unpair.iSup_comp] ** Qed", + "informal": "" + }, + { + "formal": "dist_self_mul_left ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b9 : SeminormedCommGroup E inst\u271d : SeminormedCommGroup F a\u271d a\u2081 a\u2082 b\u271d b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d a b : E \u22a2 dist (a * b) a = \u2016b\u2016 ** rw [dist_comm, dist_self_mul_right] ** Qed", + "informal": "" + }, + { + "formal": "Filter.tendsto_cocompact_mul_left ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Monoid M inst\u271d : ContinuousMul M a b : M ha : b * a = 1 \u22a2 Tendsto (fun x => a * x) (cocompact M) (cocompact M) ** refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_left b)) ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Monoid M inst\u271d : ContinuousMul M a b : M ha : b * a = 1 \u22a2 Tendsto ((fun b_1 => b * b_1) \u2218 fun x => a * x) (cocompact M) (cocompact M) ** simp only [comp_mul_left, ha, one_mul] ** \u03b9 : Type u_1 \u03b1 : Type u_2 X : Type u_3 M : Type u_4 N : Type u_5 inst\u271d\u00b3 : TopologicalSpace X inst\u271d\u00b2 : TopologicalSpace M inst\u271d\u00b9 : Monoid M inst\u271d : ContinuousMul M a b : M ha : b * a = 1 \u22a2 Tendsto (fun x => x) (cocompact M) (cocompact M) ** exact Filter.tendsto_id ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app ** C : Type u_1 inst\u271d : Category.{?u.41687, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X \u22a2 U = (Opens.map f.base).obj (op ((openFunctor H).obj U)).unop ** ext ** case h.h C : Type u_1 inst\u271d : Category.{?u.41687, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X x\u271d : \u2191\u2191X \u22a2 x\u271d \u2208 \u2191U \u2194 x\u271d \u2208 \u2191((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop) ** dsimp [openFunctor, IsOpenMap.functor] ** case h.h C : Type u_1 inst\u271d : Category.{?u.41687, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X x\u271d : \u2191\u2191X \u22a2 x\u271d \u2208 \u2191U \u2194 x\u271d \u2208 \u2191f.base \u207b\u00b9' (\u2191f.base '' \u2191U) ** rw [Set.preimage_image_eq _ H.base_open.inj] ** C : Type u_1 inst\u271d : Category.{u_3, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X \u22a2 invApp H U \u226b f.c.app (op ((openFunctor H).obj U)) = X.presheaf.map (eqToHom (_ : op U = op ((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop))) ** rw [invApp, Category.assoc, IsIso.inv_hom_id, Category.comp_id] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.SplittingFieldAux.adjoin_rootSet ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n \u22a2 Algebra.adjoin K (rootSet f (SplittingFieldAux (Nat.succ n) f)) = \u22a4 ** have hndf : f.natDegree \u2260 0 := by intro h; rw [h] at hfn; cases hfn ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 \u22a2 Algebra.adjoin K (rootSet f (SplittingFieldAux (Nat.succ n) f)) = \u22a4 ** have hfn0 : f \u2260 0 := by intro h; rw [h] at hndf; exact hndf rfl ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 \u22a2 Algebra.adjoin K (rootSet f (SplittingFieldAux (Nat.succ n) f)) = \u22a4 ** have hmf0 : map (algebraMap K (SplittingFieldAux n.succ f)) f \u2260 0 := map_ne_zero hfn0 ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 hmf0 : map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f \u2260 0 \u22a2 Algebra.adjoin K (rootSet f (SplittingFieldAux (Nat.succ n) f)) = \u22a4 ** rw [rootSet_def, aroots_def] ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 hmf0 : map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f \u2260 0 \u22a2 Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap K (SplittingFieldAux (Nat.succ n) f)) f))) = \u22a4 ** rw [algebraMap_succ, \u2190map_map, \u2190X_sub_C_mul_removeFactor _ hndf, Polynomial.map_mul] at hmf0 \u22a2 ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 hmf0 : map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (X - \u2191C (AdjoinRoot.root (factor f))) * map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260 0 \u22a2 Algebra.adjoin K \u2191(Multiset.toFinset (roots (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (X - \u2191C (AdjoinRoot.root (factor f))) * map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f)))) = \u22a4 ** erw [roots_mul hmf0, Polynomial.map_sub, map_X, map_C, roots_X_sub_C, Multiset.toFinset_add,\n Finset.coe_union, Multiset.toFinset_singleton, Finset.coe_singleton,\n Algebra.adjoin_union_eq_adjoin_adjoin, \u2190 Set.image_singleton,\n Algebra.adjoin_algebraMap K (SplittingFieldAux n f.removeFactor),\n AdjoinRoot.adjoinRoot_eq_top, Algebra.map_top] ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 hmf0 : map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (X - \u2191C (AdjoinRoot.root (factor f))) * map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260 0 \u22a2 Subalgebra.restrictScalars K (Algebra.adjoin { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) } \u2191(Multiset.toFinset (roots (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) = \u22a4 ** have := IsScalarTower.adjoin_range_toAlgHom K (AdjoinRoot f.factor)\n (SplittingFieldAux n f.removeFactor)\n (f.removeFactor.rootSet (SplittingFieldAux n f.removeFactor)) ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 hmf0 : map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (X - \u2191C (AdjoinRoot.root (factor f))) * map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260 0 this : Subalgebra.restrictScalars K (Algebra.adjoin { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) } (rootSet (removeFactor f) (SplittingFieldAux n (removeFactor f)))) = Subalgebra.restrictScalars K (Algebra.adjoin (AdjoinRoot (factor f)) (rootSet (removeFactor f) (SplittingFieldAux n (removeFactor f)))) \u22a2 Subalgebra.restrictScalars K (Algebra.adjoin { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) } \u2191(Multiset.toFinset (roots (map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f))))) = \u22a4 ** refine this.trans ?_ ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 hfn0 : f \u2260 0 hmf0 : map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (X - \u2191C (AdjoinRoot.root (factor f))) * map (algebraMap (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) (removeFactor f) \u2260 0 this : Subalgebra.restrictScalars K (Algebra.adjoin { x // x \u2208 AlgHom.range (IsScalarTower.toAlgHom K (AdjoinRoot (factor f)) (SplittingFieldAux n (removeFactor f))) } (rootSet (removeFactor f) (SplittingFieldAux n (removeFactor f)))) = Subalgebra.restrictScalars K (Algebra.adjoin (AdjoinRoot (factor f)) (rootSet (removeFactor f) (SplittingFieldAux n (removeFactor f)))) \u22a2 Subalgebra.restrictScalars K (Algebra.adjoin (AdjoinRoot (factor f)) (rootSet (removeFactor f) (SplittingFieldAux n (removeFactor f)))) = \u22a4 ** rw [ih _ (natDegree_removeFactor' hfn), Subalgebra.restrictScalars_top] ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n \u22a2 natDegree f \u2260 0 ** intro h ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n h : natDegree f = 0 \u22a2 False ** rw [h] at hfn ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : 0 = Nat.succ n h : natDegree f = 0 \u22a2 False ** cases hfn ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 \u22a2 f \u2260 0 ** intro h ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree f \u2260 0 h : f = 0 \u22a2 False ** rw [h] at hndf ** F : Type u K\u271d\u00b9 : Type v L : Type w inst\u271d\u00b3 : Field K\u271d\u00b9 inst\u271d\u00b2 : Field L inst\u271d\u00b9 : Field F n\u271d : \u2115 K\u271d : Type u inst\u271d : Field K\u271d n : \u2115 ih : (fun n => \u2200 {K : Type u} [inst : Field K] (f : K[X]), natDegree f = n \u2192 Algebra.adjoin K (rootSet f (SplittingFieldAux n f)) = \u22a4) n K : Type u x\u271d : Field K f : K[X] hfn : natDegree f = Nat.succ n hndf : natDegree 0 \u2260 0 h : f = 0 \u22a2 False ** exact hndf rfl ** Qed", + "informal": "" + }, + { + "formal": "SzemerediRegularity.hundred_lt_pow_initialBound_mul ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 \u2191100 < 4 ^ initialBound \u03b5 l * \u03b5 ^ 5 ** rw [\u2190 rpow_nat_cast 4, \u2190 div_lt_iff (pow_pos h\u03b5 5), lt_rpow_iff_log_lt _ zero_lt_four, \u2190\n div_lt_iff, initialBound, Nat.cast_max, Nat.cast_max] ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 log (\u2191100 / \u03b5 ^ 5) / log 4 < max (\u21917) (max \u2191l \u2191(\u230alog (\u2191100 / \u03b5 ^ 5) / log 4\u230b\u208a + 1)) ** push_cast ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 log (100 / \u03b5 ^ 5) / log 4 < max 7 (max (\u2191l) (\u2191\u230alog (100 / \u03b5 ^ 5) / log 4\u230b\u208a + 1)) ** exact lt_max_of_lt_right (lt_max_of_lt_right <| Nat.lt_floor_add_one _) ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 0 < log 4 ** exact log_pos (by norm_num) ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 1 < 4 ** norm_num ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 0 < \u2191100 / \u03b5 ^ 5 ** exact div_pos (by norm_num) (pow_pos h\u03b5 5) ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Fintype \u03b1 P : Finpartition univ u : Finset \u03b1 \u03b5\u271d : \u211d l\u271d : \u2115 \u03b5 : \u211d h\u03b5 : 0 < \u03b5 l : \u2115 \u22a2 0 < \u2191100 ** norm_num ** Qed", + "informal": "" + }, + { + "formal": "Additive.isAddSubgroup_iff ** G : Type u_1 H : Type u_2 A : Type u_3 a a\u2081 a\u2082 b c : G inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G \u22a2 IsAddSubgroup s \u2192 IsSubgroup s ** rintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9 ** case mk.mk G : Type u_1 H : Type u_2 A : Type u_3 a a\u2081 a\u2082 b c : G inst\u271d\u00b9 : Group G inst\u271d : AddGroup A s : Set G h\u2083 : \u2200 {a : Additive G}, a \u2208 s \u2192 -a \u2208 s h\u2081 : 0 \u2208 s h\u2082 : \u2200 {a b : Additive G}, a \u2208 s \u2192 b \u2208 s \u2192 a + b \u2208 s \u22a2 IsSubgroup s ** exact @IsSubgroup.mk G _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083 ** Qed", + "informal": "" + }, + { + "formal": "Set.infinite_of_finite_compl ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x inst\u271d : Infinite \u03b1 s : Set \u03b1 hs : Set.Finite s\u1d9c h : Set.Finite s \u22a2 Set.Finite univ ** simpa using hs.union h ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Adjunction.rightAdjointUniq_refl ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G \u22a2 (rightAdjointUniq adj1 adj1).hom = \ud835\udfd9 G ** delta rightAdjointUniq ** C : Type u\u2081 inst\u271d\u00b9 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D F : C \u2964 D G : D \u2964 C adj1 : F \u22a3 G \u22a2 (NatIso.removeOp (leftAdjointUniq (opAdjointOpOfAdjoint G F adj1) (opAdjointOpOfAdjoint G F adj1))).hom = \ud835\udfd9 G ** simp ** Qed", + "informal": "" + }, + { + "formal": "WittVector.ghostComponent_verschiebungFun ** p : \u2115 R : Type u_1 S : Type u_2 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S x : \ud835\udd4e R n : \u2115 \u22a2 \u2191(ghostComponent (n + 1)) (verschiebungFun x) = \u2191p * \u2191(ghostComponent n) x ** simp only [ghostComponent_apply, aeval_wittPolynomial] ** p : \u2115 R : Type u_1 S : Type u_2 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S x : \ud835\udd4e R n : \u2115 \u22a2 (Finset.sum (Finset.range (n + 1 + 1)) fun i => \u2191p ^ i * coeff (verschiebungFun x) i ^ p ^ (n + 1 - i)) = \u2191p * Finset.sum (Finset.range (n + 1)) fun i => \u2191p ^ i * coeff x i ^ p ^ (n - i) ** rw [Finset.sum_range_succ', verschiebungFun_coeff, if_pos rfl, zero_pow (pow_pos hp.1.pos _),\n mul_zero, add_zero, Finset.mul_sum, Finset.sum_congr rfl] ** p : \u2115 R : Type u_1 S : Type u_2 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S x : \ud835\udd4e R n : \u2115 \u22a2 \u2200 (x_1 : \u2115), x_1 \u2208 Finset.range (n + 1) \u2192 \u2191p ^ (x_1 + 1) * coeff (verschiebungFun x) (x_1 + 1) ^ p ^ (n + 1 - (x_1 + 1)) = \u2191p * (\u2191p ^ x_1 * coeff x x_1 ^ p ^ (n - x_1)) ** rintro i - ** p : \u2115 R : Type u_1 S : Type u_2 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S x : \ud835\udd4e R n i : \u2115 \u22a2 \u2191p ^ (i + 1) * coeff (verschiebungFun x) (i + 1) ^ p ^ (n + 1 - (i + 1)) = \u2191p * (\u2191p ^ i * coeff x i ^ p ^ (n - i)) ** simp only [pow_succ, mul_assoc, verschiebungFun_coeff, if_neg (Nat.succ_ne_zero i),\n Nat.succ_sub_succ, tsub_zero] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.span_singleton_group_smul_eq ** R : Type u_1 R\u2082 : Type u_2 K : Type u_3 M : Type u_4 M\u2082 : Type u_5 V : Type u_6 S : Type u_7 inst\u271d\u00b9\u2070 : Semiring R inst\u271d\u2079 : AddCommMonoid M inst\u271d\u2078 : Module R M x\u271d : M p p' : Submodule R M inst\u271d\u2077 : Semiring R\u2082 \u03c3\u2081\u2082 : R \u2192+* R\u2082 inst\u271d\u2076 : AddCommMonoid M\u2082 inst\u271d\u2075 : Module R\u2082 M\u2082 F : Type u_8 inst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082 s t : Set M G : Type u_9 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : SMul G R inst\u271d\u00b9 : MulAction G M inst\u271d : IsScalarTower G R M g : G x : M \u22a2 span R {g \u2022 x} = span R {x} ** refine' le_antisymm (span_singleton_smul_le R g x) _ ** R : Type u_1 R\u2082 : Type u_2 K : Type u_3 M : Type u_4 M\u2082 : Type u_5 V : Type u_6 S : Type u_7 inst\u271d\u00b9\u2070 : Semiring R inst\u271d\u2079 : AddCommMonoid M inst\u271d\u2078 : Module R M x\u271d : M p p' : Submodule R M inst\u271d\u2077 : Semiring R\u2082 \u03c3\u2081\u2082 : R \u2192+* R\u2082 inst\u271d\u2076 : AddCommMonoid M\u2082 inst\u271d\u2075 : Module R\u2082 M\u2082 F : Type u_8 inst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082 s t : Set M G : Type u_9 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : SMul G R inst\u271d\u00b9 : MulAction G M inst\u271d : IsScalarTower G R M g : G x : M \u22a2 span R {x} \u2264 span R {g \u2022 x} ** convert span_singleton_smul_le R g\u207b\u00b9 (g \u2022 x) ** case h.e'_3.h.e'_6.h.e'_4 R : Type u_1 R\u2082 : Type u_2 K : Type u_3 M : Type u_4 M\u2082 : Type u_5 V : Type u_6 S : Type u_7 inst\u271d\u00b9\u2070 : Semiring R inst\u271d\u2079 : AddCommMonoid M inst\u271d\u2078 : Module R M x\u271d : M p p' : Submodule R M inst\u271d\u2077 : Semiring R\u2082 \u03c3\u2081\u2082 : R \u2192+* R\u2082 inst\u271d\u2076 : AddCommMonoid M\u2082 inst\u271d\u2075 : Module R\u2082 M\u2082 F : Type u_8 inst\u271d\u2074 : SemilinearMapClass F \u03c3\u2081\u2082 M M\u2082 s t : Set M G : Type u_9 inst\u271d\u00b3 : Group G inst\u271d\u00b2 : SMul G R inst\u271d\u00b9 : MulAction G M inst\u271d : IsScalarTower G R M g : G x : M \u22a2 x = g\u207b\u00b9 \u2022 g \u2022 x ** exact (inv_smul_smul g x).symm ** Qed", + "informal": "" + }, + { + "formal": "Std.PairingHeapImp.Heap.size_tail?_lt ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 tail? le s = some s' \u2192 size s' < size s ** simp only [Heap.tail?] ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 \u22a2 Option.map (fun x => x.snd) (deleteMin le s) = some s' \u2192 size s' < size s ** intro eq ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' \u22a2 size s' < size s ** match eq\u2082 : s.deleteMin le, eq with\n| some (a, tl), rfl => exact size_deleteMin_lt eq\u2082 ** \u03b1 : Type u_1 le : \u03b1 \u2192 \u03b1 \u2192 Bool s' s : Heap \u03b1 eq : Option.map (fun x => x.snd) (deleteMin le s) = some s' a : \u03b1 tl : Heap \u03b1 eq\u2082 : deleteMin le s = some (a, tl) \u22a2 size ((fun x => x.snd) (a, tl)) < size s ** exact size_deleteMin_lt eq\u2082 ** Qed", + "informal": "" + }, + { + "formal": "lie_lie ** R : Type u L : Type v M : Type w N : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R N inst\u271d\u00b9 : LieRingModule L N inst\u271d : LieModule R L N t : R x y z : L m n : M \u22a2 \u2045\u2045x, y\u2046, m\u2046 = \u2045x, \u2045y, m\u2046\u2046 - \u2045y, \u2045x, m\u2046\u2046 ** rw [leibniz_lie, add_sub_cancel] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.biproduct.map_eq_map' ** J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b \u22a2 map p = map' p ** ext ** case w.w J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d\u00b9 j\u271d : J \u22a2 \u03b9 (fun b => f b) j\u271d \u226b map p \u226b \u03c0 (fun b => g b) j\u271d\u00b9 = \u03b9 (fun b => f b) j\u271d \u226b map' p \u226b \u03c0 (fun b => g b) j\u271d\u00b9 ** simp only [Discrete.natTrans_app, Limits.IsColimit.\u03b9_map_assoc, Limits.IsLimit.map_\u03c0,\n Category.assoc, \u2190 Bicone.toCone_\u03c0_app_mk, \u2190 biproduct.bicone_\u03c0, \u2190 Bicone.toCocone_\u03b9_app_mk,\n \u2190 biproduct.bicone_\u03b9] ** case w.w J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d\u00b9 j\u271d : J \u22a2 (Bicone.toCocone (bicone fun b => f b)).\u03b9.app { as := j\u271d } \u226b (Bicone.toCone (bicone fun b => f b)).\u03c0.app { as := j\u271d\u00b9 } \u226b p j\u271d\u00b9 = p j\u271d \u226b (Bicone.toCocone (bicone fun b => g b)).\u03b9.app { as := j\u271d } \u226b (Bicone.toCone (bicone fun b => g b)).\u03c0.app { as := j\u271d\u00b9 } ** dsimp ** case w.w J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d\u00b9 j\u271d : J \u22a2 \u03b9 (fun b => f b) j\u271d \u226b \u03c0 (fun b => f b) j\u271d\u00b9 \u226b p j\u271d\u00b9 = p j\u271d \u226b \u03b9 (fun b => g b) j\u271d \u226b \u03c0 (fun b => g b) j\u271d\u00b9 ** rw [biproduct.\u03b9_\u03c0_assoc, biproduct.\u03b9_\u03c0] ** case w.w J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d\u00b9 j\u271d : J \u22a2 (if h : j\u271d = j\u271d\u00b9 then eqToHom (_ : f j\u271d = f j\u271d\u00b9) else 0) \u226b p j\u271d\u00b9 = p j\u271d \u226b if h : j\u271d = j\u271d\u00b9 then eqToHom (_ : g j\u271d = g j\u271d\u00b9) else 0 ** split_ifs with h ** case pos J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d\u00b9 j\u271d : J h : j\u271d = j\u271d\u00b9 \u22a2 eqToHom (_ : f j\u271d = f j\u271d\u00b9) \u226b p j\u271d\u00b9 = p j\u271d \u226b eqToHom (_ : g j\u271d = g j\u271d\u00b9) ** subst h ** case pos J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d : J \u22a2 eqToHom (_ : f j\u271d = f j\u271d) \u226b p j\u271d = p j\u271d \u226b eqToHom (_ : g j\u271d = g j\u271d) ** rw [eqToHom_refl, Category.id_comp] ** case pos J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d : J \u22a2 p j\u271d = p j\u271d \u226b eqToHom (_ : g j\u271d = g j\u271d) ** erw [Category.comp_id] ** case neg J : Type w K : Type u_1 C : Type u inst\u271d\u00b3 : Category.{v, u} C inst\u271d\u00b2 : HasZeroMorphisms C f g : J \u2192 C inst\u271d\u00b9 : HasBiproduct f inst\u271d : HasBiproduct g p : (b : J) \u2192 f b \u27f6 g b j\u271d\u00b9 j\u271d : J h : \u00acj\u271d = j\u271d\u00b9 \u22a2 0 \u226b p j\u271d\u00b9 = p j\u271d \u226b 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "hausdorffMeasure_of_dimH_lt ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X s : Set X d : \u211d\u22650 h : dimH s < \u2191d \u22a2 \u2191\u2191\u03bcH[\u2191d] s = 0 ** rw [dimH_def] at h ** \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X s : Set X d : \u211d\u22650 h\u271d : dimH s < \u2191d h : \u2a06 d, \u2a06 (_ : \u2191\u2191\u03bcH[\u2191d] s = \u22a4), \u2191d < \u2191d \u22a2 \u2191\u2191\u03bcH[\u2191d] s = 0 ** rcases ENNReal.lt_iff_exists_nnreal_btwn.1 h with \u27e8d', hsd', hd'd\u27e9 ** case intro.intro \u03b9 : Type u_1 X : Type u_2 Y : Type u_3 inst\u271d\u00b3 : EMetricSpace X inst\u271d\u00b2 : EMetricSpace Y inst\u271d\u00b9 : MeasurableSpace X inst\u271d : BorelSpace X s : Set X d : \u211d\u22650 h\u271d : dimH s < \u2191d h : \u2a06 d, \u2a06 (_ : \u2191\u2191\u03bcH[\u2191d] s = \u22a4), \u2191d < \u2191d d' : \u211d\u22650 hsd' : \u2a06 d, \u2a06 (_ : \u2191\u2191\u03bcH[\u2191d] s = \u22a4), \u2191d < \u2191d' hd'd : \u2191d' < \u2191d \u22a2 \u2191\u2191\u03bcH[\u2191d] s = 0 ** exact (hausdorffMeasure_zero_or_top hd'd s).resolve_right fun h\u2082 => hsd'.not_le <|\n le_iSup\u2082 (\u03b1 := \u211d\u22650\u221e) d' h\u2082 ** Qed", + "informal": "" + }, + { + "formal": "Valuation.map_eq_of_sub_lt ** K : Type u_1 F : Type u_2 R : Type u_3 inst\u271d\u00b3 : DivisionRing K \u0393\u2080 : Type u_4 \u0393'\u2080 : Type u_5 \u0393''\u2080 : Type u_6 inst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 v : Valuation R \u0393\u2080 x y z : R h : \u2191v (y - x) < \u2191v x \u22a2 \u2191v y = \u2191v x ** have := Valuation.map_add_of_distinct_val v (ne_of_gt h).symm ** K : Type u_1 F : Type u_2 R : Type u_3 inst\u271d\u00b3 : DivisionRing K \u0393\u2080 : Type u_4 \u0393'\u2080 : Type u_5 \u0393''\u2080 : Type u_6 inst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 v : Valuation R \u0393\u2080 x y z : R h : \u2191v (y - x) < \u2191v x this : \u2191v (y - x + x) = max (\u2191v (y - x)) (\u2191v x) \u22a2 \u2191v y = \u2191v x ** rw [max_eq_right (le_of_lt h)] at this ** K : Type u_1 F : Type u_2 R : Type u_3 inst\u271d\u00b3 : DivisionRing K \u0393\u2080 : Type u_4 \u0393'\u2080 : Type u_5 \u0393''\u2080 : Type u_6 inst\u271d\u00b2 : LinearOrderedCommMonoidWithZero \u0393''\u2080 inst\u271d\u00b9 : Ring R inst\u271d : LinearOrderedCommGroupWithZero \u0393\u2080 v : Valuation R \u0393\u2080 x y z : R h : \u2191v (y - x) < \u2191v x this : \u2191v (y - x + x) = \u2191v x \u22a2 \u2191v y = \u2191v x ** simpa using this ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.IntegrableOn.integrable_of_ae_not_mem_eq_zero ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 f x = 0 \u22a2 Integrable f ** rw [\u2190 integrableOn_univ] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 f x = 0 \u22a2 IntegrableOn f univ ** apply hf.of_ae_diff_eq_zero nullMeasurableSet_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : NormedAddCommGroup E f g : \u03b1 \u2192 E s t : Set \u03b1 \u03bc \u03bd : Measure \u03b1 hf : IntegrableOn f s h't : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, \u00acx \u2208 s \u2192 f x = 0 \u22a2 \u2200\u1d50 (x : \u03b1) \u2202\u03bc, x \u2208 univ \\ s \u2192 f x = 0 ** filter_upwards [h't] with x hx h'x using hx h'x.2 ** Qed", + "informal": "" + }, + { + "formal": "span_gcd ** R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y : R \u22a2 Ideal.span {gcd x y} = Ideal.span {x, y} ** obtain \u27e8d, hd\u27e9 := IsPrincipalIdealRing.principal (span ({x, y} : Set R)) ** case mk.intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Submodule.span R {d} \u22a2 Ideal.span {gcd x y} = Ideal.span {x, y} ** rw [submodule_span_eq] at hd ** case mk.intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 Ideal.span {gcd x y} = Ideal.span {x, y} ** rw [hd] ** case mk.intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 Ideal.span {gcd x y} = Ideal.span {d} ** suffices Associated d (gcd x y) by\n obtain \u27e8D, HD\u27e9 := this\n rw [\u2190 HD]\n exact span_singleton_mul_right_unit D.isUnit _ ** case mk.intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 Associated d (gcd x y) ** apply associated_of_dvd_dvd ** R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} this : Associated d (gcd x y) \u22a2 Ideal.span {gcd x y} = Ideal.span {d} ** obtain \u27e8D, HD\u27e9 := this ** case intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} D : R\u02e3 HD : d * \u2191D = gcd x y \u22a2 Ideal.span {gcd x y} = Ideal.span {d} ** rw [\u2190 HD] ** case intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} D : R\u02e3 HD : d * \u2191D = gcd x y \u22a2 Ideal.span {d * \u2191D} = Ideal.span {d} ** exact span_singleton_mul_right_unit D.isUnit _ ** case mk.intro.hab R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 d \u2223 gcd x y ** rw [dvd_gcd_iff] ** case mk.intro.hab R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 d \u2223 x \u2227 d \u2223 y ** constructor <;> rw [\u2190 Ideal.mem_span_singleton, \u2190 hd, Ideal.mem_span_pair] ** case mk.intro.hab.left R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 \u2203 a b, a * x + b * y = x ** use 1, 0 ** case h R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 1 * x + 0 * y = x ** rw [one_mul, zero_mul, add_zero] ** case mk.intro.hab.right R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 \u2203 a b, a * x + b * y = y ** use 0, 1 ** case h R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 0 * x + 1 * y = y ** rw [one_mul, zero_mul, zero_add] ** case mk.intro.hba R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 gcd x y \u2223 d ** obtain \u27e8r, s, rfl\u27e9 : \u2203 r s, r * x + s * y = d := by\n rw [\u2190 Ideal.mem_span_pair, hd, Ideal.mem_span_singleton] ** case mk.intro.hba.intro.intro R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y r s : R hd : Ideal.span {x, y} = Ideal.span {r * x + s * y} \u22a2 gcd x y \u2223 r * x + s * y ** apply dvd_add <;> apply dvd_mul_of_dvd_right ** case mk.intro.hba.intro.intro.h\u2081.h R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y r s : R hd : Ideal.span {x, y} = Ideal.span {r * x + s * y} \u22a2 gcd x y \u2223 x case mk.intro.hba.intro.intro.h\u2082.h R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y r s : R hd : Ideal.span {x, y} = Ideal.span {r * x + s * y} \u22a2 gcd x y \u2223 y ** exacts [gcd_dvd_left x y, gcd_dvd_right x y] ** R : Type u M : Type v inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : IsDomain R inst\u271d\u00b9 : IsPrincipalIdealRing R inst\u271d : GCDMonoid R x y d : R hd : Ideal.span {x, y} = Ideal.span {d} \u22a2 \u2203 r s, r * x + s * y = d ** rw [\u2190 Ideal.mem_span_pair, hd, Ideal.mem_span_singleton] ** Qed", + "informal": "" + }, + { + "formal": "Nat.Partrec.Code.hG ** \u22a2 Primrec Nat.Partrec.Code.G ** have a := (Primrec.ofNat (\u2115 \u00d7 Code)).comp (Primrec.list_length (\u03b1 := List (Option \u2115))) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) \u22a2 Primrec Nat.Partrec.Code.G ** have k := Primrec.fst.comp a ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1 \u22a2 Primrec Nat.Partrec.Code.G ** refine' Primrec.option_some.comp (Primrec.list_map (Primrec.list_range.comp k) (_ : Primrec _)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a)).1 \u22a2 Primrec fun p => (fun a n => Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x) (fun cf cg x x => let z := (unpair n).1; Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y) Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair n).1; let m := (unpair n).2; do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1))) p.1 p.2 ** replace k := k.comp (Primrec.fst (\u03b2 := \u2115)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 \u22a2 Primrec fun p => (fun a n => Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x) (fun cf cg x x => let z := (unpair n).1; Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y) Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair n).1; let m := (unpair n).2; do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1))) p.1 p.2 ** have n := Primrec.snd (\u03b1 := List (List (Option \u2115))) (\u03b2 := \u2115) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n : Primrec Prod.snd \u22a2 Primrec fun p => (fun a n => Nat.casesOn (ofNat (\u2115 \u00d7 Code) (List.length a)).1 Option.none fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length a)).2 (some 0) (some (Nat.succ n)) (some (unpair n).1) (some (unpair n).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) n let y \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) n Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) x) (fun cf cg x x => let z := (unpair n).1; Nat.casesOn (unpair n).2 (Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z y) Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair n).1; let m := (unpair n).2; do let x \u2190 Nat.Partrec.Code.lup a ((ofNat (\u2115 \u00d7 Code) (List.length a)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a (k', (ofNat (\u2115 \u00d7 Code) (List.length a)).2) (Nat.pair z (m + 1))) p.1 p.2 ** refine' Primrec.nat_casesOn k (_root_.Primrec.const Option.none) (_ : Primrec _) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n : Primrec Prod.snd \u22a2 Primrec fun p => (fun p n => (fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1) (some (unpair p.2).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x) (fun cf cg x x => let z := (unpair p.2).1; Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair p.2).1; let m := (unpair p.2).2; do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1))) n) p.1 p.2 ** have k := k.comp (Primrec.fst (\u03b2 := \u2115)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 \u22a2 Primrec fun p => (fun p n => (fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1) (some (unpair p.2).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x) (fun cf cg x x => let z := (unpair p.2).1; Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair p.2).1; let m := (unpair p.2).2; do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1))) n) p.1 p.2 ** have n := n.comp (Primrec.fst (\u03b2 := \u2115)) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 \u22a2 Primrec fun p => (fun p n => (fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1) (some (unpair p.2).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x) (fun cf cg x x => let z := (unpair p.2).1; Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair p.2).1; let m := (unpair p.2).2; do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1))) n) p.1 p.2 ** have k' := Primrec.snd (\u03b1 := List (List (Option \u2115)) \u00d7 \u2115) (\u03b2 := \u2115) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd \u22a2 Primrec fun p => (fun p n => (fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1) (some (unpair p.2).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x) (fun cf cg x x => let z := (unpair p.2).1; Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair p.2).1; let m := (unpair p.2).2; do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1))) n) p.1 p.2 ** have c := Primrec.snd.comp (a.comp <| (Primrec.fst (\u03b2 := \u2115)).comp (Primrec.fst (\u03b2 := \u2115))) ** a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun p => (fun p n => (fun k' => Code.recOn (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2 (some 0) (some (Nat.succ p.2)) (some (unpair p.2).1) (some (unpair p.2).2) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) p.2 let y \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 some (Nat.pair x y)) (fun cf cg x x => do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) p.2 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) x) (fun cf cg x x => let z := (unpair p.2).1; Nat.casesOn (unpair p.2).2 (Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) z) fun y => do let i \u2190 Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cg) (Nat.pair z (Nat.pair y i))) fun cf x => let z := (unpair p.2).1; let m := (unpair p.2).2; do let x \u2190 Nat.Partrec.Code.lup p.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1)).1, cf) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup p.1 (k', (ofNat (\u2115 \u00d7 Code) (List.length p.1)).2) (Nat.pair z (m + 1))) n) p.1 p.2 ** apply\n Nat.Partrec.Code.rec_prim c\n (_root_.Primrec.const (some 0))\n (Primrec.option_some.comp (_root_.Primrec.succ.comp n))\n (Primrec.option_some.comp (Primrec.fst.comp <| Primrec.unpair.comp n))\n (Primrec.option_some.comp (Primrec.snd.comp <| Primrec.unpair.comp n)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2 let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** have L := (Primrec.fst.comp Primrec.fst).comp\n (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2 let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2 let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2 let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2 let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** have cg := (Primrec.fst.comp Primrec.snd).comp\n (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) a.1.1.2 let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cf).pair n) ?_ ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec\u2082 fun a x => do let y \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 some (Nat.pair x y) ** unfold Primrec\u2082 ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun p => Option.map (fun y => Nat.pair p.2 y) (Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) p.1.1.1.2) ** refine Primrec.option_map ((hlup.comp <| L.pair <| (k.pair cg).pair n).comp Primrec.fst) ?_ ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec\u2082 fun p y => Nat.pair p.2 y ** unfold Primrec\u2082 ** case hpr a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun p => (fun p y => Nat.pair p.2 y) p.1 p.2 ** exact Primrec\u2082.natPair.comp (Primrec.snd.comp Primrec.fst) Primrec.snd ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have L := (Primrec.fst.comp Primrec.fst).comp\n (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** have cg := (Primrec.fst.comp Primrec.snd).comp\n (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun a => do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) a.1.1.2 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** refine Primrec.option_bind (hlup.comp <| L.pair <| (k.pair cg).pair n) ?_ ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec\u2082 fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x ** unfold Primrec\u2082 ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun p => (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2 ** have h :=\n hlup.comp ((L.comp Primrec.fst).pair <| ((k.pair cf).comp Primrec.fst).pair Primrec.snd) ** case hco a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 h : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).1 (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.1 (a.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).1, a.1.2.1), a.2).2.2 \u22a2 Primrec fun p => (fun a x => Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) x) p.1 p.2 ** exact h ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have L := (Primrec.fst.comp Primrec.fst).comp\n (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have cg := (Primrec.fst.comp Primrec.snd).comp\n (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** have z := Primrec.fst.comp (Primrec.unpair.comp n) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z : Primrec fun a => (unpair a.1.1.2).1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; Nat.casesOn (unpair a.1.1.2).2 (Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) z) fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair z (Nat.pair y i)) ** refine'\n Primrec.nat_casesOn (Primrec.snd.comp (Primrec.unpair.comp n))\n (hlup.comp <| L.pair <| (k.pair cf).pair z)\n (_ : Primrec _) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z : Primrec fun a => (unpair a.1.1.2).1 \u22a2 Primrec fun p => (fun a n => (fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i))) n) p.1 p.2 ** have L := L.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 \u22a2 Primrec fun p => (fun a n => (fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i))) n) p.1 p.2 ** have z := z.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 \u22a2 Primrec fun p => (fun a n => (fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i))) n) p.1 p.2 ** have y := Primrec.snd\n (\u03b1 := ((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) (\u03b2 := \u2115) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd \u22a2 Primrec fun p => (fun a n => (fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i))) n) p.1 p.2 ** have h\u2081 := hlup.comp <| L.pair <| (((k'.pair c).comp Primrec.fst).comp Primrec.fst).pair\n (Primrec\u2082.natPair.comp z y) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 \u22a2 Primrec fun p => (fun a n => (fun y => do let i \u2190 Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 y) Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.2.1) (Nat.pair (unpair a.1.1.2).1 (Nat.pair y i))) n) p.1 p.2 ** refine' Primrec.option_bind h\u2081 (_ : Primrec _) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 \u22a2 Primrec fun p => (fun p i => Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i))) p.1 p.2 ** have z := z.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y : Primrec Prod.snd h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 \u22a2 Primrec fun p => (fun p i => Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i))) p.1 p.2 ** have y := y.comp (Primrec.fst (\u03b2 := \u2115)) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y\u271d : Primrec Prod.snd h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 y : Primrec fun a => a.1.2 \u22a2 Primrec fun p => (fun p i => Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i))) p.1 p.2 ** have i := Primrec.snd\n (\u03b1 := (((List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115) \u00d7 Code \u00d7 Code \u00d7 Option \u2115 \u00d7 Option \u2115) \u00d7 \u2115)\n (\u03b2 := \u2115) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y\u271d : Primrec Prod.snd h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 y : Primrec fun a => a.1.2 i : Primrec Prod.snd \u22a2 Primrec fun p => (fun p i => Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i))) p.1 p.2 ** have h\u2082 := hlup.comp ((L.comp Primrec.fst).pair <|\n ((k.pair cg).comp <| Primrec.fst.comp Primrec.fst).pair <|\n Primrec\u2082.natPair.comp z <| Primrec\u2082.natPair.comp y i) ** case hpc a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L\u271d : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 cg : Primrec fun a => a.2.2.1 z\u271d\u00b9 : Primrec fun a => (unpair a.1.1.2).1 L : Primrec fun a => a.1.1.1.1 z\u271d : Primrec fun a => (unpair a.1.1.1.2).1 y\u271d : Primrec Prod.snd h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.1 (a.1.1.1.1, (a.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1)).2), Nat.pair (unpair a.1.1.1.2).1 a.2).2.2 z : Primrec fun a => (unpair a.1.1.1.1.2).1 y : Primrec fun a => a.1.2 i : Primrec Prod.snd h\u2082 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1), Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).1 (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1), Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.1 (a.1.1.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1.1.1)).1, a.1.1.2.2.1), Nat.pair (unpair a.1.1.1.1.2).1 (Nat.pair a.1.2 a.2)).2.2 \u22a2 Primrec fun p => (fun p i => Nat.Partrec.Code.lup p.1.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).1, p.1.2.2.1) (Nat.pair (unpair p.1.1.1.2).1 (Nat.pair p.2 i))) p.1 p.2 ** exact h\u2082 ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have L := (Primrec.fst.comp Primrec.fst).comp\n (Primrec.fst (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have k := k.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have n := n.comp (Primrec.fst (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have cf := Primrec.fst.comp (Primrec.snd (\u03b1 := (List (List (Option \u2115)) \u00d7 \u2115) \u00d7 \u2115)\n (\u03b2 := Code \u00d7 Option \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have z := Primrec.fst.comp (Primrec.unpair.comp n) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have m := Primrec.snd.comp (Primrec.unpair.comp n) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m : Primrec fun a => (unpair a.1.1.2).2 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** have h\u2081 := hlup.comp <| L.pair <| (k.pair cf).pair (Primrec\u2082.natPair.comp z m) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m : Primrec fun a => (unpair a.1.1.2).2 h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 \u22a2 Primrec fun a => let z := (unpair a.1.1.2).1; let m := (unpair a.1.1.2).2; do let x \u2190 Nat.Partrec.Code.lup a.1.1.1 ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1) (Nat.pair z m) Nat.casesOn x (some m) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair z (m + 1)) ** refine' Primrec.option_bind h\u2081 (_ : Primrec _) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m : Primrec fun a => (unpair a.1.1.2).2 h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 \u22a2 Primrec fun p => (fun a x => Nat.casesOn x (some (unpair a.1.1.2).2) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1))) p.1 p.2 ** have m := m.comp (Primrec.fst (\u03b2 := \u2115)) ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m\u271d : Primrec fun a => (unpair a.1.1.2).2 h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 m : Primrec fun a => (unpair a.1.1.1.2).2 \u22a2 Primrec fun p => (fun a x => Nat.casesOn x (some (unpair a.1.1.2).2) fun x => Nat.Partrec.Code.lup a.1.1.1 (a.1.2, (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).2) (Nat.pair (unpair a.1.1.2).1 ((unpair a.1.1.2).2 + 1))) p.1 p.2 ** refine Primrec.nat_casesOn Primrec.snd (Primrec.option_some.comp m) ?_ ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m\u271d : Primrec fun a => (unpair a.1.1.2).2 h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 m : Primrec fun a => (unpair a.1.1.1.2).2 \u22a2 Primrec\u2082 fun p n => (fun x => Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2) (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1))) n ** unfold Primrec\u2082 ** case hrf a : Primrec fun a => ofNat (\u2115 \u00d7 Code) (List.length a) k\u271d\u00b9 : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1)).1 n\u271d\u00b9 : Primrec Prod.snd k\u271d : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).1 n\u271d : Primrec fun a => a.1.2 k' : Primrec Prod.snd c : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1)).2 L : Primrec fun a => a.1.1.1 k : Primrec fun a => (ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1 n : Primrec fun a => a.1.1.2 cf : Primrec fun a => a.2.1 z : Primrec fun a => (unpair a.1.1.2).1 m\u271d : Primrec fun a => (unpair a.1.1.2).2 h\u2081 : Primrec fun a => Nat.Partrec.Code.lup (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.1 (a.1.1.1, ((ofNat (\u2115 \u00d7 Code) (List.length a.1.1.1)).1, a.2.1), Nat.pair (unpair a.1.1.2).1 (unpair a.1.1.2).2).2.2 m : Primrec fun a => (unpair a.1.1.1.2).2 \u22a2 Primrec fun p => (fun p n => (fun x => Nat.Partrec.Code.lup p.1.1.1.1 (p.1.1.2, (ofNat (\u2115 \u00d7 Code) (List.length p.1.1.1.1)).2) (Nat.pair (unpair p.1.1.1.2).1 ((unpair p.1.1.1.2).2 + 1))) n) p.1 p.2 ** exact (hlup.comp ((L.comp Primrec.fst).pair <|\n ((k'.pair c).comp <| Primrec.fst.comp Primrec.fst).pair\n (Primrec\u2082.natPair.comp (z.comp Primrec.fst) (_root_.Primrec.succ.comp m)))).comp\n Primrec.fst ** Qed", + "informal": "" + }, + { + "formal": "ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2074 : Fintype \u03b9 inst\u271d\u00b9\u00b3 : Fintype \u03b9' inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' g : ContinuousMultilinearMap \ud835\udd5c E\u2081 G f : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i \u22a2 \u2016compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)\u2016 = \u2016g\u2016 ** apply le_antisymm (g.norm_compContinuous_linearIsometry_le fun i => (f i).toLinearIsometry) ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2074 : Fintype \u03b9 inst\u271d\u00b9\u00b3 : Fintype \u03b9' inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' g : ContinuousMultilinearMap \ud835\udd5c E\u2081 G f : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i \u22a2 \u2016g\u2016 \u2264 \u2016compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))\u2016 ** have : g = (g.compContinuousLinearMap fun i => (f i : E i \u2192L[\ud835\udd5c] E\u2081 i)).compContinuousLinearMap\n fun i => ((f i).symm : E\u2081 i \u2192L[\ud835\udd5c] E i) := by\n ext1 m\n simp only [compContinuousLinearMap_apply, LinearIsometryEquiv.coe_coe'',\n LinearIsometryEquiv.apply_symm_apply] ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2074 : Fintype \u03b9 inst\u271d\u00b9\u00b3 : Fintype \u03b9' inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' g : ContinuousMultilinearMap \ud835\udd5c E\u2081 G f : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) \u22a2 \u2016g\u2016 \u2264 \u2016compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))\u2016 ** conv_lhs => rw [this] ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2074 : Fintype \u03b9 inst\u271d\u00b9\u00b3 : Fintype \u03b9' inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' g : ContinuousMultilinearMap \ud835\udd5c E\u2081 G f : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i this : g = compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) \u22a2 \u2016compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)\u2016 \u2264 \u2016compContinuousLinearMap g fun i => LinearIsometry.toContinuousLinearMap (LinearIsometryEquiv.toLinearIsometry (f i))\u2016 ** apply (g.compContinuousLinearMap fun i =>\n (f i : E i \u2192L[\ud835\udd5c] E\u2081 i)).norm_compContinuous_linearIsometry_le\n fun i => (f i).symm.toLinearIsometry ** \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2074 : Fintype \u03b9 inst\u271d\u00b9\u00b3 : Fintype \u03b9' inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' g : ContinuousMultilinearMap \ud835\udd5c E\u2081 G f : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i \u22a2 g = compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv) ** ext1 m ** case H \ud835\udd5c : Type u \u03b9 : Type v \u03b9' : Type v' n : \u2115 E : \u03b9 \u2192 Type wE E\u2081 : \u03b9 \u2192 Type wE\u2081 E' : \u03b9' \u2192 Type wE' Ei : Fin (Nat.succ n) \u2192 Type wEi G : Type wG G' : Type wG' inst\u271d\u00b9\u2074 : Fintype \u03b9 inst\u271d\u00b9\u00b3 : Fintype \u03b9' inst\u271d\u00b9\u00b2 : NontriviallyNormedField \ud835\udd5c inst\u271d\u00b9\u00b9 : (i : \u03b9) \u2192 NormedAddCommGroup (E i) inst\u271d\u00b9\u2070 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E i) inst\u271d\u2079 : (i : \u03b9) \u2192 NormedAddCommGroup (E\u2081 i) inst\u271d\u2078 : (i : \u03b9) \u2192 NormedSpace \ud835\udd5c (E\u2081 i) inst\u271d\u2077 : (i : \u03b9') \u2192 NormedAddCommGroup (E' i) inst\u271d\u2076 : (i : \u03b9') \u2192 NormedSpace \ud835\udd5c (E' i) inst\u271d\u2075 : (i : Fin (Nat.succ n)) \u2192 NormedAddCommGroup (Ei i) inst\u271d\u2074 : (i : Fin (Nat.succ n)) \u2192 NormedSpace \ud835\udd5c (Ei i) inst\u271d\u00b3 : NormedAddCommGroup G inst\u271d\u00b2 : NormedSpace \ud835\udd5c G inst\u271d\u00b9 : NormedAddCommGroup G' inst\u271d : NormedSpace \ud835\udd5c G' g : ContinuousMultilinearMap \ud835\udd5c E\u2081 G f : (i : \u03b9) \u2192 E i \u2243\u2097\u1d62[\ud835\udd5c] E\u2081 i m : (i : \u03b9) \u2192 E\u2081 i \u22a2 \u2191g m = \u2191(compContinuousLinearMap (compContinuousLinearMap g fun i => \u2191(ContinuousLinearEquiv.mk (f i).toLinearEquiv)) fun i => \u2191(ContinuousLinearEquiv.mk (LinearIsometryEquiv.symm (f i)).toLinearEquiv)) m ** simp only [compContinuousLinearMap_apply, LinearIsometryEquiv.coe_coe'',\n LinearIsometryEquiv.apply_symm_apply] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.Concrete.multiequalizer_ext ** C : Type u inst\u271d\u2075 : Category.{v, u} C inst\u271d\u2074 : ConcreteCategory C J : Type w inst\u271d\u00b3 : SmallCategory J F : J \u2964 C inst\u271d\u00b2 : PreservesLimit F (forget C) I : MulticospanIndex C inst\u271d\u00b9 : HasMultiequalizer I inst\u271d : PreservesLimit (MulticospanIndex.multicospan I) (forget C) x y : (forget C).obj (multiequalizer I) h : \u2200 (t : I.L), \u2191(Multiequalizer.\u03b9 I t) x = \u2191(Multiequalizer.\u03b9 I t) y \u22a2 x = y ** apply Concrete.limit_ext ** case a C : Type u inst\u271d\u2075 : Category.{v, u} C inst\u271d\u2074 : ConcreteCategory C J : Type w inst\u271d\u00b3 : SmallCategory J F : J \u2964 C inst\u271d\u00b2 : PreservesLimit F (forget C) I : MulticospanIndex C inst\u271d\u00b9 : HasMultiequalizer I inst\u271d : PreservesLimit (MulticospanIndex.multicospan I) (forget C) x y : (forget C).obj (multiequalizer I) h : \u2200 (t : I.L), \u2191(Multiequalizer.\u03b9 I t) x = \u2191(Multiequalizer.\u03b9 I t) y \u22a2 \u2200 (j : WalkingMulticospan I.fstTo I.sndTo), \u2191(limit.\u03c0 (MulticospanIndex.multicospan I) j) x = \u2191(limit.\u03c0 (MulticospanIndex.multicospan I) j) y ** rintro (a | b) ** case a.left C : Type u inst\u271d\u2075 : Category.{v, u} C inst\u271d\u2074 : ConcreteCategory C J : Type w inst\u271d\u00b3 : SmallCategory J F : J \u2964 C inst\u271d\u00b2 : PreservesLimit F (forget C) I : MulticospanIndex C inst\u271d\u00b9 : HasMultiequalizer I inst\u271d : PreservesLimit (MulticospanIndex.multicospan I) (forget C) x y : (forget C).obj (multiequalizer I) h : \u2200 (t : I.L), \u2191(Multiequalizer.\u03b9 I t) x = \u2191(Multiequalizer.\u03b9 I t) y a : I.L \u22a2 \u2191(limit.\u03c0 (MulticospanIndex.multicospan I) (WalkingMulticospan.left a)) x = \u2191(limit.\u03c0 (MulticospanIndex.multicospan I) (WalkingMulticospan.left a)) y ** apply h ** case a.right C : Type u inst\u271d\u2075 : Category.{v, u} C inst\u271d\u2074 : ConcreteCategory C J : Type w inst\u271d\u00b3 : SmallCategory J F : J \u2964 C inst\u271d\u00b2 : PreservesLimit F (forget C) I : MulticospanIndex C inst\u271d\u00b9 : HasMultiequalizer I inst\u271d : PreservesLimit (MulticospanIndex.multicospan I) (forget C) x y : (forget C).obj (multiequalizer I) h : \u2200 (t : I.L), \u2191(Multiequalizer.\u03b9 I t) x = \u2191(Multiequalizer.\u03b9 I t) y b : I.R \u22a2 \u2191(limit.\u03c0 (MulticospanIndex.multicospan I) (WalkingMulticospan.right b)) x = \u2191(limit.\u03c0 (MulticospanIndex.multicospan I) (WalkingMulticospan.right b)) y ** rw [\u2190 limit.w I.multicospan (WalkingMulticospan.Hom.fst b), comp_apply, comp_apply] ** case a.right C : Type u inst\u271d\u2075 : Category.{v, u} C inst\u271d\u2074 : ConcreteCategory C J : Type w inst\u271d\u00b3 : SmallCategory J F : J \u2964 C inst\u271d\u00b2 : PreservesLimit F (forget C) I : MulticospanIndex C inst\u271d\u00b9 : HasMultiequalizer I inst\u271d : PreservesLimit (MulticospanIndex.multicospan I) (forget C) x y : (forget C).obj (multiequalizer I) h : \u2200 (t : I.L), \u2191(Multiequalizer.\u03b9 I t) x = \u2191(Multiequalizer.\u03b9 I t) y b : I.R \u22a2 \u2191((MulticospanIndex.multicospan I).map (WalkingMulticospan.Hom.fst b)) (\u2191(limit.\u03c0 (MulticospanIndex.multicospan I) (WalkingMulticospan.left (MulticospanIndex.fstTo I b))) x) = \u2191((MulticospanIndex.multicospan I).map (WalkingMulticospan.Hom.fst b)) (\u2191(limit.\u03c0 (MulticospanIndex.multicospan I) (WalkingMulticospan.left (MulticospanIndex.fstTo I b))) y) ** simp [h] ** Qed", + "informal": "" + }, + { + "formal": "AddLECancellable.tsub_add_eq_add_tsub ** \u03b1 : Type u_1 inst\u271d\u2075 : AddCommSemigroup \u03b1 inst\u271d\u2074 : PartialOrder \u03b1 inst\u271d\u00b3 : ExistsAddOfLE \u03b1 inst\u271d\u00b2 : CovariantClass \u03b1 \u03b1 (fun x x_1 => x + x_1) fun x x_1 => x \u2264 x_1 inst\u271d\u00b9 : Sub \u03b1 inst\u271d : OrderedSub \u03b1 a b c d : \u03b1 hb : AddLECancellable b h : b \u2264 a \u22a2 a - b + c = a + c - b ** rw [add_comm a, hb.add_tsub_assoc_of_le h, add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Rat.cast_nonpos ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 K : Type u_5 inst\u271d : LinearOrderedField K n : \u211a \u22a2 \u2191n \u2264 0 \u2194 n \u2264 0 ** norm_cast ** Qed", + "informal": "" + }, + { + "formal": "Orientation.two_zsmul_oangle_neg_left ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V \u22a2 2 \u2022 oangle o (-x) y = 2 \u2022 oangle o x y ** by_cases hx : x = 0 ** case pos V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V hx : x = 0 \u22a2 2 \u2022 oangle o (-x) y = 2 \u2022 oangle o x y ** simp [hx] ** case neg V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V hx : \u00acx = 0 \u22a2 2 \u2022 oangle o (-x) y = 2 \u2022 oangle o x y ** by_cases hy : y = 0 ** case pos V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V hx : \u00acx = 0 hy : y = 0 \u22a2 2 \u2022 oangle o (-x) y = 2 \u2022 oangle o x y ** simp [hy] ** case neg V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V hx : \u00acx = 0 hy : \u00acy = 0 \u22a2 2 \u2022 oangle o (-x) y = 2 \u2022 oangle o x y ** simp [o.oangle_neg_left hx hy] ** Qed", + "informal": "" + }, + { + "formal": "PrimeSpectrum.zeroLocus_bUnion ** R : Type u S : Type v inst\u271d\u00b9 : CommRing R inst\u271d : CommRing S s : Set (Set R) \u22a2 zeroLocus (\u22c3 s' \u2208 s, s') = \u22c2 s' \u2208 s, zeroLocus s' ** simp only [zeroLocus_iUnion] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.invOf_submatrix_equiv_eq ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u2076 : Fintype n inst\u271d\u2075 : DecidableEq n inst\u271d\u2074 : CommRing \u03b1 A\u271d B : Matrix n n \u03b1 inst\u271d\u00b3 : Fintype m inst\u271d\u00b2 : DecidableEq m A : Matrix m m \u03b1 e\u2081 e\u2082 : n \u2243 m inst\u271d\u00b9 : Invertible A inst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082) \u22a2 \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) = submatrix \u215fA \u2191e\u2082 \u2191e\u2081 ** letI := submatrixEquivInvertible A e\u2081 e\u2082 ** l : Type u_1 m : Type u n : Type u' \u03b1 : Type v inst\u271d\u2076 : Fintype n inst\u271d\u2075 : DecidableEq n inst\u271d\u2074 : CommRing \u03b1 A\u271d B : Matrix n n \u03b1 inst\u271d\u00b3 : Fintype m inst\u271d\u00b2 : DecidableEq m A : Matrix m m \u03b1 e\u2081 e\u2082 : n \u2243 m inst\u271d\u00b9 : Invertible A inst\u271d : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082) this : Invertible (submatrix A \u2191e\u2081 \u2191e\u2082) := submatrixEquivInvertible A e\u2081 e\u2082 \u22a2 \u215f(submatrix A \u2191e\u2081 \u2191e\u2082) = submatrix \u215fA \u2191e\u2082 \u2191e\u2081 ** convert (rfl : \u215f (A.submatrix e\u2081 e\u2082) = _) ** Qed", + "informal": "" + }, + { + "formal": "Ideal.map_jacobson_of_surjective ** R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f \u22a2 RingHom.ker f \u2264 I \u2192 map f (jacobson I) = jacobson (map f I) ** intro h ** R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I \u22a2 map f (jacobson I) = jacobson (map f I) ** unfold Ideal.jacobson ** R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I \u22a2 map f (sInf {J | I \u2264 J \u2227 IsMaximal J}) = sInf {J | map f I \u2264 J \u2227 IsMaximal J} ** have : \u2200 J \u2208 { J : Ideal R | I \u2264 J \u2227 J.IsMaximal }, RingHom.ker f \u2264 J :=\n fun J hJ => le_trans h hJ.left ** R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J \u22a2 map f (sInf {J | I \u2264 J \u2227 IsMaximal J}) = sInf {J | map f I \u2264 J \u2227 IsMaximal J} ** refine Trans.trans (map_sInf hf this) (le_antisymm ?_ ?_) ** case refine_1 R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J \u22a2 sInf (map f '' {J | I \u2264 J \u2227 IsMaximal J}) \u2264 sInf {J | map f I \u2264 J \u2227 IsMaximal J} ** refine'\n sInf_le_sInf fun J hJ =>\n \u27e8comap f J, \u27e8\u27e8le_comap_of_map_le hJ.1, _\u27e9, map_comap_of_surjective f hf J\u27e9\u27e9 ** case refine_1 R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J J : Ideal S hJ : J \u2208 {J | map f I \u2264 J \u2227 IsMaximal J} \u22a2 IsMaximal (comap f J) ** haveI : J.IsMaximal := hJ.right ** case refine_1 R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this\u271d : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J J : Ideal S hJ : J \u2208 {J | map f I \u2264 J \u2227 IsMaximal J} this : IsMaximal J \u22a2 IsMaximal (comap f J) ** exact comap_isMaximal_of_surjective f hf ** case refine_2 R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J \u22a2 sInf {J | map f I \u2264 J \u2227 IsMaximal J} \u2264 sInf (map f '' {J | I \u2264 J \u2227 IsMaximal J}) ** refine' sInf_le_sInf_of_subset_insert_top fun j hj => hj.recOn fun J hJ => _ ** case refine_2 R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J j : Ideal S hj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J} J : Ideal R hJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j \u22a2 j \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J} ** rw [\u2190 hJ.2] ** case refine_2 R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J j : Ideal S hj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J} J : Ideal R hJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j \u22a2 map f J \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J} ** cases' map_eq_top_or_isMaximal_of_surjective f hf hJ.left.right with htop hmax ** case refine_2.inl R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J j : Ideal S hj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J} J : Ideal R hJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j htop : map f J = \u22a4 \u22a2 map f J \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J} ** exact htop.symm \u25b8 Set.mem_insert \u22a4 _ ** case refine_2.inr R : Type u S : Type v inst\u271d\u00b9 : Ring R inst\u271d : Ring S I : Ideal R f : R \u2192+* S hf : Function.Surjective \u2191f h : RingHom.ker f \u2264 I this : \u2200 (J : Ideal R), J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2192 RingHom.ker f \u2264 J j : Ideal S hj : j \u2208 map f '' {J | I \u2264 J \u2227 IsMaximal J} J : Ideal R hJ : J \u2208 {J | I \u2264 J \u2227 IsMaximal J} \u2227 map f J = j hmax : IsMaximal (map f J) \u22a2 map f J \u2208 insert \u22a4 {J | map f I \u2264 J \u2227 IsMaximal J} ** exact Set.mem_insert_of_mem \u22a4 \u27e8map_mono hJ.1.1, hmax\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.biprod.isoProd_hom ** J : Type w C : Type u inst\u271d\u00b2 : Category.{v, u} C inst\u271d\u00b9 : HasZeroMorphisms C P Q X Y : C inst\u271d : HasBinaryBiproduct X Y \u22a2 (isoProd X Y).hom = prod.lift fst snd ** ext <;> simp [biprod.isoProd] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.lifts_iff_coeff_lifts ** R : Type u inst\u271d\u00b9 : Semiring R S : Type v inst\u271d : Semiring S f : R \u2192+* S p : S[X] \u22a2 p \u2208 lifts f \u2194 \u2200 (n : \u2115), coeff p n \u2208 Set.range \u2191f ** rw [lifts_iff_ringHom_rangeS, mem_map_rangeS f] ** R : Type u inst\u271d\u00b9 : Semiring R S : Type v inst\u271d : Semiring S f : R \u2192+* S p : S[X] \u22a2 (\u2200 (n : \u2115), coeff p n \u2208 RingHom.rangeS f) \u2194 \u2200 (n : \u2115), coeff p n \u2208 Set.range \u2191f ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.IsComplement.equiv_mul_left ** G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G hST : IsComplement S T hHT : IsComplement (\u2191H) T hSK : IsComplement S \u2191K h : { x // x \u2208 H } g : G \u22a2 \u2191(equiv hHT) (\u2191h * g) = (h * (\u2191(equiv hHT) g).1, (\u2191(equiv hHT) g).2) ** have : (hHT.equiv (h * g)).snd = (hHT.equiv g).snd :=\n hHT.equiv_snd_eq_iff_rightCosetEquivalence.2\n (by simp [RightCosetEquivalence, rightCoset_eq_iff]) ** G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G hST : IsComplement S T hHT : IsComplement (\u2191H) T hSK : IsComplement S \u2191K h : { x // x \u2208 H } g : G this : (\u2191(equiv hHT) (\u2191h * g)).2 = (\u2191(equiv hHT) g).2 \u22a2 \u2191(equiv hHT) (\u2191h * g) = (h * (\u2191(equiv hHT) g).1, (\u2191(equiv hHT) g).2) ** ext ** G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G hST : IsComplement S T hHT : IsComplement (\u2191H) T hSK : IsComplement S \u2191K h : { x // x \u2208 H } g : G \u22a2 RightCosetEquivalence (\u2191H) (\u2191h * g) g ** simp [RightCosetEquivalence, rightCoset_eq_iff] ** case h\u2081.a G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G hST : IsComplement S T hHT : IsComplement (\u2191H) T hSK : IsComplement S \u2191K h : { x // x \u2208 H } g : G this : (\u2191(equiv hHT) (\u2191h * g)).2 = (\u2191(equiv hHT) g).2 \u22a2 \u2191(\u2191(equiv hHT) (\u2191h * g)).1 = \u2191(h * (\u2191(equiv hHT) g).1, (\u2191(equiv hHT) g).2).1 ** rw [coe_mul, equiv_fst_eq_mul_inv, this, equiv_fst_eq_mul_inv, mul_assoc] ** case h\u2082.a G : Type u_1 inst\u271d : Group G H K : Subgroup G S T : Set G hST : IsComplement S T hHT : IsComplement (\u2191H) T hSK : IsComplement S \u2191K h : { x // x \u2208 H } g : G this : (\u2191(equiv hHT) (\u2191h * g)).2 = (\u2191(equiv hHT) g).2 \u22a2 \u2191(\u2191(equiv hHT) (\u2191h * g)).2 = \u2191(h * (\u2191(equiv hHT) g).1, (\u2191(equiv hHT) g).2).2 ** rw [this] ** Qed", + "informal": "" + }, + { + "formal": "RatFunc.laurent_C ** R : Type u inst\u271d : CommRing R hdomain : IsDomain R r s : R p q : R[X] f : RatFunc R x : R \u22a2 \u2191(laurent r) (\u2191C x) = \u2191C x ** rw [\u2190 algebraMap_C, laurent_algebraMap, taylor_C] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Lp.edist_toLp_toLp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : Mem\u2112p f p hg : Mem\u2112p g p \u22a2 edist (Mem\u2112p.toLp f hf) (Mem\u2112p.toLp g hg) = snorm (f - g) p \u03bc ** rw [edist_def] ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedAddCommGroup G f g : \u03b1 \u2192 E hf : Mem\u2112p f p hg : Mem\u2112p g p \u22a2 snorm (\u2191\u2191(Mem\u2112p.toLp f hf) - \u2191\u2191(Mem\u2112p.toLp g hg)) p \u03bc = snorm (f - g) p \u03bc ** exact snorm_congr_ae (hf.coeFn_toLp.sub hg.coeFn_toLp) ** Qed", + "informal": "" + }, + { + "formal": "Std.RBNode.RedRed.balance1 ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n \u22a2 \u2203 c, Balanced (balance1 l v r) c (n + 1) ** unfold balance1 ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n \u22a2 \u2203 c, Balanced (match l, v, r with | node red (node red a x b) y c, z, d => node red (node black a x b) y (node black c z d) | node red a x (node red b y c), z, d => node red (node black a x b) y (node black c z d) | a, x, b => node black a x b) c (n + 1) ** split ** case h_1 \u03b1 : Type u_1 p : Prop n : Nat c : RBColor v : \u03b1 r : RBNode \u03b1 hr : Balanced r c n x\u271d\u00b3 : RBNode \u03b1 x\u271d\u00b2 : \u03b1 x\u271d\u00b9 a\u271d : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 hl : RedRed p (node red (node red a\u271d x\u271d b\u271d) y\u271d c\u271d) n \u22a2 \u2203 c, Balanced (node red (node black a\u271d x\u271d b\u271d) y\u271d (node black c\u271d v r)) c (n + 1) ** have .redred _ (.red ha hb) hc := hl ** case h_1 \u03b1 : Type u_1 p : Prop n : Nat c : RBColor v : \u03b1 r : RBNode \u03b1 hr : Balanced r c n x\u271d\u00b3 : RBNode \u03b1 x\u271d\u00b2 : \u03b1 x\u271d\u00b9 a\u271d\u00b9 : RBNode \u03b1 x\u271d : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 hl : RedRed p (node red (node red a\u271d\u00b9 x\u271d b\u271d) y\u271d c\u271d) n c\u2082\u271d : RBColor a\u271d : p ha : Balanced a\u271d\u00b9 black n hb : Balanced b\u271d black n hc : Balanced c\u271d c\u2082\u271d n \u22a2 \u2203 c, Balanced (node red (node black a\u271d\u00b9 x\u271d b\u271d) y\u271d (node black c\u271d v r)) c (n + 1) ** exact \u27e8_, .red (.black ha hb) (.black hc hr)\u27e9 ** case h_2 \u03b1 : Type u_1 p : Prop n : Nat c : RBColor v : \u03b1 r : RBNode \u03b1 hr : Balanced r c n x\u271d\u2074 : RBNode \u03b1 x\u271d\u00b3 : \u03b1 x\u271d\u00b2 a\u271d : RBNode \u03b1 x\u271d\u00b9 : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 x\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), a\u271d = node red a x b \u2192 False hl : RedRed p (node red a\u271d x\u271d\u00b9 (node red b\u271d y\u271d c\u271d)) n \u22a2 \u2203 c, Balanced (node red (node black a\u271d x\u271d\u00b9 b\u271d) y\u271d (node black c\u271d v r)) c (n + 1) ** have .redred _ ha (.red hb hc) := hl ** case h_2 \u03b1 : Type u_1 p : Prop n : Nat c : RBColor v : \u03b1 r : RBNode \u03b1 hr : Balanced r c n x\u271d\u2074 : RBNode \u03b1 x\u271d\u00b3 : \u03b1 x\u271d\u00b2 a\u271d\u00b9 : RBNode \u03b1 x\u271d\u00b9 : \u03b1 b\u271d : RBNode \u03b1 y\u271d : \u03b1 c\u271d : RBNode \u03b1 x\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1), a\u271d\u00b9 = node red a x b \u2192 False hl : RedRed p (node red a\u271d\u00b9 x\u271d\u00b9 (node red b\u271d y\u271d c\u271d)) n c\u2081\u271d : RBColor a\u271d : p ha : Balanced a\u271d\u00b9 c\u2081\u271d n hb : Balanced b\u271d black n hc : Balanced c\u271d black n \u22a2 \u2203 c, Balanced (node red (node black a\u271d\u00b9 x\u271d\u00b9 b\u271d) y\u271d (node black c\u271d v r)) c (n + 1) ** exact \u27e8_, .red (.black ha hb) (.black hc hr)\u27e9 ** case h_3 \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n x\u271d\u2074 : RBNode \u03b1 x\u271d\u00b3 : \u03b1 x\u271d\u00b2 : RBNode \u03b1 x\u271d\u00b9 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red (node red a x b) y c \u2192 False x\u271d : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red a x (node red b y c) \u2192 False \u22a2 \u2203 c, Balanced (node black l v r) c (n + 1) ** next H1 H2 => match hl with\n| .balanced hl => exact \u27e8_, .black hl hr\u27e9\n| .redred _ (c\u2081 := black) (c\u2082 := black) ha hb => exact \u27e8_, .black (.red ha hb) hr\u27e9\n| .redred _ (c\u2081 := red) (.red ..) _ => cases H1 _ _ _ _ _ rfl\n| .redred _ (c\u2082 := red) _ (.red ..) => cases H2 _ _ _ _ _ rfl ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n x\u271d\u00b2 : RBNode \u03b1 x\u271d\u00b9 : \u03b1 x\u271d : RBNode \u03b1 H1 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red (node red a x b) y c \u2192 False H2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red a x (node red b y c) \u2192 False \u22a2 \u2203 c, Balanced (node black l v r) c (n + 1) ** match hl with\n| .balanced hl => exact \u27e8_, .black hl hr\u27e9\n| .redred _ (c\u2081 := black) (c\u2082 := black) ha hb => exact \u27e8_, .black (.red ha hb) hr\u27e9\n| .redred _ (c\u2081 := red) (.red ..) _ => cases H1 _ _ _ _ _ rfl\n| .redred _ (c\u2082 := red) _ (.red ..) => cases H2 _ _ _ _ _ rfl ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l\u271d : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl\u271d : RedRed p l\u271d n hr : Balanced r c n x\u271d\u00b2 : RBNode \u03b1 x\u271d\u00b9 : \u03b1 x\u271d l : RBNode \u03b1 c\u271d : RBColor hl : Balanced l c\u271d n H1 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red (node red a x b) y c \u2192 False H2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), l = node red a x (node red b y c) \u2192 False \u22a2 \u2203 c, Balanced (node black l v r) c (n + 1) ** exact \u27e8_, .black hl hr\u27e9 ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n x\u271d\u00b3 : RBNode \u03b1 x\u271d\u00b2 : \u03b1 x\u271d\u00b9 a\u271d\u00b9 b\u271d : RBNode \u03b1 x\u271d : \u03b1 a\u271d : p ha : Balanced a\u271d\u00b9 black n hb : Balanced b\u271d black n H1 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), node red a\u271d\u00b9 x\u271d b\u271d = node red (node red a x b) y c \u2192 False H2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), node red a\u271d\u00b9 x\u271d b\u271d = node red a x (node red b y c) \u2192 False \u22a2 \u2203 c, Balanced (node black (node red a\u271d\u00b9 x\u271d b\u271d) v r) c (n + 1) ** exact \u27e8_, .black (.red ha hb) hr\u27e9 ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n x\u271d\u00b3 : RBNode \u03b1 x\u271d\u00b2 : \u03b1 x\u271d\u00b9 l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d b\u271d : RBNode \u03b1 c\u2082\u271d : RBColor x\u271d : \u03b1 a\u271d\u00b3 : p a\u271d\u00b2 : Balanced l\u271d black n a\u271d\u00b9 : Balanced r\u271d black n a\u271d : Balanced b\u271d c\u2082\u271d n H1 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), node red (node red l\u271d v\u271d r\u271d) x\u271d b\u271d = node red (node red a x b) y c \u2192 False H2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), node red (node red l\u271d v\u271d r\u271d) x\u271d b\u271d = node red a x (node red b y c) \u2192 False \u22a2 \u2203 c, Balanced (node black (node red (node red l\u271d v\u271d r\u271d) x\u271d b\u271d) v r) c (n + 1) ** cases H1 _ _ _ _ _ rfl ** \u03b1 : Type u_1 p : Prop n : Nat c : RBColor l : RBNode \u03b1 v : \u03b1 r : RBNode \u03b1 hl : RedRed p l n hr : Balanced r c n x\u271d\u00b3 : RBNode \u03b1 x\u271d\u00b2 : \u03b1 x\u271d\u00b9 a\u271d\u2074 : RBNode \u03b1 c\u2081\u271d : RBColor l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 x\u271d : \u03b1 a\u271d\u00b3 : p a\u271d\u00b2 : Balanced a\u271d\u2074 c\u2081\u271d n a\u271d\u00b9 : Balanced l\u271d black n a\u271d : Balanced r\u271d black n H1 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), node red a\u271d\u2074 x\u271d (node red l\u271d v\u271d r\u271d) = node red (node red a x b) y c \u2192 False H2 : \u2200 (a : RBNode \u03b1) (x : \u03b1) (b : RBNode \u03b1) (y : \u03b1) (c : RBNode \u03b1), node red a\u271d\u2074 x\u271d (node red l\u271d v\u271d r\u271d) = node red a x (node red b y c) \u2192 False \u22a2 \u2203 c, Balanced (node black (node red a\u271d\u2074 x\u271d (node red l\u271d v\u271d r\u271d)) v r) c (n + 1) ** cases H2 _ _ _ _ _ rfl ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.zero_mem_polar ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : NormedCommRing \ud835\udd5c inst\u271d\u00b3 : AddCommMonoid E inst\u271d\u00b2 : AddCommMonoid F inst\u271d\u00b9 : Module \ud835\udd5c E inst\u271d : Module \ud835\udd5c F B : E \u2192\u2097[\ud835\udd5c] F \u2192\u2097[\ud835\udd5c] \ud835\udd5c s : Set E x\u271d\u00b9 : E x\u271d : x\u271d\u00b9 \u2208 s \u22a2 \u2016\u2191(\u2191B x\u271d\u00b9) 0\u2016 \u2264 1 ** simp only [map_zero, norm_zero, zero_le_one] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.eval\u2082_at_one ** R : Type u S\u271d : Type v T : Type w \u03b9 : Type y a b : R m n : \u2115 inst\u271d\u00b9 : Semiring R p q r : R[X] x : R S : Type u_1 inst\u271d : Semiring S f : R \u2192+* S \u22a2 eval\u2082 f 1 p = \u2191f (eval 1 p) ** convert eval\u2082_at_apply (p := p) f 1 ** case h.e'_2.h.e'_6 R : Type u S\u271d : Type v T : Type w \u03b9 : Type y a b : R m n : \u2115 inst\u271d\u00b9 : Semiring R p q r : R[X] x : R S : Type u_1 inst\u271d : Semiring S f : R \u2192+* S \u22a2 1 = \u2191f 1 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.restriction_one ** R : Type u S : Type u_1 inst\u271d : Ring R i : \u2115 \u22a2 \u2191(coeff (restriction 1) i) = \u2191(coeff 1 i) ** rw [coeff_restriction', coeff_one, coeff_one] ** R : Type u S : Type u_1 inst\u271d : Ring R i : \u2115 \u22a2 (if i = 0 then 1 else 0) = \u2191(if i = 0 then 1 else 0) ** split_ifs <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "quadraticChar_odd_prime ** F : Type u_1 inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Fintype F inst\u271d\u00b9 : DecidableEq F hF : ringChar F \u2260 2 p : \u2115 inst\u271d : Fact (Nat.Prime p) hp\u2081 : p \u2260 2 hp\u2082 : ringChar F \u2260 p \u22a2 \u2191(quadraticChar F) \u2191p = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191\u03c7\u2084 \u2191(Fintype.card F)) * \u2191(Fintype.card F)) ** rw [\u2190 quadraticChar_neg_one hF] ** F : Type u_1 inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Fintype F inst\u271d\u00b9 : DecidableEq F hF : ringChar F \u2260 2 p : \u2115 inst\u271d : Fact (Nat.Prime p) hp\u2081 : p \u2260 2 hp\u2082 : ringChar F \u2260 p \u22a2 \u2191(quadraticChar F) \u2191p = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F)) ** have h := quadraticChar_card_card hF (ne_of_eq_of_ne (ringChar_zmod_n p) hp\u2081)\n (ne_of_eq_of_ne (ringChar_zmod_n p) hp\u2082.symm) ** F : Type u_1 inst\u271d\u00b3 : Field F inst\u271d\u00b2 : Fintype F inst\u271d\u00b9 : DecidableEq F hF : ringChar F \u2260 2 p : \u2115 inst\u271d : Fact (Nat.Prime p) hp\u2081 : p \u2260 2 hp\u2082 : ringChar F \u2260 p h : \u2191(quadraticChar F) \u2191(Fintype.card (ZMod p)) = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F)) \u22a2 \u2191(quadraticChar F) \u2191p = \u2191(quadraticChar (ZMod p)) (\u2191(\u2191(quadraticChar F) (-1)) * \u2191(Fintype.card F)) ** rwa [card p] at h ** Qed", + "informal": "" + }, + { + "formal": "Nat.Ico_insert_succ_left ** a b c : \u2115 h : a < b \u22a2 insert a (Ico (succ a) b) = Ico a b ** rw [Ico_succ_left, \u2190 Ioo_insert_left h] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.coeff_hermite_self ** n : \u2115 \u22a2 coeff (hermite n) n = 1 ** induction' n with n ih ** case zero \u22a2 coeff (hermite Nat.zero) Nat.zero = 1 ** apply coeff_C ** case succ n : \u2115 ih : coeff (hermite n) n = 1 \u22a2 coeff (hermite (Nat.succ n)) (Nat.succ n) = 1 ** rw [coeff_hermite_succ_succ, ih, coeff_hermite_of_lt, mul_zero, sub_zero] ** case succ n : \u2115 ih : coeff (hermite n) n = 1 \u22a2 n < n + 2 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.IsUnitTrinomial.irreducible_of_coprime' ** p q : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) \u22a2 Irreducible p ** refine' hp.irreducible_of_coprime fun q hq hq' => _ ** p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p \u22a2 IsUnit q ** suffices \u00ac0 < q.natDegree by\n rcases hq with \u27e8p, rfl\u27e9\n replace hp := hp.leadingCoeff_isUnit\n rw [leadingCoeff_mul] at hp\n replace hp := isUnit_of_mul_isUnit_left hp\n rw [not_lt, le_zero_iff] at this\n rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, \u2190 this] ** p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p \u22a2 \u00ac0 < natDegree q ** intro hq'' ** p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < natDegree q \u22a2 False ** rw [natDegree_pos_iff_degree_pos] at hq'' ** p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < degree q \u22a2 False ** rw [\u2190 degree_map_eq_of_injective (algebraMap \u2124 \u2102).injective_int] at hq'' ** p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) \u22a2 False ** cases' Complex.exists_root hq'' with z hz ** case intro p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) z : \u2102 hz : IsRoot (map (algebraMap \u2124 \u2102) q) z \u22a2 False ** rw [IsRoot, eval_map, \u2190 aeval_def] at hz ** case intro p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) z : \u2102 hz : \u2191(aeval z) q = 0 \u22a2 False ** refine' h z \u27e8_, _\u27e9 ** p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p this : \u00ac0 < natDegree q \u22a2 IsUnit q ** rcases hq with \u27e8p, rfl\u27e9 ** case intro q\u271d q : \u2124[X] this : \u00ac0 < natDegree q p : \u2124[X] hp : IsUnitTrinomial (q * p) h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0) hq' : q \u2223 mirror (q * p) \u22a2 IsUnit q ** replace hp := hp.leadingCoeff_isUnit ** case intro q\u271d q : \u2124[X] this : \u00ac0 < natDegree q p : \u2124[X] h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0) hq' : q \u2223 mirror (q * p) hp : IsUnit (leadingCoeff (q * p)) \u22a2 IsUnit q ** rw [leadingCoeff_mul] at hp ** case intro q\u271d q : \u2124[X] this : \u00ac0 < natDegree q p : \u2124[X] h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0) hq' : q \u2223 mirror (q * p) hp : IsUnit (leadingCoeff q * leadingCoeff p) \u22a2 IsUnit q ** replace hp := isUnit_of_mul_isUnit_left hp ** case intro q\u271d q : \u2124[X] this : \u00ac0 < natDegree q p : \u2124[X] h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0) hq' : q \u2223 mirror (q * p) hp : IsUnit (leadingCoeff q) \u22a2 IsUnit q ** rw [not_lt, le_zero_iff] at this ** case intro q\u271d q : \u2124[X] this : natDegree q = 0 p : \u2124[X] h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (q * p) = 0 \u2227 \u2191(aeval z) (mirror (q * p)) = 0) hq' : q \u2223 mirror (q * p) hp : IsUnit (leadingCoeff q) \u22a2 IsUnit q ** rwa [eq_C_of_natDegree_eq_zero this, isUnit_C, \u2190 this] ** case intro.refine'_1 p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) z : \u2102 hz : \u2191(aeval z) q = 0 \u22a2 \u2191(aeval z) p = 0 ** cases' hq with g' hg' ** case intro.refine'_1.intro p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq' : q \u2223 mirror p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) z : \u2102 hz : \u2191(aeval z) q = 0 g' : \u2124[X] hg' : p = q * g' \u22a2 \u2191(aeval z) p = 0 ** rw [hg', aeval_mul, hz, zero_mul] ** case intro.refine'_2 p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq' : q \u2223 mirror p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) z : \u2102 hz : \u2191(aeval z) q = 0 \u22a2 \u2191(aeval z) (mirror p) = 0 ** cases' hq' with g' hg' ** case intro.refine'_2.intro p q\u271d : \u2124[X] hp : IsUnitTrinomial p h : \u2200 (z : \u2102), \u00ac(\u2191(aeval z) p = 0 \u2227 \u2191(aeval z) (mirror p) = 0) q : \u2124[X] hq : q \u2223 p hq'' : 0 < degree (map (algebraMap \u2124 \u2102) q) z : \u2102 hz : \u2191(aeval z) q = 0 g' : \u2124[X] hg' : mirror p = q * g' \u22a2 \u2191(aeval z) (mirror p) = 0 ** rw [hg', aeval_mul, hz, zero_mul] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.det_comm' ** R : Type u_1 inst\u271d\u00b9\u00b9 : CommRing R M\u271d : Type u_2 inst\u271d\u00b9\u2070 : AddCommGroup M\u271d inst\u271d\u2079 : Module R M\u271d M'\u271d : Type u_3 inst\u271d\u2078 : AddCommGroup M'\u271d inst\u271d\u2077 : Module R M'\u271d \u03b9 : Type u_4 inst\u271d\u2076 : DecidableEq \u03b9 inst\u271d\u2075 : Fintype \u03b9 e : Basis \u03b9 R M\u271d A : Type u_5 inst\u271d\u2074 : CommRing A m : Type u_6 n : Type u_7 inst\u271d\u00b3 : Fintype m inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq m inst\u271d : DecidableEq n M : Matrix n m A N M' : Matrix m n A hMM' : M * M' = 1 hM'M : M' * M = 1 \u22a2 det (M * N) = det (N * M) ** nontriviality A ** R : Type u_1 inst\u271d\u00b9\u00b9 : CommRing R M\u271d : Type u_2 inst\u271d\u00b9\u2070 : AddCommGroup M\u271d inst\u271d\u2079 : Module R M\u271d M'\u271d : Type u_3 inst\u271d\u2078 : AddCommGroup M'\u271d inst\u271d\u2077 : Module R M'\u271d \u03b9 : Type u_4 inst\u271d\u2076 : DecidableEq \u03b9 inst\u271d\u2075 : Fintype \u03b9 e : Basis \u03b9 R M\u271d A : Type u_5 inst\u271d\u2074 : CommRing A m : Type u_6 n : Type u_7 inst\u271d\u00b3 : Fintype m inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq m inst\u271d : DecidableEq n M : Matrix n m A N M' : Matrix m n A hMM' : M * M' = 1 hM'M : M' * M = 1 \u271d : Nontrivial A \u22a2 det (M * N) = det (N * M) ** let e := indexEquivOfInv hMM' hM'M ** R : Type u_1 inst\u271d\u00b9\u00b9 : CommRing R M\u271d : Type u_2 inst\u271d\u00b9\u2070 : AddCommGroup M\u271d inst\u271d\u2079 : Module R M\u271d M'\u271d : Type u_3 inst\u271d\u2078 : AddCommGroup M'\u271d inst\u271d\u2077 : Module R M'\u271d \u03b9 : Type u_4 inst\u271d\u2076 : DecidableEq \u03b9 inst\u271d\u2075 : Fintype \u03b9 e\u271d : Basis \u03b9 R M\u271d A : Type u_5 inst\u271d\u2074 : CommRing A m : Type u_6 n : Type u_7 inst\u271d\u00b3 : Fintype m inst\u271d\u00b2 : Fintype n inst\u271d\u00b9 : DecidableEq m inst\u271d : DecidableEq n M : Matrix n m A N M' : Matrix m n A hMM' : M * M' = 1 hM'M : M' * M = 1 \u271d : Nontrivial A e : n \u2243 m := indexEquivOfInv hMM' hM'M \u22a2 det (M * N) = det (N * M) ** rw [\u2190 det_submatrix_equiv_self e, \u2190 submatrix_mul_equiv _ _ _ (Equiv.refl n) _, det_comm,\n submatrix_mul_equiv, Equiv.coe_refl, submatrix_id_id] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.blockDiagonal_one ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 p : Type u_5 q : Type u_6 m' : o \u2192 Type u_7 n' : o \u2192 Type u_8 p' : o \u2192 Type u_9 R : Type u_10 S : Type u_11 \u03b1 : Type u_12 \u03b2 : Type u_13 inst\u271d\u2074 : DecidableEq o inst\u271d\u00b3 : Zero \u03b1 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : DecidableEq m inst\u271d : One \u03b1 \u22a2 (blockDiagonal fun x => diagonal fun x => 1) = diagonal fun x => 1 ** rw [blockDiagonal_diagonal] ** Qed", + "informal": "" + }, + { + "formal": "covariant_le_of_covariant_lt ** M : Type u_1 N : Type u_2 \u03bc : M \u2192 N \u2192 N r : N \u2192 N \u2192 Prop inst\u271d : PartialOrder N \u22a2 (Covariant M N \u03bc fun x x_1 => x < x_1) \u2192 Covariant M N \u03bc fun x x_1 => x \u2264 x_1 ** intro h a b c bc ** M : Type u_1 N : Type u_2 \u03bc : M \u2192 N \u2192 N r : N \u2192 N \u2192 Prop inst\u271d : PartialOrder N h : Covariant M N \u03bc fun x x_1 => x < x_1 a : M b c : N bc : b \u2264 c \u22a2 \u03bc a b \u2264 \u03bc a c ** rcases bc.eq_or_lt with (rfl | bc) ** case inl M : Type u_1 N : Type u_2 \u03bc : M \u2192 N \u2192 N r : N \u2192 N \u2192 Prop inst\u271d : PartialOrder N h : Covariant M N \u03bc fun x x_1 => x < x_1 a : M b : N bc : b \u2264 b \u22a2 \u03bc a b \u2264 \u03bc a b ** exact le_rfl ** case inr M : Type u_1 N : Type u_2 \u03bc : M \u2192 N \u2192 N r : N \u2192 N \u2192 Prop inst\u271d : PartialOrder N h : Covariant M N \u03bc fun x x_1 => x < x_1 a : M b c : N bc\u271d : b \u2264 c bc : b < c \u22a2 \u03bc a b \u2264 \u03bc a c ** exact (h _ bc).le ** Qed", + "informal": "" + }, + { + "formal": "ofMul_image_zpowers_eq_zmultiples_ofMul ** G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G \u22a2 \u2191Additive.ofMul '' \u2191(Subgroup.zpowers x) = \u2191(AddSubgroup.zmultiples (\u2191Additive.ofMul x)) ** ext y ** case h G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G \u22a2 y \u2208 \u2191Additive.ofMul '' \u2191(Subgroup.zpowers x) \u2194 y \u2208 \u2191(AddSubgroup.zmultiples (\u2191Additive.ofMul x)) ** constructor ** case h.mp G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G \u22a2 y \u2208 \u2191Additive.ofMul '' \u2191(Subgroup.zpowers x) \u2192 y \u2208 \u2191(AddSubgroup.zmultiples (\u2191Additive.ofMul x)) ** rintro \u27e8z, \u27e8m, hm\u27e9, hz2\u27e9 ** case h.mp.intro.intro.intro G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G z : G hz2 : \u2191Additive.ofMul z = y m : \u2124 hm : (fun x x_1 => x ^ x_1) x m = z \u22a2 y \u2208 \u2191(AddSubgroup.zmultiples (\u2191Additive.ofMul x)) ** use m ** case h G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G z : G hz2 : \u2191Additive.ofMul z = y m : \u2124 hm : (fun x x_1 => x ^ x_1) x m = z \u22a2 (fun x_1 => x_1 \u2022 \u2191Additive.ofMul x) m = y ** simp only at * ** case h G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G z : G hz2 : \u2191Additive.ofMul z = y m : \u2124 hm : x ^ m = z \u22a2 m \u2022 \u2191Additive.ofMul x = y ** rwa [\u2190 ofMul_zpow, hm] ** case h.mpr G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G \u22a2 y \u2208 \u2191(AddSubgroup.zmultiples (\u2191Additive.ofMul x)) \u2192 y \u2208 \u2191Additive.ofMul '' \u2191(Subgroup.zpowers x) ** rintro \u27e8n, hn\u27e9 ** case h.mpr.intro G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G n : \u2124 hn : (fun x_1 => x_1 \u2022 \u2191Additive.ofMul x) n = y \u22a2 y \u2208 \u2191Additive.ofMul '' \u2191(Subgroup.zpowers x) ** refine' \u27e8x ^ n, \u27e8n, rfl\u27e9, _\u27e9 ** case h.mpr.intro G : Type u_1 inst\u271d\u00b2 : Group G A : Type u_2 inst\u271d\u00b9 : AddGroup A N : Type u_3 inst\u271d : Group N x : G y : Additive G n : \u2124 hn : (fun x_1 => x_1 \u2022 \u2191Additive.ofMul x) n = y \u22a2 \u2191Additive.ofMul (x ^ n) = y ** rwa [ofMul_zpow] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.HasCoequalizersOfHasPushoutsAndBinaryCoproducts.pushoutInl_eq_pushout_inr ** C : Type u inst\u271d\u00b3 : Category.{v, u} C D : Type u' inst\u271d\u00b2 : Category.{v', u'} D G : C \u2964 D inst\u271d\u00b9 : HasBinaryCoproducts C inst\u271d : HasPushouts C F : WalkingParallelPair \u2964 C \u22a2 pushoutInl F = pushout.inr ** convert (whisker_eq Limits.coprod.inl pushout.condition :\n (_ : F.obj _ \u27f6 constructCoequalizer _) = _) <;> simp ** Qed", + "informal": "" + }, + { + "formal": "SMulCommClass.of_mclosure_eq_top ** M : Type u_1 A : Type u_2 B : Type u_3 N : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : SMul N \u03b1 inst\u271d : MulAction M \u03b1 s : Set M htop : Submonoid.closure s = \u22a4 hs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z \u22a2 SMulCommClass M N \u03b1 ** refine' \u27e8fun x => Submonoid.induction_of_closure_eq_top_left htop x _ _\u27e9 ** case refine'_1 M : Type u_1 A : Type u_2 B : Type u_3 N : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : SMul N \u03b1 inst\u271d : MulAction M \u03b1 s : Set M htop : Submonoid.closure s = \u22a4 hs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z x : M \u22a2 \u2200 (n : N) (a : \u03b1), 1 \u2022 n \u2022 a = n \u2022 1 \u2022 a ** intro y z ** case refine'_1 M : Type u_1 A : Type u_2 B : Type u_3 N : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : SMul N \u03b1 inst\u271d : MulAction M \u03b1 s : Set M htop : Submonoid.closure s = \u22a4 hs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z x : M y : N z : \u03b1 \u22a2 1 \u2022 y \u2022 z = y \u2022 1 \u2022 z ** rw [one_smul, one_smul] ** case refine'_2 M : Type u_1 A : Type u_2 B : Type u_3 N : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : SMul N \u03b1 inst\u271d : MulAction M \u03b1 s : Set M htop : Submonoid.closure s = \u22a4 hs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z x : M \u22a2 \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), (\u2200 (n : N) (a : \u03b1), y \u2022 n \u2022 a = n \u2022 y \u2022 a) \u2192 \u2200 (n : N) (a : \u03b1), (x * y) \u2022 n \u2022 a = n \u2022 (x * y) \u2022 a ** clear x ** case refine'_2 M : Type u_1 A : Type u_2 B : Type u_3 N : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : SMul N \u03b1 inst\u271d : MulAction M \u03b1 s : Set M htop : Submonoid.closure s = \u22a4 hs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z \u22a2 \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : M), (\u2200 (n : N) (a : \u03b1), y \u2022 n \u2022 a = n \u2022 y \u2022 a) \u2192 \u2200 (n : N) (a : \u03b1), (x * y) \u2022 n \u2022 a = n \u2022 (x * y) \u2022 a ** intro x hx x' hx' y z ** case refine'_2 M : Type u_1 A : Type u_2 B : Type u_3 N : Type u_4 \u03b1 : Type u_5 inst\u271d\u00b2 : Monoid M inst\u271d\u00b9 : SMul N \u03b1 inst\u271d : MulAction M \u03b1 s : Set M htop : Submonoid.closure s = \u22a4 hs : \u2200 (x : M), x \u2208 s \u2192 \u2200 (y : N) (z : \u03b1), x \u2022 y \u2022 z = y \u2022 x \u2022 z x : M hx : x \u2208 s x' : M hx' : \u2200 (n : N) (a : \u03b1), x' \u2022 n \u2022 a = n \u2022 x' \u2022 a y : N z : \u03b1 \u22a2 (x * x') \u2022 y \u2022 z = y \u2022 (x * x') \u2022 z ** rw [mul_smul, mul_smul, hx', hs x hx] ** Qed", + "informal": "" + }, + { + "formal": "innerDualCone_univ ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup H inst\u271d : InnerProductSpace \u211d H s t : Set H \u22a2 innerDualCone univ = 0 ** suffices \u2200 x : H, x \u2208 (univ : Set H).innerDualCone \u2192 x = 0 by\n apply SetLike.coe_injective\n exact eq_singleton_iff_unique_mem.mpr \u27e8fun x _ => (inner_zero_right _).ge, this\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup H inst\u271d : InnerProductSpace \u211d H s t : Set H \u22a2 \u2200 (x : H), x \u2208 innerDualCone univ \u2192 x = 0 ** exact fun x hx => by simpa [\u2190 real_inner_self_nonpos] using hx (-x) (mem_univ _) ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup H inst\u271d : InnerProductSpace \u211d H s t : Set H this : \u2200 (x : H), x \u2208 innerDualCone univ \u2192 x = 0 \u22a2 innerDualCone univ = 0 ** apply SetLike.coe_injective ** case a \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup H inst\u271d : InnerProductSpace \u211d H s t : Set H this : \u2200 (x : H), x \u2208 innerDualCone univ \u2192 x = 0 \u22a2 \u2191(innerDualCone univ) = \u21910 ** exact eq_singleton_iff_unique_mem.mpr \u27e8fun x _ => (inner_zero_right _).ge, this\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 H : Type u_5 inst\u271d\u00b9 : NormedAddCommGroup H inst\u271d : InnerProductSpace \u211d H s t : Set H x : H hx : x \u2208 innerDualCone univ \u22a2 x = 0 ** simpa [\u2190 real_inner_self_nonpos] using hx (-x) (mem_univ _) ** Qed", + "informal": "" + }, + { + "formal": "IsLocalizedModule.fromLocalizedModule'_add ** R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M \u22a2 \u2200 (m m' : M) (s s' : { x // x \u2208 S }), fromLocalizedModule' S f (LocalizedModule.mk m s + LocalizedModule.mk m' s') = fromLocalizedModule' S f (LocalizedModule.mk m s) + fromLocalizedModule' S f (LocalizedModule.mk m' s') ** intro a a' b b' ** R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 fromLocalizedModule' S f (LocalizedModule.mk a b + LocalizedModule.mk a' b') = fromLocalizedModule' S f (LocalizedModule.mk a b) + fromLocalizedModule' S f (LocalizedModule.mk a' b') ** simp only [LocalizedModule.mk_add_mk, fromLocalizedModule'_mk] ** R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191(b * b'))))\u207b\u00b9 (\u2191f (b' \u2022 a + b \u2022 a')) = \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b)))\u207b\u00b9 (\u2191f a) + \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b')))\u207b\u00b9 (\u2191f a') ** erw [Module.End_algebraMap_isUnit_inv_apply_eq_iff, smul_add, \u2190map_smul, \u2190map_smul,\n \u2190map_smul, map_add] ** R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191f (b' \u2022 a) + \u2191f (b \u2022 a') = \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b)))\u207b\u00b9 (\u2191f (\u2191(b * b') \u2022 a)) + \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b')))\u207b\u00b9 (\u2191(b * b') \u2022 \u2191f a') ** congr 1 ** case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191f (b' \u2022 a) = \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b)))\u207b\u00b9 (\u2191f (\u2191(b * b') \u2022 a)) case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191f (b \u2022 a') = \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b')))\u207b\u00b9 (\u2191(b * b') \u2022 \u2191f a') ** all_goals rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff'] ** case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191f (b \u2022 a') = \u2191\u2191(IsUnit.unit (_ : IsUnit (\u2191(algebraMap R (Module.End R M')) \u2191b')))\u207b\u00b9 (\u2191(b * b') \u2022 \u2191f a') ** rw [Module.End_algebraMap_isUnit_inv_apply_eq_iff'] ** case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191f (\u2191(b * b') \u2022 a) = \u2191b \u2022 \u2191f (b' \u2022 a) ** erw [mul_smul, f.map_smul] ** case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191b \u2022 \u2191f (\u2191b' \u2022 a) = \u2191b \u2022 \u2191f (b' \u2022 a) ** rfl ** case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191(b * b') \u2022 \u2191f a' = \u2191b' \u2022 \u2191f (b \u2022 a') ** erw [mul_comm, f.map_smul, mul_smul] ** case e_a R : Type u_1 inst\u271d\u2077 : CommRing R S : Submonoid R M : Type u_2 M' : Type u_3 M'' : Type u_4 inst\u271d\u2076 : AddCommMonoid M inst\u271d\u2075 : AddCommMonoid M' inst\u271d\u2074 : AddCommMonoid M'' inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R M' inst\u271d\u00b9 : Module R M'' f : M \u2192\u2097[R] M' g : M \u2192\u2097[R] M'' inst\u271d : IsLocalizedModule S f x y : LocalizedModule S M a a' : M b b' : { x // x \u2208 S } \u22a2 \u2191b' \u2022 \u2191b \u2022 \u2191f a' = \u2191b' \u2022 \u2191(Submonoid.subtype S) b \u2022 \u2191f a' ** rfl ** Qed", + "informal": "" + }, + { + "formal": "BooleanRing.inf_assoc ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b2 : BooleanRing \u03b1 inst\u271d\u00b9 : BooleanRing \u03b2 inst\u271d : BooleanRing \u03b3 a b c : \u03b1 \u22a2 a * b * c = a * (b * c) ** ring ** Qed", + "informal": "" + }, + { + "formal": "WithBot.preimage_coe_Ioc ** \u03b1 : Type u_1 inst\u271d : PartialOrder \u03b1 a b : \u03b1 \u22a2 some \u207b\u00b9' Ioc \u2191a \u2191b = Ioc a b ** simp [\u2190 Ioi_inter_Iic] ** Qed", + "informal": "" + }, + { + "formal": "mem_extChartAt_source ** \ud835\udd5c : Type u_1 E : Type u_2 M : Type u_3 H : Type u_4 E' : Type u_5 M' : Type u_6 H' : Type u_7 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : TopologicalSpace H inst\u271d\u2076 : TopologicalSpace M f f' : LocalHomeomorph M H I : ModelWithCorners \ud835\udd5c E H inst\u271d\u2075 : NormedAddCommGroup E' inst\u271d\u2074 : NormedSpace \ud835\udd5c E' inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' I' : ModelWithCorners \ud835\udd5c E' H' x : M s t : Set M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : ChartedSpace H' M' \u22a2 x \u2208 (extChartAt I x).source ** simp only [extChartAt_source, mem_chart_source] ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.Substructure.FG.map ** L : Language M : Type u_1 inst\u271d\u00b9 : Structure L M N : Type u_2 inst\u271d : Structure L N f : M \u2192[L] N s : Substructure L M hs : FG s t : Set M ht : Set.Finite t \u2227 LowerAdjoint.toFun (closure L) t = s \u22a2 LowerAdjoint.toFun (closure L) (\u2191f '' t) = Substructure.map f s ** rw [closure_image, ht.2] ** Qed", + "informal": "" + }, + { + "formal": "SemiconjBy.inv_inv_symm_iff ** G : Type u_1 inst\u271d : DivisionMonoid G a x y : G \u22a2 (a\u207b\u00b9 * x\u207b\u00b9)\u207b\u00b9 = (y\u207b\u00b9 * a\u207b\u00b9)\u207b\u00b9 \u2194 SemiconjBy a y x ** rw [mul_inv_rev, mul_inv_rev, inv_inv, inv_inv, inv_inv, eq_comm, SemiconjBy] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.NatIso.cancel_natIso_inv_right ** C : Type u\u2081 inst\u271d\u00b2 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b9 : Category.{v\u2082, u\u2082} D E : Type u\u2083 inst\u271d : Category.{v\u2083, u\u2083} E F G : C \u2964 D \u03b1 : F \u2245 G X : D Y : C f f' : X \u27f6 G.obj Y \u22a2 f \u226b \u03b1.inv.app Y = f' \u226b \u03b1.inv.app Y \u2194 f = f' ** simp only [cancel_mono, refl] ** Qed", + "informal": "" + }, + { + "formal": "multiplicity_normalizedFactorsEquivSpanNormalizedFactors_symm_eq_multiplicity ** R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing A inst\u271d\u2078 : Field K inst\u271d\u2077 : IsDomain A inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsPrincipalIdealRing R inst\u271d\u2074 : DecidableEq R inst\u271d\u00b3 : DecidableEq (Ideal R) inst\u271d\u00b2 : NormalizationMonoid R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 r : R hr : r \u2260 0 I : \u2191{I | I \u2208 normalizedFactors (span {r})} \u22a2 multiplicity (\u2191(\u2191(normalizedFactorsEquivSpanNormalizedFactors hr).symm I)) r = multiplicity (\u2191I) (span {r}) ** obtain \u27e8x, hx\u27e9 := (normalizedFactorsEquivSpanNormalizedFactors hr).surjective I ** case intro R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing A inst\u271d\u2078 : Field K inst\u271d\u2077 : IsDomain A inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsPrincipalIdealRing R inst\u271d\u2074 : DecidableEq R inst\u271d\u00b3 : DecidableEq (Ideal R) inst\u271d\u00b2 : NormalizationMonoid R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 r : R hr : r \u2260 0 I : \u2191{I | I \u2208 normalizedFactors (span {r})} x : \u2191{d | d \u2208 normalizedFactors r} hx : \u2191(normalizedFactorsEquivSpanNormalizedFactors hr) x = I \u22a2 multiplicity (\u2191(\u2191(normalizedFactorsEquivSpanNormalizedFactors hr).symm I)) r = multiplicity (\u2191I) (span {r}) ** obtain \u27e8a, ha\u27e9 := x ** case intro.mk R : Type u_1 A : Type u_2 K : Type u_3 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : CommRing A inst\u271d\u2078 : Field K inst\u271d\u2077 : IsDomain A inst\u271d\u2076 : IsDomain R inst\u271d\u2075 : IsPrincipalIdealRing R inst\u271d\u2074 : DecidableEq R inst\u271d\u00b3 : DecidableEq (Ideal R) inst\u271d\u00b2 : NormalizationMonoid R inst\u271d\u00b9 : DecidableRel fun x x_1 => x \u2223 x_1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 r : R hr : r \u2260 0 I : \u2191{I | I \u2208 normalizedFactors (span {r})} a : R ha : a \u2208 {d | d \u2208 normalizedFactors r} hx : \u2191(normalizedFactorsEquivSpanNormalizedFactors hr) { val := a, property := ha } = I \u22a2 multiplicity (\u2191(\u2191(normalizedFactorsEquivSpanNormalizedFactors hr).symm I)) r = multiplicity (\u2191I) (span {r}) ** rw [hx.symm, Equiv.symm_apply_apply, Subtype.coe_mk,\n multiplicity_normalizedFactorsEquivSpanNormalizedFactors_eq_multiplicity hr ha, hx] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.condexp_prod_ae_eq_integral_condDistrib ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2078 : TopologicalSpace \u03a9 inst\u271d\u2077 : MeasurableSpace \u03a9 inst\u271d\u2076 : PolishSpace \u03a9 inst\u271d\u2075 : BorelSpace \u03a9 inst\u271d\u2074 : Nonempty \u03a9 inst\u271d\u00b3 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u00b2 : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F hX : Measurable X hY : AEMeasurable Y hf : StronglyMeasurable f hf_int : Integrable fun a => f (X a, Y a) \u22a2 Integrable f ** rwa [integrable_map_measure hf.aestronglyMeasurable (hX.aemeasurable.prod_mk hY)] ** Qed", + "informal": "" + }, + { + "formal": "ZFSet.mem_pair ** x y z : ZFSet \u22a2 x \u2208 {y, z} \u2194 x = y \u2228 x = z ** simp ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.root_mul_right_of_isRoot ** R : Type u S : Type v T : Type w \u03b9 : Type y a b : R m n : \u2115 inst\u271d\u00b9 : CommSemiring R p\u271d q\u271d : R[X] x : R inst\u271d : CommSemiring S f : R \u2192+* S p q : R[X] H : IsRoot p a \u22a2 IsRoot (p * q) a ** rw [IsRoot, eval_mul, IsRoot.def.1 H, zero_mul] ** Qed", + "informal": "" + }, + { + "formal": "Fintype.card_of_finset' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Set \u03b1 s : Finset \u03b1 H : \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 p inst\u271d : Fintype \u2191p \u22a2 card \u2191p = Finset.card s ** rw [\u2190 card_ofFinset s H] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Set \u03b1 s : Finset \u03b1 H : \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 p inst\u271d : Fintype \u2191p \u22a2 card \u2191p = card \u2191p ** congr ** case h.e_2.h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p : Set \u03b1 s : Finset \u03b1 H : \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 p inst\u271d : Fintype \u2191p \u22a2 inst\u271d = ofFinset s H ** apply Subsingleton.elim ** Qed", + "informal": "" + }, + { + "formal": "Submodule.ClosedComplemented.isClosed ** R : Type u_1 inst\u271d\u2078 : Ring R M : Type u_2 inst\u271d\u2077 : TopologicalSpace M inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M\u2082 : Type u_3 inst\u271d\u2074 : TopologicalSpace M\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : TopologicalAddGroup M inst\u271d : T1Space M p : Submodule R M h : ClosedComplemented p \u22a2 IsClosed \u2191p ** rcases h with \u27e8f, hf\u27e9 ** case intro R : Type u_1 inst\u271d\u2078 : Ring R M : Type u_2 inst\u271d\u2077 : TopologicalSpace M inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M\u2082 : Type u_3 inst\u271d\u2074 : TopologicalSpace M\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : TopologicalAddGroup M inst\u271d : T1Space M p : Submodule R M f : M \u2192L[R] { x // x \u2208 p } hf : \u2200 (x : { x // x \u2208 p }), \u2191f \u2191x = x \u22a2 IsClosed \u2191p ** have : ker (id R M - p.subtypeL.comp f) = p := LinearMap.ker_id_sub_eq_of_proj hf ** case intro R : Type u_1 inst\u271d\u2078 : Ring R M : Type u_2 inst\u271d\u2077 : TopologicalSpace M inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M\u2082 : Type u_3 inst\u271d\u2074 : TopologicalSpace M\u2082 inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : TopologicalAddGroup M inst\u271d : T1Space M p : Submodule R M f : M \u2192L[R] { x // x \u2208 p } hf : \u2200 (x : { x // x \u2208 p }), \u2191f \u2191x = x this : LinearMap.ker (ContinuousLinearMap.id R M - comp (subtypeL p) f) = p \u22a2 IsClosed \u2191p ** exact this \u25b8 isClosed_ker _ ** Qed", + "informal": "" + }, + { + "formal": "sdiff_symmDiff_right ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03c0 : \u03b9 \u2192 Type u_4 inst\u271d : GeneralizedBooleanAlgebra \u03b1 a b c d : \u03b1 \u22a2 b \\ a \u2206 b = a \u2293 b ** rw [symmDiff_comm, inf_comm, sdiff_symmDiff_left] ** Qed", + "informal": "" + }, + { + "formal": "Real.tan_lt_tan_of_nonneg_of_lt_pi_div_two ** x y : \u211d hx\u2081 : 0 \u2264 x hy\u2082 : y < \u03c0 / 2 hxy : x < y \u22a2 tan x < tan y ** rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos] ** x y : \u211d hx\u2081 : 0 \u2264 x hy\u2082 : y < \u03c0 / 2 hxy : x < y \u22a2 sin x / cos x < sin y / cos y ** exact\n div_lt_div (sin_lt_sin_of_lt_of_le_pi_div_two (by linarith) (le_of_lt hy\u2082) hxy)\n (cos_le_cos_of_nonneg_of_le_pi hx\u2081 (by linarith) (le_of_lt hxy))\n (sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith))\n (cos_pos_of_mem_Ioo \u27e8by linarith, hy\u2082\u27e9) ** x y : \u211d hx\u2081 : 0 \u2264 x hy\u2082 : y < \u03c0 / 2 hxy : x < y \u22a2 -(\u03c0 / 2) \u2264 x ** linarith ** x y : \u211d hx\u2081 : 0 \u2264 x hy\u2082 : y < \u03c0 / 2 hxy : x < y \u22a2 y \u2264 \u03c0 ** linarith ** x y : \u211d hx\u2081 : 0 \u2264 x hy\u2082 : y < \u03c0 / 2 hxy : x < y \u22a2 0 \u2264 y ** linarith ** x y : \u211d hx\u2081 : 0 \u2264 x hy\u2082 : y < \u03c0 / 2 hxy : x < y \u22a2 -(\u03c0 / 2) < y ** linarith ** Qed", + "informal": "" + }, + { + "formal": "Array.get?_set_eq ** \u03b1 : Type u_1 a : Array \u03b1 i : Fin (size a) v : \u03b1 \u22a2 (set a i v)[i.val]? = some v ** simp [getElem?_pos, i.2] ** Qed", + "informal": "" + }, + { + "formal": "Nat.pow_length_le_mul_ofDigits ** n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 \u22a2 (b + 2) ^ List.length l \u2264 (b + 2) * ofDigits (b + 2) l ** rw [\u2190 List.dropLast_append_getLast hl] ** n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 \u22a2 (b + 2) ^ List.length (List.dropLast l ++ [List.getLast l hl]) \u2264 (b + 2) * ofDigits (b + 2) (List.dropLast l ++ [List.getLast l hl]) ** simp only [List.length_append, List.length, zero_add, List.length_dropLast, ofDigits_append,\n List.length_dropLast, ofDigits_singleton, add_comm (l.length - 1), pow_add, pow_one] ** n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 \u22a2 (b + 2) * (b + 2) ^ (List.length l - 1) \u2264 (b + 2) * (ofDigits (b + 2) (List.dropLast l) + (b + 2) ^ (List.length l - 1) * List.getLast l hl) ** apply Nat.mul_le_mul_left ** case h n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 \u22a2 (b + 2) ^ (List.length l - 1) \u2264 ofDigits (b + 2) (List.dropLast l) + (b + 2) ^ (List.length l - 1) * List.getLast l hl ** refine' le_trans _ (Nat.le_add_left _ _) ** case h n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 \u22a2 (b + 2) ^ (List.length l - 1) \u2264 (b + 2) ^ (List.length l - 1) * List.getLast l hl ** have : 0 < l.getLast hl := by rwa [pos_iff_ne_zero] ** case h n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 this : 0 < List.getLast l hl \u22a2 (b + 2) ^ (List.length l - 1) \u2264 (b + 2) ^ (List.length l - 1) * List.getLast l hl ** convert Nat.mul_le_mul_left ((b + 2) ^ (l.length - 1)) this using 1 ** case h.e'_3 n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 this : 0 < List.getLast l hl \u22a2 (b + 2) ^ (List.length l - 1) = (b + 2) ^ (List.length l - 1) * succ 0 ** rw [Nat.mul_one] ** n b : \u2115 l : List \u2115 hl : l \u2260 [] hl2 : List.getLast l hl \u2260 0 \u22a2 0 < List.getLast l hl ** rwa [pos_iff_ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "Set.zero_mem_smul_set_iff ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : Zero \u03b1 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2 s : Set \u03b1 t : Set \u03b2 inst\u271d : NoZeroSMulDivisors \u03b1 \u03b2 a : \u03b1 ha : a \u2260 0 \u22a2 0 \u2208 a \u2022 t \u2194 0 \u2208 t ** refine' \u27e8_, zero_mem_smul_set\u27e9 ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : Zero \u03b1 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2 s : Set \u03b1 t : Set \u03b2 inst\u271d : NoZeroSMulDivisors \u03b1 \u03b2 a : \u03b1 ha : a \u2260 0 \u22a2 0 \u2208 a \u2022 t \u2192 0 \u2208 t ** rintro \u27e8b, hb, h\u27e9 ** case intro.intro F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 inst\u271d\u00b3 : Zero \u03b1 inst\u271d\u00b2 : Zero \u03b2 inst\u271d\u00b9 : SMulWithZero \u03b1 \u03b2 s : Set \u03b1 t : Set \u03b2 inst\u271d : NoZeroSMulDivisors \u03b1 \u03b2 a : \u03b1 ha : a \u2260 0 b : \u03b2 hb : b \u2208 t h : (fun x => a \u2022 x) b = 0 \u22a2 0 \u2208 t ** rwa [(eq_zero_or_eq_zero_of_smul_eq_zero h).resolve_left ha] at hb ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.X_pow_sub_X_sub_one_irreducible ** n : \u2115 hn1 : n \u2260 1 \u22a2 Irreducible (X ^ n - X - 1) ** by_cases hn0 : n = 0 ** case neg n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 \u22a2 Irreducible (X ^ n - X - 1) ** have hn : 1 < n := Nat.one_lt_iff_ne_zero_and_ne_one.mpr \u27e8hn0, hn1\u27e9 ** case neg n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n \u22a2 Irreducible (X ^ n - X - 1) ** have hp : (X ^ n - X - 1 : \u2124[X]) = trinomial 0 1 n (-1) (-1) 1 := by\n simp only [trinomial, C_neg, C_1]; ring ** case neg n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 \u22a2 Irreducible (X ^ n - X - 1) ** rw [hp] ** case neg n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 \u22a2 Irreducible (trinomial 0 1 n (-1) (-1) 1) ** apply IsUnitTrinomial.irreducible_of_coprime' \u27e80, 1, n, zero_lt_one, hn, -1, -1, 1, rfl\u27e9 ** case neg n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 \u22a2 \u2200 (z : \u2102), \u00ac(\u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 \u2227 \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0) ** rintro z \u27e8h1, h2\u27e9 ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 False ** apply X_pow_sub_X_sub_one_irreducible_aux z ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 z ^ ?m.20827 = z + 1 \u2227 z ^ ?m.20827 + z ^ 2 = 0 n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 \u2115 ** rw [trinomial_mirror zero_lt_one hn (-1 : \u2124\u02e3).ne_zero (1 : \u2124\u02e3).ne_zero] at h2 ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h2 : \u2191(aeval z) (trinomial 0 (n - 1 + 0) n \u21911 \u2191(-1) \u2191(-1)) = 0 \u22a2 z ^ ?m.20827 = z + 1 \u2227 z ^ ?m.20827 + z ^ 2 = 0 n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 \u2115 ** simp_rw [trinomial, aeval_add, aeval_mul, aeval_X_pow, aeval_C,\n Units.val_neg, Units.val_one, map_neg, map_one] at h1 h2 ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h1 : -1 * z ^ 0 + -1 * z ^ 1 + 1 * z ^ n = 0 h2 : 1 * z ^ 0 + -1 * z ^ (n - 1 + 0) + -1 * z ^ n = 0 \u22a2 z ^ ?m.20827 = z + 1 \u2227 z ^ ?m.20827 + z ^ 2 = 0 n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 \u2115 ** replace h1 : z ^ n = z + 1 := by linear_combination h1 ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h2 : 1 * z ^ 0 + -1 * z ^ (n - 1 + 0) + -1 * z ^ n = 0 h1 : z ^ n = z + 1 \u22a2 z ^ ?m.20827 = z + 1 \u2227 z ^ ?m.20827 + z ^ 2 = 0 n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 \u2115 ** replace h2 := mul_eq_zero_of_left h2 z ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h1 : z ^ n = z + 1 h2 : (1 * z ^ 0 + -1 * z ^ (n - 1 + 0) + -1 * z ^ n) * z = 0 \u22a2 z ^ ?m.20827 = z + 1 \u2227 z ^ ?m.20827 + z ^ 2 = 0 n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 \u2115 ** rw [add_mul, add_mul, add_zero, mul_assoc (-1 : \u2102), \u2190 pow_succ', Nat.sub_add_cancel hn.le] at h2 ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h1 : z ^ n = z + 1 h2 : 1 * z ^ 0 * z + -1 * z ^ n + -1 * z ^ n * z = 0 \u22a2 z ^ ?m.20827 = z + 1 \u2227 z ^ ?m.20827 + z ^ 2 = 0 n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1 : \u2191(aeval z) (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911) = 0 h2 : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 \u22a2 \u2115 ** rw [h1] at h2 \u22a2 ** case neg.intro n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h1 : z ^ n = z + 1 h2 : 1 * z ^ 0 * z + -1 * (z + 1) + -1 * (z + 1) * z = 0 \u22a2 z + 1 = z + 1 \u2227 z + 1 + z ^ 2 = 0 ** exact \u27e8rfl, by linear_combination -h2\u27e9 ** case pos n : \u2115 hn1 : n \u2260 1 hn0 : n = 0 \u22a2 Irreducible (X ^ n - X - 1) ** rw [hn0, pow_zero, sub_sub, add_comm, \u2190 sub_sub, sub_self, zero_sub] ** case pos n : \u2115 hn1 : n \u2260 1 hn0 : n = 0 \u22a2 Irreducible (-X) ** exact Associated.irreducible \u27e8-1, mul_neg_one X\u27e9 irreducible_X ** n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n \u22a2 X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 ** simp only [trinomial, C_neg, C_1] ** n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n \u22a2 X ^ n - X - 1 = -1 * X ^ 0 + -1 * X ^ 1 + 1 * X ^ n ** ring ** n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h1 : -1 * z ^ 0 + -1 * z ^ 1 + 1 * z ^ n = 0 h2 : 1 * z ^ 0 + -1 * z ^ (n - 1 + 0) + -1 * z ^ n = 0 \u22a2 z ^ n = z + 1 ** linear_combination h1 ** n : \u2115 hn1 : n \u2260 1 hn0 : \u00acn = 0 hn : 1 < n hp : X ^ n - X - 1 = trinomial 0 1 n (-1) (-1) 1 z : \u2102 h1\u271d : \u2191(aeval z) (\u2191C \u2191(-1) * X ^ 0 + \u2191C \u2191(-1) * X ^ 1 + \u2191C \u21911 * X ^ n) = 0 h2\u271d : \u2191(aeval z) (mirror (trinomial 0 1 n \u2191(-1) \u2191(-1) \u21911)) = 0 h1 : z ^ n = z + 1 h2 : 1 * z ^ 0 * z + -1 * (z + 1) + -1 * (z + 1) * z = 0 \u22a2 z + 1 + z ^ 2 = 0 ** linear_combination -h2 ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.sub_mem_ker_iff ** R : Type u_1 R\u2081 : Type u_2 R\u2082 : Type u_3 R\u2083 : Type u_4 R\u2084 : Type u_5 S : Type u_6 K : Type u_7 K\u2082 : Type u_8 M : Type u_9 M' : Type u_10 M\u2081 : Type u_11 M\u2082 : Type u_12 M\u2083 : Type u_13 M\u2084 : Type u_14 N : Type u_15 N\u2082 : Type u_16 \u03b9 : Type u_17 V : Type u_18 V\u2082 : Type u_19 inst\u271d\u2079 : Ring R inst\u271d\u2078 : Ring R\u2082 inst\u271d\u2077 : Ring R\u2083 inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : AddCommGroup M\u2082 inst\u271d\u2074 : AddCommGroup M\u2083 inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Module R\u2082 M\u2082 inst\u271d\u00b9 : Module R\u2083 M\u2083 \u03c4\u2081\u2082 : R \u2192+* R\u2082 \u03c4\u2082\u2083 : R\u2082 \u2192+* R\u2083 \u03c4\u2081\u2083 : R \u2192+* R\u2083 inst\u271d : RingHomCompTriple \u03c4\u2081\u2082 \u03c4\u2082\u2083 \u03c4\u2081\u2083 F : Type u_20 sc : SemilinearMapClass F \u03c4\u2081\u2082 M M\u2082 f : F x y : M \u22a2 x - y \u2208 ker f \u2194 \u2191f x = \u2191f y ** rw [mem_ker, map_sub, sub_eq_zero] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.ProjIsoSpecTopComponent.ToSpec.preimage_eq ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f \u22a2 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) = {x | \u2191x \u2208 (pbo f) \u2293 pbo a} ** ext1 y ** case h R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace \u22a2 y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) \u2194 y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a} ** constructor <;> intro hy ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) \u22a2 y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a} ** refine' \u27e8y.2, _\u27e9 ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) \u22a2 \u2191y \u2208 \u2191(pbo a) ** rw [Set.mem_preimage, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] at hy ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal \u22a2 \u2191y \u2208 \u2191(pbo a) ** rw [ProjectiveSpectrum.mem_coe_basicOpen] ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal \u22a2 \u00aca \u2208 (\u2191y).asHomogeneousIdeal ** intro a_mem_y ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal \u22a2 False ** apply hy ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal \u22a2 Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal ** rw [toFun, mem_carrier_iff, HomogeneousLocalization.val_mk'', Subtype.coe_mk] ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal \u22a2 Localization.mk a { val := \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den, property := (_ : \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den \u2208 Submonoid.powers f) } \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) ** dsimp ** case h.mp R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 (toFun f y).asIdeal a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal \u22a2 Localization.mk a { val := b, property := b_mem2 } \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) ** rcases b_mem2 with \u27e8k, hk\u27e9 ** case h.mp.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k\u271d : \u2115 a_mem : a \u2208 \ud835\udc9c k\u271d b_mem1 : b \u2208 \ud835\udc9c k\u271d y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal k : \u2115 hk : (fun x x_1 => x ^ x_1) f k = b hy : \u00acQuotient.mk'' { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 (toFun f y).asIdeal \u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) ** dsimp at hk ** case h.mp.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k\u271d : \u2115 a_mem : a \u2208 \ud835\udc9c k\u271d b_mem1 : b \u2208 \ud835\udc9c k\u271d y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal k : \u2115 hk : f ^ k = b hy : \u00acQuotient.mk'' { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 (toFun f y).asIdeal \u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) ** simp only [show (mk a \u27e8b, \u27e8k, hk\u27e9\u27e9 : Away f) =\n Localization.mk 1 (\u27e8f ^ k, \u27e8_, rfl\u27e9\u27e9 : Submonoid.powers f) * mk a 1 by\n rw [mk_mul, one_mul, mul_one]; congr; rw [hk]] ** case h.mp.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k\u271d : \u2115 a_mem : a \u2208 \ud835\udc9c k\u271d b_mem1 : b \u2208 \ud835\udc9c k\u271d y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal k : \u2115 hk : f ^ k = b hy : \u00acQuotient.mk'' { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 (toFun f y).asIdeal \u22a2 Localization.mk 1 { val := f ^ k, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ k) } * Localization.mk a 1 \u2208 span (\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) ** exact Ideal.mul_mem_left _ _ (Ideal.subset_span \u27e8_, a_mem_y, rfl\u27e9) ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k\u271d : \u2115 a_mem : a \u2208 \ud835\udc9c k\u271d b_mem1 : b \u2208 \ud835\udc9c k\u271d y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal k : \u2115 hk : f ^ k = b hy : \u00acQuotient.mk'' { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 (toFun f y).asIdeal \u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } = Localization.mk 1 { val := f ^ k, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ k) } * Localization.mk a 1 ** rw [mk_mul, one_mul, mul_one] ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k\u271d : \u2115 a_mem : a \u2208 \ud835\udc9c k\u271d b_mem1 : b \u2208 \ud835\udc9c k\u271d y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal k : \u2115 hk : f ^ k = b hy : \u00acQuotient.mk'' { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 (toFun f y).asIdeal \u22a2 Localization.mk a { val := b, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } = Localization.mk a { val := f ^ k, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = f ^ k) } ** congr ** case e_y.e_val R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k\u271d : \u2115 a_mem : a \u2208 \ud835\udc9c k\u271d b_mem1 : b \u2208 \ud835\udc9c k\u271d y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace a_mem_y : a \u2208 (\u2191y).asHomogeneousIdeal k : \u2115 hk : f ^ k = b hy : \u00acQuotient.mk'' { deg := k\u271d, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = b) } \u2208 (toFun f y).asIdeal \u22a2 b = f ^ k ** rw [hk] ** case h.mpr R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : y \u2208 {x | \u2191x \u2208 (pbo f) \u2293 pbo a} \u22a2 y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) ** change y.1 \u2208 ProjectiveSpectrum.basicOpen \ud835\udc9c f \u2293 ProjectiveSpectrum.basicOpen \ud835\udc9c a at hy ** case h.mpr R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy : \u2191y \u2208 (pbo f) \u2293 pbo a \u22a2 y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) ** rcases hy with \u27e8hy1, hy2\u27e9 ** case h.mpr.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u2191y \u2208 \u2191(pbo f) hy2 : \u2191y \u2208 \u2191(pbo a) \u22a2 y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) ** rw [ProjectiveSpectrum.mem_coe_basicOpen] at hy1 hy2 ** case h.mpr.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal \u22a2 y \u2208 toFun f \u207b\u00b9' \u2191(sbo Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) ** rw [Set.mem_preimage, toFun, SetLike.mem_coe, PrimeSpectrum.mem_basicOpen] ** case h.mpr.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal \u22a2 \u00acQuotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 { asIdeal := carrier y, IsPrime := (_ : IsPrime (carrier y)) }.asIdeal ** intro rid ** case h.mpr.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 { asIdeal := carrier y, IsPrime := (_ : IsPrime (carrier y)) }.asIdeal \u22a2 False ** dsimp at rid ** case h.mpr.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y \u22a2 False ** rcases MemCarrier.clear_denominator _ rid with \u27e8c, N, acd, eq1\u27e9 ** case h.mpr.intro.intro.intro.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A eq1 : f ^ N \u2022 HomogeneousLocalization.val (Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) = \u2191(algebraMap A (Away f)) (\u2211 i in attach c.support, acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)) \u22a2 False ** rw [Algebra.smul_def] at eq1 ** case h.mpr.intro.intro.intro.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A eq1 : \u2191(algebraMap A (Localization (Submonoid.powers f))) (f ^ N) * HomogeneousLocalization.val (Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }) = \u2191(algebraMap A (Away f)) (\u2211 i in attach c.support, acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal)) \u22a2 False ** change Localization.mk (f ^ N) 1 * Localization.mk _ _ = Localization.mk _ _ at eq1 ** case h.mpr.intro.intro.intro.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A eq1 : Localization.mk (f ^ N) 1 * Localization.mk \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.num { val := \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den, property := (_ : \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den \u2208 Submonoid.powers f) } = Localization.mk (\u2191(algebraMap A A) (\u2211 i in attach c.support, acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) 1 \u22a2 False ** rw [mk_mul, one_mul, mk_eq_mk', IsLocalization.eq] at eq1 ** case h.mpr.intro.intro.intro.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A eq1 : \u2203 c_1, \u2191c_1 * (\u21911 * (f ^ N * \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.num)) = \u2191c_1 * (\u2191{ val := \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den, property := (_ : \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den \u2208 Submonoid.powers f) } * \u2191(algebraMap A A) (\u2211 i in attach c.support, acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u22a2 False ** rcases eq1 with \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9 ** case h.mpr.intro.intro.intro.intro.intro.mk.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : \u2191{ val := (fun x x_1 => x ^ x_1) f M, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } * (\u21911 * (f ^ N * \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.num)) = \u2191{ val := (fun x x_1 => x ^ x_1) f M, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } * (\u2191{ val := \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den, property := (_ : \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den \u2208 Submonoid.powers f) } * \u2191(algebraMap A A) (\u2211 i in attach c.support, acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u22a2 False ** rw [Submonoid.coe_one, one_mul] at eq1 ** case h.mpr.intro.intro.intro.intro.intro.mk.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : \u2191{ val := (fun x x_1 => x ^ x_1) f M, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } * (f ^ N * \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.num) = \u2191{ val := (fun x x_1 => x ^ x_1) f M, property := (_ : \u2203 y, (fun x x_1 => x ^ x_1) f y = (fun x x_1 => x ^ x_1) f M) } * (\u2191{ val := \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den, property := (_ : \u2191{ deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 }.den \u2208 Submonoid.powers f) } * \u2191(algebraMap A A) (\u2211 i in attach c.support, acd (\u2191c \u2191i) (_ : \u2191c \u2191i \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u22a2 False ** simp only [Subtype.coe_mk] at eq1 ** case h.mpr.intro.intro.intro.intro.intro.mk.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u22a2 False ** have : a * f ^ N * f ^ M \u2208 y.val.asHomogeneousIdeal.toIdeal := by\n rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1]\n refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _))\n generalize_proofs h\u2081 h\u2082; exact (Classical.choose_spec h\u2082).1 ** case h.mpr.intro.intro.intro.intro.intro.mk.intro R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) this : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal \u22a2 False ** rcases y.1.isPrime.mem_or_mem this with (H1 | H3) ** case h.mpr.intro.intro.intro.intro.intro.mk.intro.inl R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) this : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H1 : a * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal \u22a2 False case h.mpr.intro.intro.intro.intro.intro.mk.intro.inr R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) this : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H3 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal \u22a2 False ** rcases y.1.isPrime.mem_or_mem H1 with (H1 | H2) ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u22a2 a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal ** rw [mul_comm _ (f ^ N), mul_comm _ (f ^ M), eq1] ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u22a2 f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal ** refine' mul_mem_left _ _ (mul_mem_left _ _ (sum_mem _ fun i _ => mul_mem_left _ _ _)) ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) i : { x // x \u2208 c.support } x\u271d : i \u2208 attach c.support \u22a2 Exists.choose (_ : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal ** generalize_proofs h\u2081 h\u2082 ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) i : { x // x \u2208 c.support } x\u271d : i \u2208 attach c.support h\u2081 : AddSubmonoidClass (Submodule R A) A h\u2082 : \u2191\u2191i \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal \u22a2 Exists.choose h\u2082 \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal ** exact (Classical.choose_spec h\u2082).1 ** case h.mpr.intro.intro.intro.intro.intro.mk.intro.inl.inl R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) this : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H1\u271d : a * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H1 : a \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal \u22a2 False ** exact hy2 H1 ** case h.mpr.intro.intro.intro.intro.intro.mk.intro.inl.inr R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) this : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H1 : a * f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H2 : f ^ N \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal \u22a2 False ** exact y.2 (y.1.isPrime.mem_of_pow_mem N H2) ** case h.mpr.intro.intro.intro.intro.intro.mk.intro.inr R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommRing R inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : Algebra R A \ud835\udc9c : \u2115 \u2192 Submodule R A inst\u271d : GradedAlgebra \ud835\udc9c f : A m : \u2115 f_deg : f \u2208 \ud835\udc9c m x : \u2191(LocallyRingedSpace.toTopCat (LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f))))) a b : A k : \u2115 a_mem : a \u2208 \ud835\udc9c k b_mem1 : b \u2208 \ud835\udc9c k b_mem2 : b \u2208 Submonoid.powers f y : \u2191\u2191(LocallyRingedSpace.restrict Proj (_ : OpenEmbedding \u2191(Opens.inclusion (pbo f)))).toSheafedSpace.toPresheafedSpace hy1 : \u00acf \u2208 (\u2191y).asHomogeneousIdeal hy2 : \u00aca \u2208 (\u2191y).asHomogeneousIdeal rid : Quotient.mk'' { deg := k, num := { val := a, property := a_mem }, den := { val := b, property := b_mem1 }, den_mem := b_mem2 } \u2208 carrier y c : \u2191(\u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal) \u2192\u2080 Away f N : \u2115 acd : (y_1 : Away f) \u2192 y_1 \u2208 image (\u2191c) c.support \u2192 A M : \u2115 eq1 : f ^ M * (f ^ N * a) = f ^ M * (b * \u2191(algebraMap A A) (\u2211 x in attach c.support, acd (\u2191c \u2191x) (_ : \u2191c \u2191x \u2208 image (\u2191c) c.support) * Exists.choose (_ : \u2191\u2191x \u2208 \u2191(algebraMap A (Away f)) '' \u2191(\u2191y).asHomogeneousIdeal))) this : a * f ^ N * f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal H3 : f ^ M \u2208 HomogeneousIdeal.toIdeal (\u2191y).asHomogeneousIdeal \u22a2 False ** exact y.2 (y.1.isPrime.mem_of_pow_mem M H3) ** Qed", + "informal": "" + }, + { + "formal": "DirectSum.decompose_lhom_ext ** \u03b9 : Type u_1 R : Type u_2 M : Type u_3 \u03c3 : Type u_4 inst\u271d\u2076 : DecidableEq \u03b9 inst\u271d\u2075 : Semiring R inst\u271d\u2074 : AddCommMonoid M inst\u271d\u00b3 : Module R M \u2133 : \u03b9 \u2192 Submodule R M inst\u271d\u00b2 : Decomposition \u2133 N : Type u_5 inst\u271d\u00b9 : AddCommMonoid N inst\u271d : Module R N f g : M \u2192\u2097[R] N h : \u2200 (i : \u03b9), LinearMap.comp f (Submodule.subtype (\u2133 i)) = LinearMap.comp g (Submodule.subtype (\u2133 i)) i : \u03b9 \u22a2 LinearMap.comp (LinearMap.comp f \u2191(LinearEquiv.symm (decomposeLinearEquiv \u2133))) (lof R \u03b9 (fun i => { x // x \u2208 \u2133 i }) i) = LinearMap.comp (LinearMap.comp g \u2191(LinearEquiv.symm (decomposeLinearEquiv \u2133))) (lof R \u03b9 (fun i => { x // x \u2208 \u2133 i }) i) ** simp_rw [LinearMap.comp_assoc, decomposeLinearEquiv_symm_comp_lof \u2133 i, h] ** Qed", + "informal": "" + }, + { + "formal": "Set.iUnion_setOf ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 P : \u03b9 \u2192 \u03b1 \u2192 Prop \u22a2 \u22c3 i, {x | P i x} = {x | \u2203 i, P i x} ** ext ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Sort u_4 \u03b9' : Sort u_5 \u03b9\u2082 : Sort u_6 \u03ba : \u03b9 \u2192 Sort u_7 \u03ba\u2081 : \u03b9 \u2192 Sort u_8 \u03ba\u2082 : \u03b9 \u2192 Sort u_9 \u03ba' : \u03b9' \u2192 Sort u_10 P : \u03b9 \u2192 \u03b1 \u2192 Prop x\u271d : \u03b1 \u22a2 x\u271d \u2208 \u22c3 i, {x | P i x} \u2194 x\u271d \u2208 {x | \u2203 i, P i x} ** exact mem_iUnion ** Qed", + "informal": "" + }, + { + "formal": "Finset.filter_eq_empty_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 p q : \u03b1 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q s : Finset \u03b1 \u22a2 filter p s = \u2205 \u2194 \u2200 \u2983x : \u03b1\u2984, x \u2208 s \u2192 \u00acp x ** simp [Finset.ext_iff] ** Qed", + "informal": "" + }, + { + "formal": "continuous_sigma_map ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 \u03c3 : \u03b9 \u2192 Type u_7 \u03c4 : \u03ba \u2192 Type u_8 inst\u271d\u00b2 : (i : \u03b9) \u2192 TopologicalSpace (\u03c3 i) inst\u271d\u00b9 : (k : \u03ba) \u2192 TopologicalSpace (\u03c4 k) inst\u271d : TopologicalSpace \u03b1 f\u2081 : \u03b9 \u2192 \u03ba f\u2082 : (i : \u03b9) \u2192 \u03c3 i \u2192 \u03c4 (f\u2081 i) \u22a2 (\u2200 (i : \u03b9), Continuous fun a => Sigma.map f\u2081 f\u2082 { fst := i, snd := a }) \u2194 \u2200 (i : \u03b9), Continuous (f\u2082 i) ** simp only [Sigma.map, embedding_sigmaMk.continuous_iff, comp] ** Qed", + "informal": "" + }, + { + "formal": "Ring.inverse_pow ** R : Type u_1 S : Type u_2 M : Type u_3 inst\u271d : MonoidWithZero M r : M \u22a2 inverse r ^ 0 = inverse (r ^ 0) ** rw [pow_zero, pow_zero, Ring.inverse_one] ** R : Type u_1 S : Type u_2 M : Type u_3 inst\u271d : MonoidWithZero M r : M n : \u2115 \u22a2 inverse r ^ (n + 1) = inverse (r ^ (n + 1)) ** rw [pow_succ, pow_succ', Ring.mul_inverse_rev' ((Commute.refl r).pow_left n),\n Ring.inverse_pow r n] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.prod_rightUnitor_inv_naturality ** C : Type u inst\u271d\u00b2 : Category.{v, u} C X Y : C inst\u271d\u00b9 : HasTerminal C inst\u271d : HasBinaryProducts C f : X \u27f6 Y \u22a2 (prod.rightUnitor X).inv \u226b prod.map f (\ud835\udfd9 (\u22a4_ C)) = f \u226b (prod.rightUnitor Y).inv ** rw [Iso.inv_comp_eq, \u2190 Category.assoc, Iso.eq_comp_inv, prod.rightUnitor_hom_naturality] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp ** C : Type u_1 inst\u271d : Category.{?u.36087, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X \u22a2 (Opens.map f.base).obj (op ((openFunctor H).obj U)).unop = U ** ext ** case h.h C : Type u_1 inst\u271d : Category.{?u.36087, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X x\u271d : \u2191\u2191X \u22a2 x\u271d \u2208 \u2191((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop) \u2194 x\u271d \u2208 \u2191U ** dsimp [openFunctor, IsOpenMap.functor] ** case h.h C : Type u_1 inst\u271d : Category.{?u.36087, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X x\u271d : \u2191\u2191X \u22a2 x\u271d \u2208 \u2191f.base \u207b\u00b9' (\u2191f.base '' \u2191U) \u2194 x\u271d \u2208 \u2191U ** rw [Set.preimage_image_eq _ H.base_open.inj] ** C : Type u_1 inst\u271d : Category.{u_3, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X \u22a2 inv (invApp H U) = f.c.app (op ((openFunctor H).obj U)) \u226b X.presheaf.map (eqToHom (_ : op ((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop) = op U)) ** rw [\u2190 cancel_epi (H.invApp U), IsIso.hom_inv_id] ** C : Type u_1 inst\u271d : Category.{u_3, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X \u22a2 \ud835\udfd9 (X.presheaf.obj (op U)) = invApp H U \u226b f.c.app (op ((openFunctor H).obj U)) \u226b X.presheaf.map (eqToHom (_ : op ((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop) = op U)) ** delta invApp ** C : Type u_1 inst\u271d : Category.{u_3, u_1} C X Y : PresheafedSpace C f : X \u27f6 Y H : IsOpenImmersion f U : Opens \u2191\u2191X \u22a2 \ud835\udfd9 (X.presheaf.obj (op U)) = (X.presheaf.map (eqToHom (_ : op U = op ((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop))) \u226b inv (f.c.app (op ((openFunctor H).obj U)))) \u226b f.c.app (op ((openFunctor H).obj U)) \u226b X.presheaf.map (eqToHom (_ : op ((Opens.map f.base).obj (op ((openFunctor H).obj U)).unop) = op U)) ** simp [\u2190 Functor.map_comp] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.derivative_nat_cast_mul ** R : Type u S : Type v T : Type w \u03b9 : Type y A : Type z a b : R n\u271d : \u2115 inst\u271d : Semiring R n : \u2115 f : R[X] \u22a2 \u2191derivative (\u2191n * f) = \u2191n * \u2191derivative f ** simp ** Qed", + "informal": "" + }, + { + "formal": "OrthonormalBasis.orthonormal_adjustToOrientation ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \u211d E \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 ne : Nonempty \u03b9 e f : OrthonormalBasis \u03b9 \u211d E x : Orientation \u211d E \u03b9 \u22a2 Orthonormal \u211d \u2191(Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) ** apply e.orthonormal.orthonormal_of_forall_eq_or_eq_neg ** E : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \u211d E \u03b9 : Type u_2 inst\u271d\u00b9 : Fintype \u03b9 inst\u271d : DecidableEq \u03b9 ne : Nonempty \u03b9 e f : OrthonormalBasis \u03b9 \u211d E x : Orientation \u211d E \u03b9 \u22a2 \u2200 (i : \u03b9), \u2191(Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) i = \u2191e i \u2228 \u2191(Basis.adjustToOrientation (OrthonormalBasis.toBasis e) x) i = -\u2191e i ** simpa using e.toBasis.adjustToOrientation_apply_eq_or_eq_neg x ** Qed", + "informal": "" + }, + { + "formal": "Multiset.sum_map_singleton ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 s : Multiset \u03b1 \u22a2 sum (map (fun a => {a}) 0) = 0 ** simp ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 s : Multiset \u03b1 \u22a2 \u2200 \u2983a : \u03b1\u2984 {s : Multiset \u03b1}, sum (map (fun a => {a}) s) = s \u2192 sum (map (fun a => {a}) (a ::\u2098 s)) = a ::\u2098 s ** simp ** Qed", + "informal": "" + }, + { + "formal": "LieModule.mem_ker ** R : Type u L : Type v M : Type w N : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup N inst\u271d\u00b2 : Module R N inst\u271d\u00b9 : LieRingModule L N inst\u271d : LieModule R L N x : L \u22a2 x \u2208 LieModule.ker R L M \u2194 \u2200 (m : M), \u2045x, m\u2046 = 0 ** simp only [LieModule.ker, LieHom.mem_ker, LinearMap.ext_iff, LinearMap.zero_apply,\n toEndomorphism_apply_apply] ** Qed", + "informal": "" + }, + { + "formal": "Padic.norm_eq_pow_val ** p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] \u22a2 x \u2260 0 \u2192 \u2016x\u2016 = \u2191p ^ (-valuation x) ** refine Quotient.inductionOn' x fun f hf => ?_ ** p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 \u22a2 \u2016Quotient.mk'' f\u2016 = \u2191p ^ (-valuation (Quotient.mk'' f)) ** change (PadicSeq.norm _ : \u211d) = (p : \u211d) ^ (-PadicSeq.valuation _) ** p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 \u22a2 \u2191(PadicSeq.norm f) = \u2191p ^ (-PadicSeq.valuation f) ** rw [PadicSeq.norm_eq_pow_val] ** p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 \u22a2 \u2191(\u2191p ^ (-PadicSeq.valuation f)) = \u2191p ^ (-PadicSeq.valuation f) ** rw [Rat.cast_zpow, Rat.cast_coe_nat] ** p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 \u22a2 \u00acf \u2248 0 ** apply CauSeq.not_limZero_of_not_congr_zero ** case hf p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 \u22a2 \u00acf - 0 \u2248 0 ** intro hf' ** case hf p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 hf' : f - 0 \u2248 0 \u22a2 False ** apply hf ** case hf p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 hf' : f - 0 \u2248 0 \u22a2 Quotient.mk'' f = 0 ** apply Quotient.sound ** case hf.a p : \u2115 hp : Fact (Nat.Prime p) x : \u211a_[p] f : CauSeq \u211a (padicNorm p) hf : Quotient.mk'' f \u2260 0 hf' : f - 0 \u2248 0 \u22a2 f \u2248 const (padicNorm p) 0 ** simpa using hf' ** Qed", + "informal": "" + }, + { + "formal": "String.data_drop ** s : String n : Nat \u22a2 (drop s n).data = List.drop n s.data ** rw [drop_eq] ** Qed", + "informal": "" + }, + { + "formal": "Nat.cast_add ** R : Type u_1 inst\u271d : AddMonoidWithOne R m n : \u2115 \u22a2 \u2191(m + n) = \u2191m + \u2191n ** induction n <;> simp [add_succ, add_assoc, Nat.add_zero, Nat.cast_one, Nat.cast_zero, *] ** Qed", + "informal": "" + }, + { + "formal": "Submodule.span_pow ** \u03b9 : Sort u\u03b9 R : Type u inst\u271d\u00b2 : CommSemiring R A : Type v inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A S T : Set A M N P Q : Submodule R A m n : A s : Set A \u22a2 span R s ^ 0 = span R (s ^ 0) ** rw [pow_zero, pow_zero, one_eq_span_one_set] ** \u03b9 : Sort u\u03b9 R : Type u inst\u271d\u00b2 : CommSemiring R A : Type v inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R A S T : Set A M N P Q : Submodule R A m n\u271d : A s : Set A n : \u2115 \u22a2 span R s ^ (n + 1) = span R (s ^ (n + 1)) ** rw [pow_succ, pow_succ, span_pow s n, span_mul_span] ** Qed", + "informal": "" + }, + { + "formal": "intervalIntegral.integral_deriv_comp_mul_deriv' ** \u03b9 : Type u_1 \ud835\udd5c : Type u_2 E : Type u_3 F : Type u_4 A : Type u_5 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : CompleteSpace E inst\u271d : NormedSpace \u211d E f\u271d\u00b9 : \u211d \u2192 E g'\u271d g\u271d \u03c6 : \u211d \u2192 \u211d f\u271d f'\u271d : \u211d \u2192 E a b : \u211d f f' g g' : \u211d \u2192 \u211d hf : ContinuousOn f [[a, b]] hff' : \u2200 (x : \u211d), x \u2208 Ioo (min a b) (max a b) \u2192 HasDerivWithinAt f (f' x) (Ioi x) x hf' : ContinuousOn f' [[a, b]] hg : ContinuousOn g [[f a, f b]] hgg' : \u2200 (x : \u211d), x \u2208 Ioo (min (f a) (f b)) (max (f a) (f b)) \u2192 HasDerivWithinAt g (g' x) (Ioi x) x hg' : ContinuousOn g' (f '' [[a, b]]) \u22a2 \u222b (x : \u211d) in a..b, (g' \u2218 f) x * f' x = (g \u2218 f) b - (g \u2218 f) a ** simpa [mul_comm] using integral_deriv_comp_smul_deriv' hf hff' hf' hg hgg' hg' ** Qed", + "informal": "" + }, + { + "formal": "DiscreteValuationRing.eq_unit_mul_pow_irreducible ** R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : DiscreteValuationRing R x : R hx : x \u2260 0 \u03d6 : R hirr : Irreducible \u03d6 \u22a2 \u2203 n u, x = \u2191u * \u03d6 ^ n ** obtain \u27e8n, hn\u27e9 := associated_pow_irreducible hx hirr ** case intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : DiscreteValuationRing R x : R hx : x \u2260 0 \u03d6 : R hirr : Irreducible \u03d6 n : \u2115 hn : Associated x (\u03d6 ^ n) \u22a2 \u2203 n u, x = \u2191u * \u03d6 ^ n ** obtain \u27e8u, rfl\u27e9 := hn.symm ** case intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : DiscreteValuationRing R \u03d6 : R hirr : Irreducible \u03d6 n : \u2115 u : R\u02e3 hx : \u03d6 ^ n * \u2191u \u2260 0 hn : Associated (\u03d6 ^ n * \u2191u) (\u03d6 ^ n) \u22a2 \u2203 n_1 u_1, \u03d6 ^ n * \u2191u = \u2191u_1 * \u03d6 ^ n_1 ** use n, u ** case h R : Type u_1 inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : IsDomain R inst\u271d : DiscreteValuationRing R \u03d6 : R hirr : Irreducible \u03d6 n : \u2115 u : R\u02e3 hx : \u03d6 ^ n * \u2191u \u2260 0 hn : Associated (\u03d6 ^ n * \u2191u) (\u03d6 ^ n) \u22a2 \u03d6 ^ n * \u2191u = \u2191u * \u03d6 ^ n ** apply mul_comm ** Qed", + "informal": "" + }, + { + "formal": "Complex.exp_ne_zero ** x y : \u2102 h : cexp x = 0 \u22a2 0 = 1 ** rw [\u2190 exp_zero, \u2190 add_neg_self x, exp_add, h] ** x y : \u2102 h : cexp x = 0 \u22a2 x + -x = 0 * cexp (-x) ** simp ** Qed", + "informal": "" + }, + { + "formal": "LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_pi ** \u03b1 : Type u_1 inst\u271d\u2075 : LinearOrder \u03b1 E : Type u_2 inst\u271d\u2074 : PseudoEMetricSpace E V : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : NormedSpace \u211d V inst\u271d\u00b9 : FiniteDimensional \u211d V \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 f : \u211d \u2192 \u03b9 \u2192 \u211d s : Set \u211d h : LocallyBoundedVariationOn f s \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x ** have A : \u2200 i : \u03b9, LipschitzWith 1 fun x : \u03b9 \u2192 \u211d => x i := fun i => LipschitzWith.eval i ** \u03b1 : Type u_1 inst\u271d\u2075 : LinearOrder \u03b1 E : Type u_2 inst\u271d\u2074 : PseudoEMetricSpace E V : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : NormedSpace \u211d V inst\u271d\u00b9 : FiniteDimensional \u211d V \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 f : \u211d \u2192 \u03b9 \u2192 \u211d s : Set \u211d h : LocallyBoundedVariationOn f s A : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x ** have : \u2200 i : \u03b9, \u2200\u1d50 x, x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x : \u211d => f x i) s x := fun i \u21a6 by\n apply ae_differentiableWithinAt_of_mem_real\n exact LipschitzWith.comp_locallyBoundedVariationOn (A i) h ** \u03b1 : Type u_1 inst\u271d\u2075 : LinearOrder \u03b1 E : Type u_2 inst\u271d\u2074 : PseudoEMetricSpace E V : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : NormedSpace \u211d V inst\u271d\u00b9 : FiniteDimensional \u211d V \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 f : \u211d \u2192 \u03b9 \u2192 \u211d s : Set \u211d h : LocallyBoundedVariationOn f s A : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i this : \u2200 (i : \u03b9), \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d f s x ** filter_upwards [ae_all_iff.2 this] with x hx xs ** case h \u03b1 : Type u_1 inst\u271d\u2075 : LinearOrder \u03b1 E : Type u_2 inst\u271d\u2074 : PseudoEMetricSpace E V : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : NormedSpace \u211d V inst\u271d\u00b9 : FiniteDimensional \u211d V \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 f : \u211d \u2192 \u03b9 \u2192 \u211d s : Set \u211d h : LocallyBoundedVariationOn f s A : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i this : \u2200 (i : \u03b9), \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x x : \u211d hx : \u2200 (i : \u03b9), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x xs : x \u2208 s \u22a2 DifferentiableWithinAt \u211d f s x ** exact differentiableWithinAt_pi.2 fun i => hx i xs ** \u03b1 : Type u_1 inst\u271d\u2075 : LinearOrder \u03b1 E : Type u_2 inst\u271d\u2074 : PseudoEMetricSpace E V : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : NormedSpace \u211d V inst\u271d\u00b9 : FiniteDimensional \u211d V \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 f : \u211d \u2192 \u03b9 \u2192 \u211d s : Set \u211d h : LocallyBoundedVariationOn f s A : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i i : \u03b9 \u22a2 \u2200\u1d50 (x : \u211d), x \u2208 s \u2192 DifferentiableWithinAt \u211d (fun x => f x i) s x ** apply ae_differentiableWithinAt_of_mem_real ** case h \u03b1 : Type u_1 inst\u271d\u2075 : LinearOrder \u03b1 E : Type u_2 inst\u271d\u2074 : PseudoEMetricSpace E V : Type u_3 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : NormedSpace \u211d V inst\u271d\u00b9 : FiniteDimensional \u211d V \u03b9 : Type u_4 inst\u271d : Fintype \u03b9 f : \u211d \u2192 \u03b9 \u2192 \u211d s : Set \u211d h : LocallyBoundedVariationOn f s A : \u2200 (i : \u03b9), LipschitzWith 1 fun x => x i i : \u03b9 \u22a2 LocallyBoundedVariationOn (fun x => f x i) s ** exact LipschitzWith.comp_locallyBoundedVariationOn (A i) h ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.Cofork.coequalizer_ext ** C : Type u inst\u271d : Category.{v, u} C X Y : C f g : X \u27f6 Y s : Cofork f g W : C k l : s.pt \u27f6 W h : \u03c0 s \u226b k = \u03c0 s \u226b l \u22a2 s.\u03b9.app zero \u226b k = s.\u03b9.app zero \u226b l ** simp only [s.app_zero_eq_comp_\u03c0_left, Category.assoc, h] ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.mk_bool ** \u03b1 \u03b2 : Type u \u22a2 #Bool = 2 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Path.delayReflRight_one ** X : Type u inst\u271d : TopologicalSpace X x y : X \u03b3 : Path x y \u22a2 delayReflRight 1 \u03b3 = \u03b3 ** ext t ** case a.h X : Type u inst\u271d : TopologicalSpace X x y : X \u03b3 : Path x y t : \u2191I \u22a2 \u2191(delayReflRight 1 \u03b3) t = \u2191\u03b3 t ** exact congr_arg \u03b3 (qRight_one_right t) ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.zero_powerlt ** \u03b1 \u03b2 : Type u a : Cardinal.{u_1} h : a \u2260 0 \u22a2 0 ^< a = 1 ** apply (powerlt_le.2 fun c _ => zero_power_le _).antisymm ** \u03b1 \u03b2 : Type u a : Cardinal.{u_1} h : a \u2260 0 \u22a2 1 \u2264 0 ^< a ** rw [\u2190 power_zero] ** \u03b1 \u03b2 : Type u a : Cardinal.{u_1} h : a \u2260 0 \u22a2 ?m.858249 ^ 0 \u2264 0 ^< a \u03b1 \u03b2 : Type u a : Cardinal.{u_1} h : a \u2260 0 \u22a2 Cardinal.{u_1} ** exact le_powerlt 0 (pos_iff_ne_zero.2 h) ** Qed", + "informal": "" + }, + { + "formal": "Complex.norm_max_aux\u2083 ** E : Type u inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E F : Type v inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u2102 F f : \u2102 \u2192 F z w : \u2102 r : \u211d hr : dist w z = r hd : DiffContOnCl \u2102 f (ball z r) hz : IsMaxOn (norm \u2218 f) (ball z r) z \u22a2 \u2016f w\u2016 = \u2016f z\u2016 ** subst r ** E : Type u inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E F : Type v inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u2102 F f : \u2102 \u2192 F z w : \u2102 hd : DiffContOnCl \u2102 f (ball z (dist w z)) hz : IsMaxOn (norm \u2218 f) (ball z (dist w z)) z \u22a2 \u2016f w\u2016 = \u2016f z\u2016 ** rcases eq_or_ne w z with (rfl | hne) ** case inr E : Type u inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E F : Type v inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u2102 F f : \u2102 \u2192 F z w : \u2102 hd : DiffContOnCl \u2102 f (ball z (dist w z)) hz : IsMaxOn (norm \u2218 f) (ball z (dist w z)) z hne : w \u2260 z \u22a2 \u2016f w\u2016 = \u2016f z\u2016 ** rw [\u2190 dist_ne_zero] at hne ** case inr E : Type u inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E F : Type v inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u2102 F f : \u2102 \u2192 F z w : \u2102 hd : DiffContOnCl \u2102 f (ball z (dist w z)) hz : IsMaxOn (norm \u2218 f) (ball z (dist w z)) z hne\u271d : w \u2260 z hne : dist w z \u2260 0 \u22a2 \u2016f w\u2016 = \u2016f z\u2016 ** exact norm_max_aux\u2082 hd (closure_ball z hne \u25b8 hz.closure hd.continuousOn.norm) ** case inl E : Type u inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u2102 E F : Type v inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u2102 F f : \u2102 \u2192 F w : \u2102 hd : DiffContOnCl \u2102 f (ball w (dist w w)) hz : IsMaxOn (norm \u2218 f) (ball w (dist w w)) w \u22a2 \u2016f w\u2016 = \u2016f w\u2016 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.preserves_desc_mapCocone ** C : Type u\u2081 inst\u271d\u00b3 : Category.{v\u2081, u\u2081} C D : Type u\u2082 inst\u271d\u00b2 : Category.{v\u2082, u\u2082} D G : C \u2964 D J : Type w inst\u271d\u00b9 : Category.{w', w} J F : J \u2964 C inst\u271d : PreservesColimit F G c\u2081 c\u2082 : Cocone F t : IsColimit c\u2081 \u22a2 \u2200 (j : J), (G.mapCocone c\u2081).\u03b9.app j \u226b G.map (IsColimit.desc t c\u2082) = (G.mapCocone c\u2082).\u03b9.app j ** simp [\u2190 G.map_comp] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_iSup' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x \u22a2 \u222b\u207b (a : \u03b1), \u2a06 n, f n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** simp_rw [\u2190 iSup_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** let p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun _ f' => Monotone f' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** have hp : \u2200\u1d50 x \u2202\u03bc, p x fun i => f i x := h_mono ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_mono : Monotone (aeSeq hf p) \u22a2 \u222b\u207b (a : \u03b1), iSup (fun i => f i) a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rw [lintegral_congr_ae (aeSeq.iSup hf hp).symm] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_mono : Monotone (aeSeq hf p) \u22a2 \u222b\u207b (a : \u03b1), iSup (fun n => aeSeq hf (fun x => p x) n) a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** simp_rw [iSup_apply] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_mono : Monotone (aeSeq hf p) \u22a2 \u222b\u207b (a : \u03b1), \u2a06 i, aeSeq hf (fun x f' => Monotone f') i a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** rw [@lintegral_iSup _ _ \u03bc _ (aeSeq.measurable hf p) h_ae_seq_mono] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_mono : Monotone (aeSeq hf p) \u22a2 \u2a06 n, \u222b\u207b (a : \u03b1), aeSeq hf p n a \u2202\u03bc = \u2a06 n, \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** congr ** case e_s \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x h_ae_seq_mono : Monotone (aeSeq hf p) \u22a2 (fun n => \u222b\u207b (a : \u03b1), aeSeq hf p n a \u2202\u03bc) = fun n => \u222b\u207b (a : \u03b1), f n a \u2202\u03bc ** exact funext fun n => lintegral_congr_ae (aeSeq.aeSeq_n_eq_fun_n_ae hf hp n) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x \u22a2 Monotone (aeSeq hf p) ** intro n m hnm x ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x n m : \u2115 hnm : n \u2264 m x : \u03b1 \u22a2 aeSeq hf p n x \u2264 aeSeq hf p m x ** by_cases hx : x \u2208 aeSeqSet hf p ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x n m : \u2115 hnm : n \u2264 m x : \u03b1 hx : x \u2208 aeSeqSet hf p \u22a2 aeSeq hf p n x \u2264 aeSeq hf p m x ** exact aeSeq.prop_of_mem_aeSeqSet hf hx hnm ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x n m : \u2115 hnm : n \u2264 m x : \u03b1 hx : \u00acx \u2208 aeSeqSet hf p \u22a2 aeSeq hf p n x \u2264 aeSeq hf p m x ** simp only [aeSeq, hx, if_false] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m\u271d : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e hf : \u2200 (n : \u2115), AEMeasurable (f n) h_mono : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, Monotone fun n => f n x p : \u03b1 \u2192 (\u2115 \u2192 \u211d\u22650\u221e) \u2192 Prop := fun x f' => Monotone f' hp : \u2200\u1d50 (x : \u03b1) \u2202\u03bc, p x fun i => f i x n m : \u2115 hnm : n \u2264 m x : \u03b1 hx : \u00acx \u2208 aeSeqSet hf p \u22a2 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) \u2264 Nonempty.some (_ : Nonempty \u211d\u22650\u221e) ** exact le_rfl ** Qed", + "informal": "" + }, + { + "formal": "Set.biUnion_diff_biUnion_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f \u22a2 (\u22c3 i \u2208 s, f i) \\ \u22c3 i \u2208 t, f i = \u22c3 i \u2208 s \\ t, f i ** refine'\n (biUnion_diff_biUnion_subset f s t).antisymm\n (iUnion\u2082_subset fun i hi a ha => (mem_diff _).2 \u27e8mem_biUnion hi.1 ha, _\u27e9) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f i : \u03b9 hi : i \u2208 s \\ t a : \u03b1 ha : a \u2208 f i \u22a2 \u00aca \u2208 \u22c3 x \u2208 t, f x ** rw [mem_iUnion\u2082] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f i : \u03b9 hi : i \u2208 s \\ t a : \u03b1 ha : a \u2208 f i \u22a2 \u00ac\u2203 i j, a \u2208 f i ** rintro \u27e8j, hj, haj\u27e9 ** case intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 \u03b9' : Type u_5 \u03ba : Sort u_6 r p q : \u03b1 \u2192 \u03b1 \u2192 Prop s t : Set \u03b9 f : \u03b9 \u2192 Set \u03b1 h : PairwiseDisjoint (s \u222a t) f i : \u03b9 hi : i \u2208 s \\ t a : \u03b1 ha : a \u2208 f i j : \u03b9 hj : j \u2208 t haj : a \u2208 f j \u22a2 False ** exact (h (Or.inl hi.1) (Or.inr hj) (ne_of_mem_of_not_mem hj hi.2).symm).le_bot \u27e8ha, haj\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.X_pow_mul_monomial ** R : Type u a b : R m n\u271d : \u2115 inst\u271d : Semiring R p q : R[X] k n : \u2115 r : R \u22a2 X ^ k * \u2191(monomial n) r = \u2191(monomial (n + k)) r ** rw [X_pow_mul, monomial_mul_X_pow] ** Qed", + "informal": "" + }, + { + "formal": "StrictConvexOn.congr ** \ud835\udd5c : Type u_1 E : Type u_2 \u03b2 : Type u_3 s : Set E f g : E \u2192 \u03b2 inst\u271d\u2074 : OrderedSemiring \ud835\udd5c inst\u271d\u00b3 : SMul \ud835\udd5c E inst\u271d\u00b2 : AddCommMonoid E inst\u271d\u00b9 : OrderedAddCommMonoid \u03b2 inst\u271d : SMul \ud835\udd5c \u03b2 hf : StrictConvexOn \ud835\udd5c s f hfg : EqOn f g s x : E hx : x \u2208 s y : E hy : y \u2208 s hxy : x \u2260 y a b : \ud835\udd5c ha : 0 < a hb : 0 < b hab : a + b = 1 \u22a2 g (a \u2022 x + b \u2022 y) < a \u2022 g x + b \u2022 g y ** simpa only [\u2190 hfg hx, \u2190 hfg hy, \u2190 hfg (hf.1 hx hy ha.le hb.le hab)] using\n hf.2 hx hy hxy ha hb hab ** Qed", + "informal": "" + }, + { + "formal": "norm_pow_le_mul_norm ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b9 : SeminormedCommGroup E inst\u271d : SeminormedCommGroup F a\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d n : \u2115 a : E \u22a2 \u2016a ^ n\u2016 \u2264 \u2191n * \u2016a\u2016 ** induction' n with n ih ** case succ \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b9 : SeminormedCommGroup E inst\u271d : SeminormedCommGroup F a\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d a : E n : \u2115 ih : \u2016a ^ n\u2016 \u2264 \u2191n * \u2016a\u2016 \u22a2 \u2016a ^ Nat.succ n\u2016 \u2264 \u2191(Nat.succ n) * \u2016a\u2016 ** simpa only [pow_succ', Nat.cast_succ, add_mul, one_mul] using norm_mul_le_of_le ih le_rfl ** case zero \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b9 : SeminormedCommGroup E inst\u271d : SeminormedCommGroup F a\u271d a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d a : E \u22a2 \u2016a ^ Nat.zero\u2016 \u2264 \u2191Nat.zero * \u2016a\u2016 ** simp ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.embDomain_mapRange ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d\u00b9 : Zero M inst\u271d : Zero N f : \u03b1 \u21aa \u03b2 g : M \u2192 N p : \u03b1 \u2192\u2080 M hg : g 0 = 0 \u22a2 embDomain f (mapRange g hg p) = mapRange g hg (embDomain f p) ** ext a ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d\u00b9 : Zero M inst\u271d : Zero N f : \u03b1 \u21aa \u03b2 g : M \u2192 N p : \u03b1 \u2192\u2080 M hg : g 0 = 0 a : \u03b2 \u22a2 \u2191(embDomain f (mapRange g hg p)) a = \u2191(mapRange g hg (embDomain f p)) a ** by_cases h : a \u2208 Set.range f ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d\u00b9 : Zero M inst\u271d : Zero N f : \u03b1 \u21aa \u03b2 g : M \u2192 N p : \u03b1 \u2192\u2080 M hg : g 0 = 0 a : \u03b2 h : a \u2208 Set.range \u2191f \u22a2 \u2191(embDomain f (mapRange g hg p)) a = \u2191(mapRange g hg (embDomain f p)) a ** rcases h with \u27e8a', rfl\u27e9 ** case pos.intro \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d\u00b9 : Zero M inst\u271d : Zero N f : \u03b1 \u21aa \u03b2 g : M \u2192 N p : \u03b1 \u2192\u2080 M hg : g 0 = 0 a' : \u03b1 \u22a2 \u2191(embDomain f (mapRange g hg p)) (\u2191f a') = \u2191(mapRange g hg (embDomain f p)) (\u2191f a') ** rw [mapRange_apply, embDomain_apply, embDomain_apply, mapRange_apply] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 M : Type u_5 M' : Type u_6 N : Type u_7 P : Type u_8 G : Type u_9 H : Type u_10 R : Type u_11 S : Type u_12 inst\u271d\u00b9 : Zero M inst\u271d : Zero N f : \u03b1 \u21aa \u03b2 g : M \u2192 N p : \u03b1 \u2192\u2080 M hg : g 0 = 0 a : \u03b2 h : \u00aca \u2208 Set.range \u2191f \u22a2 \u2191(embDomain f (mapRange g hg p)) a = \u2191(mapRange g hg (embDomain f p)) a ** rw [mapRange_apply, embDomain_notin_range, embDomain_notin_range, \u2190 hg] <;> assumption ** Qed", + "informal": "" + }, + { + "formal": "PMF.toMeasure_eq_iff_eq_toPMF ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u2074 : Countable \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSingletonClass \u03b1 p : PMF \u03b1 \u03bc\u271d : Measure \u03b1 inst\u271d\u00b9 : IsProbabilityMeasure \u03bc\u271d \u03bc : Measure \u03b1 inst\u271d : IsProbabilityMeasure \u03bc \u22a2 toMeasure p = \u03bc \u2194 p = Measure.toPMF \u03bc ** rw [\u2190 toMeasure_inj, Measure.toPMF_toMeasure] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Walk.copy_nil ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' u u' : V hu : u = u' \u22a2 Walk.copy nil hu hu = nil ** subst_vars ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' u' : V \u22a2 Walk.copy nil (_ : u' = u') (_ : u' = u') = nil ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.JordanDecomposition.eq_of_posPart_eq_posPart ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081 = j\u2082 ** ext1 ** case posPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081.posPart = j\u2082.posPart ** exact hj ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 j\u2081.negPart = j\u2082.negPart ** rw [\u2190 toSignedMeasure_eq_toSignedMeasure_iff] ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : toSignedMeasure j\u2081 = toSignedMeasure j\u2082 \u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart ** unfold toSignedMeasure at hj' ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : Measure.toSignedMeasure j\u2081.posPart - Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.posPart - Measure.toSignedMeasure j\u2082.negPart \u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart ** simp_rw [hj, sub_right_inj] at hj' ** case negPart \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : MeasurableSpace \u03b1 j\u2081 j\u2082 : JordanDecomposition \u03b1 hj : j\u2081.posPart = j\u2082.posPart hj' : Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart \u22a2 Measure.toSignedMeasure j\u2081.negPart = Measure.toSignedMeasure j\u2082.negPart ** exact hj' ** Qed", + "informal": "" + }, + { + "formal": "Set.range_diag ** \u03b1 : Type u_1 s t : Set \u03b1 \u22a2 (range fun x => (x, x)) = diagonal \u03b1 ** ext \u27e8x, y\u27e9 ** case h.mk \u03b1 : Type u_1 s t : Set \u03b1 x y : \u03b1 \u22a2 ((x, y) \u2208 range fun x => (x, x)) \u2194 (x, y) \u2208 diagonal \u03b1 ** simp [diagonal, eq_comm] ** Qed", + "informal": "" + }, + { + "formal": "Nat.filter_range_nth_subset_insert ** p : \u2115 \u2192 Prop inst\u271d : DecidablePred p k : \u2115 \u22a2 filter p (range (nth p (k + 1))) \u2286 insert (nth p k) (filter p (range (nth p k))) ** intro a ha ** p : \u2115 \u2192 Prop inst\u271d : DecidablePred p k a : \u2115 ha : a \u2208 filter p (range (nth p (k + 1))) \u22a2 a \u2208 insert (nth p k) (filter p (range (nth p k))) ** simp only [mem_insert, mem_filter, mem_range] at ha \u22a2 ** p : \u2115 \u2192 Prop inst\u271d : DecidablePred p k a : \u2115 ha : a < nth p (k + 1) \u2227 p a \u22a2 a = nth p k \u2228 a < nth p k \u2227 p a ** exact (le_nth_of_lt_nth_succ ha.1 ha.2).eq_or_lt.imp_right fun h => \u27e8h, ha.2\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Submodule.mem_smul_top_iff ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N\u271d P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' N : Submodule R M x : { x // x \u2208 N } \u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191x \u2208 I \u2022 N ** change _ \u2194 N.subtype x \u2208 I \u2022 N ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N\u271d P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' N : Submodule R M x : { x // x \u2208 N } \u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191(Submodule.subtype N) x \u2208 I \u2022 N ** have : Submodule.map N.subtype (I \u2022 \u22a4) = I \u2022 N := by\n rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N\u271d P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' N : Submodule R M x : { x // x \u2208 N } this : map (Submodule.subtype N) (I \u2022 \u22a4) = I \u2022 N \u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191(Submodule.subtype N) x \u2208 I \u2022 N ** rw [\u2190 this] ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N\u271d P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' N : Submodule R M x : { x // x \u2208 N } this : map (Submodule.subtype N) (I \u2022 \u22a4) = I \u2022 N \u22a2 x \u2208 I \u2022 \u22a4 \u2194 \u2191(Submodule.subtype N) x \u2208 map (Submodule.subtype N) (I \u2022 \u22a4) ** exact (Function.Injective.mem_set_image N.injective_subtype).symm ** R : Type u M : Type v F : Type u_1 G : Type u_2 inst\u271d\u2074 : CommSemiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M I J : Ideal R N\u271d P : Submodule R M S : Set R T : Set M M' : Type w inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' N : Submodule R M x : { x // x \u2208 N } \u22a2 map (Submodule.subtype N) (I \u2022 \u22a4) = I \u2022 N ** rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] ** Qed", + "informal": "" + }, + { + "formal": "WittVector.ext ** p : \u2115 R : Type u_1 x y : \ud835\udd4e R h : \u2200 (n : \u2115), coeff x n = coeff y n \u22a2 x = y ** cases x ** case mk' p : \u2115 R : Type u_1 y : \ud835\udd4e R coeff\u271d : \u2115 \u2192 R h : \u2200 (n : \u2115), coeff { coeff := coeff\u271d } n = coeff y n \u22a2 { coeff := coeff\u271d } = y ** cases y ** case mk'.mk' p : \u2115 R : Type u_1 coeff\u271d\u00b9 coeff\u271d : \u2115 \u2192 R h : \u2200 (n : \u2115), coeff { coeff := coeff\u271d\u00b9 } n = coeff { coeff := coeff\u271d } n \u22a2 { coeff := coeff\u271d\u00b9 } = { coeff := coeff\u271d } ** simp only at h ** case mk'.mk' p : \u2115 R : Type u_1 coeff\u271d\u00b9 coeff\u271d : \u2115 \u2192 R h : \u2200 (n : \u2115), coeff\u271d\u00b9 n = coeff\u271d n \u22a2 { coeff := coeff\u271d\u00b9 } = { coeff := coeff\u271d } ** simp [Function.funext_iff, h] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.ContinuousMap.inner_toLp ** \u03b1 : Type u_1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : MeasureSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 \ud835\udd5c : Type u_2 inst\u271d\u00b2 : IsROrC \ud835\udd5c \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : CompactSpace \u03b1 f g : C(\u03b1, \ud835\udd5c) \u22a2 inner (\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) (\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) = \u222b (x : \u03b1), \u2191(starRingEnd ((fun x => \ud835\udd5c) x)) (\u2191f x) * \u2191g x \u2202\u03bc ** apply integral_congr_ae ** case h \u03b1 : Type u_1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : MeasureSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 \ud835\udd5c : Type u_2 inst\u271d\u00b2 : IsROrC \ud835\udd5c \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : CompactSpace \u03b1 f g : C(\u03b1, \ud835\udd5c) \u22a2 (fun a => inner (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) a) (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) a)) =\u1d50[\u03bc] fun a => \u2191(starRingEnd ((fun x => \ud835\udd5c) a)) (\u2191f a) * \u2191g a ** have hf_ae := f.coeFn_toLp (p := 2) (\ud835\udd5c := \ud835\udd5c) \u03bc ** case h \u03b1 : Type u_1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : MeasureSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 \ud835\udd5c : Type u_2 inst\u271d\u00b2 : IsROrC \ud835\udd5c \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : CompactSpace \u03b1 f g : C(\u03b1, \ud835\udd5c) hf_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191f \u22a2 (fun a => inner (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) a) (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) a)) =\u1d50[\u03bc] fun a => \u2191(starRingEnd ((fun x => \ud835\udd5c) a)) (\u2191f a) * \u2191g a ** have hg_ae := g.coeFn_toLp (p := 2) (\ud835\udd5c := \ud835\udd5c) \u03bc ** case h \u03b1 : Type u_1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : MeasureSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 \ud835\udd5c : Type u_2 inst\u271d\u00b2 : IsROrC \ud835\udd5c \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : CompactSpace \u03b1 f g : C(\u03b1, \ud835\udd5c) hf_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191f hg_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) =\u1d50[\u03bc] \u2191g \u22a2 (fun a => inner (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) a) (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) a)) =\u1d50[\u03bc] fun a => \u2191(starRingEnd ((fun x => \ud835\udd5c) a)) (\u2191f a) * \u2191g a ** filter_upwards [hf_ae, hg_ae] with _ hf hg ** case h \u03b1 : Type u_1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : MeasureSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 \ud835\udd5c : Type u_2 inst\u271d\u00b2 : IsROrC \ud835\udd5c \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : CompactSpace \u03b1 f g : C(\u03b1, \ud835\udd5c) hf_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191f hg_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) =\u1d50[\u03bc] \u2191g a\u271d : \u03b1 hf : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) a\u271d = \u2191f a\u271d hg : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) a\u271d = \u2191g a\u271d \u22a2 inner (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) a\u271d) (\u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) a\u271d) = \u2191(starRingEnd \ud835\udd5c) (\u2191f a\u271d) * \u2191g a\u271d ** rw [hf, hg] ** case h \u03b1 : Type u_1 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : MeasureSpace \u03b1 inst\u271d\u00b3 : BorelSpace \u03b1 \ud835\udd5c : Type u_2 inst\u271d\u00b2 : IsROrC \ud835\udd5c \u03bc : Measure \u03b1 inst\u271d\u00b9 : IsFiniteMeasure \u03bc inst\u271d : CompactSpace \u03b1 f g : C(\u03b1, \ud835\udd5c) hf_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) =\u1d50[\u03bc] \u2191f hg_ae : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) =\u1d50[\u03bc] \u2191g a\u271d : \u03b1 hf : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) f) a\u271d = \u2191f a\u271d hg : \u2191\u2191(\u2191(ContinuousMap.toLp 2 \u03bc \ud835\udd5c) g) a\u271d = \u2191g a\u271d \u22a2 inner (\u2191f a\u271d) (\u2191g a\u271d) = \u2191(starRingEnd \ud835\udd5c) (\u2191f a\u271d) * \u2191g a\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "Subsemiring.mem_map ** R : Type u S : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S inst\u271d : NonAssocSemiring T s\u271d : Subsemiring R f : R \u2192+* S s : Subsemiring R y : S \u22a2 y \u2208 map f s \u2194 \u2203 x, x \u2208 s \u2227 \u2191f x = y ** convert Set.mem_image_iff_bex (f := f) (s := s.carrier) (y := y) using 1 ** case h.e'_2.a R : Type u S : Type v T : Type w inst\u271d\u00b2 : NonAssocSemiring R M : Submonoid R inst\u271d\u00b9 : NonAssocSemiring S inst\u271d : NonAssocSemiring T s\u271d : Subsemiring R f : R \u2192+* S s : Subsemiring R y : S \u22a2 (\u2203 x, x \u2208 s \u2227 \u2191f x = y) \u2194 \u2203 x x_1, \u2191f x = y ** simp ** Qed", + "informal": "" + }, + { + "formal": "ONote.split_add_lt ** o e : ONote n : \u2115+ a : ONote m : \u2115 inst\u271d : NF o h : split o = (oadd e n a, m) \u22a2 repr a + \u2191m < \u03c9 ^ repr e ** cases' nf_repr_split h with h\u2081 h\u2082 ** case intro o e : ONote n : \u2115+ a : ONote m : \u2115 inst\u271d : NF o h : split o = (oadd e n a, m) h\u2081 : NF (oadd e n a) h\u2082 : repr o = repr (oadd e n a) + \u2191m \u22a2 repr a + \u2191m < \u03c9 ^ repr e ** cases' h\u2081.of_dvd_omega (split_dvd h) with e0 d ** case intro.intro o e : ONote n : \u2115+ a : ONote m : \u2115 inst\u271d : NF o h : split o = (oadd e n a, m) h\u2081 : NF (oadd e n a) h\u2082 : repr o = repr (oadd e n a) + \u2191m e0 : repr e \u2260 0 d : \u03c9 \u2223 repr a \u22a2 repr a + \u2191m < \u03c9 ^ repr e ** apply principal_add_omega_opow _ h\u2081.snd'.repr_lt (lt_of_lt_of_le (nat_lt_omega _) _) ** o e : ONote n : \u2115+ a : ONote m : \u2115 inst\u271d : NF o h : split o = (oadd e n a, m) h\u2081 : NF (oadd e n a) h\u2082 : repr o = repr (oadd e n a) + \u2191m e0 : repr e \u2260 0 d : \u03c9 \u2223 repr a \u22a2 \u03c9 \u2264 \u03c9 ^ repr e ** simpa using opow_le_opow_right omega_pos (one_le_iff_ne_zero.2 e0) ** Qed", + "informal": "" + }, + { + "formal": "rank_punit ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u22a2 Module.rank R PUnit.{u_2 + 1} = 0 ** rw [\u2190 bot_eq_zero, \u2190 le_bot_iff, Module.rank_def] ** K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u22a2 \u2a06 \u03b9, #\u2191\u2191\u03b9 \u2264 \u22a5 ** apply ciSup_le' ** case h K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R \u22a2 \u2200 (i : { s // LinearIndependent R Subtype.val }), #\u2191\u2191i \u2264 \u22a5 ** rintro \u27e8s, li\u27e9 ** case h.mk K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R s : Set PUnit.{u_2 + 1} li : LinearIndependent R Subtype.val \u22a2 #\u2191\u2191{ val := s, property := li } \u2264 \u22a5 ** rw [le_bot_iff, bot_eq_zero, Cardinal.mk_emptyCollection_iff, Subtype.coe_mk] ** case h.mk K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R s : Set PUnit.{u_2 + 1} li : LinearIndependent R Subtype.val \u22a2 s = \u2205 ** by_contra h ** case h.mk K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R s : Set PUnit.{u_2 + 1} li : LinearIndependent R Subtype.val h : \u00acs = \u2205 \u22a2 False ** obtain \u27e8a, ha\u27e9 := nonempty_iff_ne_empty.2 h ** case h.mk.intro K : Type u V V\u2081 V\u2082 V\u2083 : Type v V' V'\u2081 : Type v' V'' : Type v'' \u03b9 : Type w \u03b9' : Type w' \u03b7 : Type u\u2081' \u03c6 : \u03b7 \u2192 Type u_1 R : Type u inst\u271d\u2077 : Ring R M : Type v inst\u271d\u2076 : AddCommGroup M inst\u271d\u2075 : Module R M M' : Type v' inst\u271d\u2074 : AddCommGroup M' inst\u271d\u00b3 : Module R M' M\u2081 : Type v inst\u271d\u00b2 : AddCommGroup M\u2081 inst\u271d\u00b9 : Module R M\u2081 inst\u271d : Nontrivial R s : Set PUnit.{u_2 + 1} li : LinearIndependent R Subtype.val h : \u00acs = \u2205 a : PUnit.{u_2 + 1} ha : a \u2208 s \u22a2 False ** simpa using LinearIndependent.ne_zero (\u27e8a, ha\u27e9 : s) li ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.cardinal_mk_eq_max_lift ** \u03c3 : Type u R : Type v inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : Nonempty \u03c3 inst\u271d : Nontrivial R \u22a2 max (lift.{v, u} #(\u03c3 \u2192\u2080 \u2115)) (lift.{u, v} #R) = max (max (lift.{u, v} #R) (lift.{v, u} #\u03c3)) \u2135\u2080 ** rw [mk_finsupp_nat, max_assoc, lift_max, lift_aleph0, max_comm] ** Qed", + "informal": "" + }, + { + "formal": "wbtw_comm ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u2076 : OrderedRing R inst\u271d\u2075 : AddCommGroup V inst\u271d\u2074 : Module R V inst\u271d\u00b3 : AddTorsor V P inst\u271d\u00b2 : AddCommGroup V' inst\u271d\u00b9 : Module R V' inst\u271d : AddTorsor V' P' x y z : P \u22a2 Wbtw R x y z \u2194 Wbtw R z y x ** rw [Wbtw, Wbtw, affineSegment_comm] ** Qed", + "informal": "" + }, + { + "formal": "RatFunc.num_eq_zero_iff ** K : Type u inst\u271d : Field K x : RatFunc K h : num x = 0 \u22a2 x = 0 ** rw [\u2190 num_div_denom x, h, RingHom.map_zero, zero_div] ** Qed", + "informal": "" + }, + { + "formal": "card_perms_of_finset ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 \u22a2 \u2200 (s : Finset \u03b1), card (permsOfFinset s) = (card s)! ** rintro \u27e8\u27e8l\u27e9, hs\u27e9 ** case mk.mk \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : DecidableEq \u03b2 val\u271d : Multiset \u03b1 l : List \u03b1 hs : Multiset.Nodup (Quot.mk Setoid.r l) \u22a2 card (permsOfFinset { val := Quot.mk Setoid.r l, nodup := hs }) = (card { val := Quot.mk Setoid.r l, nodup := hs })! ** exact length_permsOfList l ** Qed", + "informal": "" + }, + { + "formal": "Nat.odd_mod_four_iff ** m n\u271d a b c d n : \u2115 \u22a2 \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3 ** decide ** m n\u271d a b c d n : \u2115 help : \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3 hn : n % 2 = 1 \u22a2 4 > 0 ** norm_num ** m n\u271d a b c d n : \u2115 help : \u2200 (m : \u2115), m < 4 \u2192 m % 2 = 1 \u2192 m = 1 \u2228 m = 3 hn : n % 2 = 1 \u22a2 2 \u2223 4 ** norm_num ** Qed", + "informal": "" + }, + { + "formal": "Sylow.smul_eq_iff_mem_normalizer ** p : \u2115 G : Type u_1 inst\u271d : Group G g : G P : Sylow p G \u22a2 g \u2022 P = P \u2194 g \u2208 normalizer \u2191P ** rw [eq_comm, SetLike.ext_iff, \u2190 inv_mem_iff (G := G) (H := normalizer P.toSubgroup),\n mem_normalizer_iff, inv_inv] ** p : \u2115 G : Type u_1 inst\u271d : Group G g : G P : Sylow p G \u22a2 (\u2200 (x : G), x \u2208 P \u2194 x \u2208 g \u2022 P) \u2194 \u2200 (h : G), h \u2208 \u2191P \u2194 g\u207b\u00b9 * h * g \u2208 \u2191P ** exact\n forall_congr' fun h =>\n iff_congr Iff.rfl\n \u27e8fun \u27e8a, b, c\u27e9 => c \u25b8 by simpa [mul_assoc] using b,\n fun hh => \u27e8(MulAut.conj g)\u207b\u00b9 h, hh, MulAut.apply_inv_self G (MulAut.conj g) h\u27e9\u27e9 ** p : \u2115 G : Type u_1 inst\u271d : Group G g : G P : Sylow p G h : G x\u271d : h \u2208 g \u2022 P a : G b : a \u2208 \u2191\u2191P c : \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) a = h \u22a2 g\u207b\u00b9 * \u2191(\u2191(MulDistribMulAction.toMonoidEnd (MulAut G) G) (\u2191MulAut.conj g)) a * g \u2208 \u2191P ** simpa [mul_assoc] using b ** Qed", + "informal": "" + }, + { + "formal": "NNReal.tendsto_pow_atTop_nhds_0_of_lt_1 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 r : \u211d\u22650 hr : r < 1 \u22a2 Tendsto (fun a => \u2191(r ^ a)) atTop (\ud835\udcdd \u21910) ** simp only [NNReal.coe_pow, NNReal.coe_zero,\n _root_.tendsto_pow_atTop_nhds_0_of_lt_1 r.coe_nonneg hr] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.PullbackCone.condition_one ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D W X Y Z : C f : X \u27f6 Z g : Y \u27f6 Z t : PullbackCone f g \u22a2 t.\u03c0.app WalkingCospan.one = fst t \u226b f ** have w := t.\u03c0.naturality WalkingCospan.Hom.inl ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D W X Y Z : C f : X \u27f6 Z g : Y \u27f6 Z t : PullbackCone f g w : ((Functor.const WalkingCospan).obj t.pt).map inl \u226b t.\u03c0.app WalkingCospan.one = t.\u03c0.app WalkingCospan.left \u226b (cospan f g).map inl \u22a2 t.\u03c0.app WalkingCospan.one = fst t \u226b f ** dsimp at w ** C : Type u inst\u271d\u00b9 : Category.{v, u} C D : Type u\u2082 inst\u271d : Category.{v\u2082, u\u2082} D W X Y Z : C f : X \u27f6 Z g : Y \u27f6 Z t : PullbackCone f g w : \ud835\udfd9 t.pt \u226b t.\u03c0.app WalkingCospan.one = fst t \u226b f \u22a2 t.\u03c0.app WalkingCospan.one = fst t \u226b f ** simpa using w ** Qed", + "informal": "" + }, + { + "formal": "Filter.map\u2082_left_comm ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 m\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 f f\u2081 f\u2082 : Filter \u03b1 g g\u2081 g\u2082 : Filter \u03b2 h h\u2081 h\u2082 : Filter \u03b3 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 u : Set \u03b3 v : Set \u03b4 a : \u03b1 b : \u03b2 c : \u03b3 m : \u03b1 \u2192 \u03b4 \u2192 \u03b5 n : \u03b2 \u2192 \u03b3 \u2192 \u03b4 m' : \u03b1 \u2192 \u03b3 \u2192 \u03b4' n' : \u03b2 \u2192 \u03b4' \u2192 \u03b5 h_left_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' b (m' a c) \u22a2 map\u2082 m f (map\u2082 n g h) = map\u2082 n' g (map\u2082 m' f h) ** rw [map\u2082_swap m', map\u2082_swap m] ** \u03b1 : Type u_1 \u03b1' : Type u_2 \u03b2 : Type u_3 \u03b2' : Type u_4 \u03b3 : Type u_5 \u03b3' : Type u_6 \u03b4 : Type u_7 \u03b4' : Type u_8 \u03b5 : Type u_9 \u03b5' : Type u_10 m\u271d : \u03b1 \u2192 \u03b2 \u2192 \u03b3 f f\u2081 f\u2082 : Filter \u03b1 g g\u2081 g\u2082 : Filter \u03b2 h h\u2081 h\u2082 : Filter \u03b3 s s\u2081 s\u2082 : Set \u03b1 t t\u2081 t\u2082 : Set \u03b2 u : Set \u03b3 v : Set \u03b4 a : \u03b1 b : \u03b2 c : \u03b3 m : \u03b1 \u2192 \u03b4 \u2192 \u03b5 n : \u03b2 \u2192 \u03b3 \u2192 \u03b4 m' : \u03b1 \u2192 \u03b3 \u2192 \u03b4' n' : \u03b2 \u2192 \u03b4' \u2192 \u03b5 h_left_comm : \u2200 (a : \u03b1) (b : \u03b2) (c : \u03b3), m a (n b c) = n' b (m' a c) \u22a2 map\u2082 (fun a b => m b a) (map\u2082 n g h) f = map\u2082 n' g (map\u2082 (fun a b => m' b a) h f) ** exact map\u2082_assoc fun _ _ _ => h_left_comm _ _ _ ** Qed", + "informal": "" + }, + { + "formal": "Real.Angle.two_zsmul_eq_iff ** \u03c8 \u03b8 : Angle \u22a2 2 \u2022 \u03c8 = 2 \u2022 \u03b8 \u2194 \u03c8 = \u03b8 \u2228 \u03c8 = \u03b8 + \u2191\u03c0 ** have : Int.natAbs 2 = 2 := rfl ** \u03c8 \u03b8 : Angle this : Int.natAbs 2 = 2 \u22a2 2 \u2022 \u03c8 = 2 \u2022 \u03b8 \u2194 \u03c8 = \u03b8 \u2228 \u03c8 = \u03b8 + \u2191\u03c0 ** rw [zsmul_eq_iff two_ne_zero, this, Fin.exists_fin_two, Fin.val_zero,\n Fin.val_one, zero_smul, coe_zero, add_zero, one_smul, Int.cast_two,\n mul_div_cancel_left (_ : \u211d) two_ne_zero] ** Qed", + "informal": "" + }, + { + "formal": "Complex.cos_sub ** x y : \u2102 \u22a2 cos (x - y) = cos x * cos y + sin x * sin y ** simp [sub_eq_add_neg, cos_add, sin_neg, cos_neg] ** Qed", + "informal": "" + }, + { + "formal": "matPolyEquiv_charmatrix ** R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R \u22a2 \u2191matPolyEquiv (charmatrix M) = X - \u2191C M ** ext k i j ** case a.a.h R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i j : n \u22a2 coeff (\u2191matPolyEquiv (charmatrix M)) k i j = coeff (X - \u2191C M) k i j ** simp only [matPolyEquiv_coeff_apply, coeff_sub, Pi.sub_apply] ** case a.a.h R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i j : n \u22a2 coeff (charmatrix M i j) k = (coeff X k - coeff (\u2191C M) k) i j ** by_cases h : i = j ** case pos R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i j : n h : i = j \u22a2 coeff (charmatrix M i j) k = (coeff X k - coeff (\u2191C M) k) i j ** subst h ** case pos R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i : n \u22a2 coeff (charmatrix M i i) k = (coeff X k - coeff (\u2191C M) k) i i ** rw [charmatrix_apply_eq, coeff_sub] ** case pos R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i : n \u22a2 coeff X k - coeff (\u2191C (M i i)) k = (coeff X k - coeff (\u2191C M) k) i i ** simp only [coeff_X, coeff_C] ** case pos R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i : n \u22a2 ((if 1 = k then 1 else 0) - if k = 0 then M i i else 0) = ((if 1 = k then 1 else 0) - if k = 0 then M else 0) i i ** split_ifs <;> simp ** case neg R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i j : n h : \u00aci = j \u22a2 coeff (charmatrix M i j) k = (coeff X k - coeff (\u2191C M) k) i j ** rw [charmatrix_apply_ne _ _ _ h, coeff_X, coeff_neg, coeff_C, coeff_C] ** case neg R : Type u inst\u271d\u00b2 : CommRing R n : Type w inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n M : Matrix n n R k : \u2115 i j : n h : \u00aci = j \u22a2 (-if k = 0 then M i j else 0) = ((if 1 = k then 1 else 0) - if k = 0 then M else 0) i j ** split_ifs <;> simp [h] ** Qed", + "informal": "" + }, + { + "formal": "Basis.coe_toDual_self ** R : Type uR M : Type uM K : Type uK V : Type uV \u03b9 : Type u\u03b9 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R M inst\u271d : DecidableEq \u03b9 b : Basis \u03b9 R M i : \u03b9 \u22a2 \u2191(toDual b) (\u2191b i) = coord b i ** ext ** case h R : Type uR M : Type uM K : Type uK V : Type uV \u03b9 : Type u\u03b9 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : Module R M inst\u271d : DecidableEq \u03b9 b : Basis \u03b9 R M i : \u03b9 x\u271d : M \u22a2 \u2191(\u2191(toDual b) (\u2191b i)) x\u271d = \u2191(coord b i) x\u271d ** apply toDual_apply_right ** Qed", + "informal": "" + }, + { + "formal": "Stream'.WSeq.head_cons ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w a : \u03b1 s : WSeq \u03b1 \u22a2 head (cons a s) = Computation.pure (some a) ** simp [head] ** Qed", + "informal": "" + }, + { + "formal": "ENNReal.iInf_mul_right' ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 a\u271d b c d : \u211d\u22650\u221e r p q : \u211d\u22650 x y z \u03b5 \u03b5\u2081 \u03b5\u2082 : \u211d\u22650\u221e s : Set \u211d\u22650\u221e \u03b9 : Sort u_4 f : \u03b9 \u2192 \u211d\u22650\u221e a : \u211d\u22650\u221e h : a = \u22a4 \u2192 \u2a05 i, f i = 0 \u2192 \u2203 i, f i = 0 h0 : a = 0 \u2192 Nonempty \u03b9 \u22a2 \u2a05 i, f i * a = (\u2a05 i, f i) * a ** simpa only [mul_comm a] using iInf_mul_left' h h0 ** Qed", + "informal": "" + }, + { + "formal": "SzemerediRegularity.energy_increment ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 \u22a2 \u2191(energy P G) + \u03b5 ^ 5 / 4 \u2264 \u2191(energy (increment hP G \u03b5) G) ** rw [coe_energy] ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 \u22a2 (\u2211 uv in offDiag P.parts, \u2191(edgeDensity G uv.1 uv.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264 \u2191(energy (increment hP G \u03b5) G) ** have h := uniform_add_nonuniform_eq_offDiag_pairs (hP := hP) h\u03b5\u2081 hP\u2087 hP\u03b1 h\u03b5.le hPG ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 h : (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.1 x.2) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4)) / \u2191(Finset.card P.parts) ^ 2 \u2264 \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u22a2 (\u2211 uv in offDiag P.parts, \u2191(edgeDensity G uv.1 uv.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264 \u2191(energy (increment hP G \u03b5) G) ** rw [add_div, mul_div_cancel_left] at h ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 h : (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.1 x.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264 \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u22a2 (\u2211 uv in offDiag P.parts, \u2191(edgeDensity G uv.1 uv.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264 \u2191(energy (increment hP G \u03b5) G) case ha \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 h : (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.1 x.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) / \u2191(Finset.card P.parts) ^ 2 \u2264 \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u22a2 \u2191(Finset.card P.parts) ^ 2 \u2260 0 ** exact h.trans (by exact_mod_cast offDiag_pairs_le_increment_energy) ** case ha \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 h : (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.1 x.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u2191(Finset.card P.parts) ^ 2 * (\u03b5 ^ 5 / 4) / \u2191(Finset.card P.parts) ^ 2 \u2264 \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u22a2 \u2191(Finset.card P.parts) ^ 2 \u2260 0 ** positivity ** \u03b1 : Type u_1 inst\u271d\u00b9 : Fintype \u03b1 P : Finpartition univ hP\u271d : IsEquipartition P G : SimpleGraph \u03b1 \u03b5 : \u211d inst\u271d : Nonempty \u03b1 hP : IsEquipartition P hP\u2087 : 7 \u2264 Finset.card P.parts h\u03b5 : 100 < 4 ^ Finset.card P.parts * \u03b5 ^ 5 hP\u03b1 : Finset.card P.parts * 16 ^ Finset.card P.parts \u2264 Fintype.card \u03b1 hPG : \u00acFinpartition.IsUniform P G \u03b5 h\u03b5\u2081 : \u03b5 \u2264 1 h : (\u2211 x in offDiag P.parts, \u2191(edgeDensity G x.1 x.2) ^ 2) / \u2191(Finset.card P.parts) ^ 2 + \u03b5 ^ 5 / 4 \u2264 \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u22a2 \u2211 x in attach (offDiag P.parts), \u2191(SzemerediRegularity.pairContrib G \u03b5 hP x) / \u2191(Finset.card (increment hP G \u03b5).parts) ^ 2 \u2264 \u2191(energy (increment hP G \u03b5) G) ** exact_mod_cast offDiag_pairs_le_increment_energy ** Qed", + "informal": "" + }, + { + "formal": "Complex.contDiff_exp ** \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 \u22a2 \u2200 {n : \u2115\u221e}, ContDiff \ud835\udd5c n cexp ** refine' @(contDiff_all_iff_nat.2 fun n => ?_) ** \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 n : \u2115 this : ContDiff \u2102 \u2191n cexp \u22a2 ContDiff \ud835\udd5c \u2191n cexp ** exact this.restrict_scalars \ud835\udd5c ** \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 n : \u2115 \u22a2 ContDiff \u2102 \u2191n cexp ** induction' n with n ihn ** case zero \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 \u22a2 ContDiff \u2102 \u2191Nat.zero cexp ** exact contDiff_zero.2 continuous_exp ** case succ \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 n : \u2115 ihn : ContDiff \u2102 \u2191n cexp \u22a2 ContDiff \u2102 \u2191(Nat.succ n) cexp ** rw [contDiff_succ_iff_deriv] ** case succ \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 n : \u2115 ihn : ContDiff \u2102 \u2191n cexp \u22a2 Differentiable \u2102 cexp \u2227 ContDiff \u2102 (\u2191n) (deriv cexp) ** use differentiable_exp ** case right \ud835\udd5c : Type u_1 inst\u271d\u00b9 : NontriviallyNormedField \ud835\udd5c inst\u271d : NormedAlgebra \ud835\udd5c \u2102 n : \u2115 ihn : ContDiff \u2102 \u2191n cexp \u22a2 ContDiff \u2102 (\u2191n) (deriv cexp) ** rwa [deriv_exp] ** Qed", + "informal": "" + }, + { + "formal": "Basis.reindexRange_repr' ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x bi : M i : \u03b9 h : \u2191b i = bi \u22a2 \u2191(\u2191(reindexRange b).repr x) { val := bi, property := (_ : \u2203 y, \u2191b y = bi) } = \u2191(\u2191b.repr x) i ** nontriviality ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x bi : M i : \u03b9 h : \u2191b i = bi \u271d : Nontrivial ((fun x => R) { val := bi, property := (_ : \u2203 y, \u2191b y = bi) }) \u22a2 \u2191(\u2191(reindexRange b).repr x) { val := bi, property := (_ : \u2203 y, \u2191b y = bi) } = \u2191(\u2191b.repr x) i ** subst h ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) \u22a2 \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) } = \u2191(\u2191b.repr x) i ** apply (b.repr_apply_eq (fun x i => b.reindexRange.repr x \u27e8b i, _\u27e9) _ _ _ x i).symm ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) \u22a2 \u2200 (x y : M), (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (x + y) = (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x + (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) y ** intro x y ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d\u00b9 : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x\u271d : M i : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x y : M \u22a2 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (x + y) = (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x + (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) y ** ext i ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d\u00b9 : \u03b9 c : R x\u271d\u00b9 : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x\u271d : M i\u271d : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i\u271d, property := (_ : \u2203 y, \u2191b y = \u2191b i\u271d) }) x y : M i : \u03b9 \u22a2 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (x + y) i = ((fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x + (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) y) i ** simp only [Pi.add_apply, LinearEquiv.map_add, Finsupp.coe_add] ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) \u22a2 \u2200 (c : R) (x : M), (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (c \u2022 x) = c \u2022 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x ** intro c x ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c\u271d : R x\u271d\u00b9 : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x\u271d : M i : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) c : R x : M \u22a2 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (c \u2022 x) = c \u2022 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x ** ext i ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d\u00b9 : \u03b9 c\u271d : R x\u271d\u00b9 : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x\u271d : M i\u271d : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i\u271d, property := (_ : \u2203 y, \u2191b y = \u2191b i\u271d) }) c : R x : M i : \u03b9 \u22a2 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (c \u2022 x) i = (c \u2022 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) x) i ** simp only [Pi.smul_apply, LinearEquiv.map_smul, Finsupp.coe_smul] ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) \u22a2 \u2200 (i : \u03b9), (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (\u2191b i) = \u2191fun\u2080 | i => 1 ** intro i ** \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d\u00b9 : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i\u271d : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i\u271d, property := (_ : \u2203 y, \u2191b y = \u2191b i\u271d) }) i : \u03b9 \u22a2 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (\u2191b i) = \u2191fun\u2080 | i => 1 ** ext j ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d\u00b9 : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i\u271d : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i\u271d, property := (_ : \u2203 y, \u2191b y = \u2191b i\u271d) }) i j : \u03b9 \u22a2 (fun x i => \u2191(\u2191(reindexRange b).repr x) { val := \u2191b i, property := (_ : \u2203 y, \u2191b y = \u2191b i) }) (\u2191b i) j = (\u2191fun\u2080 | i => 1) j ** simp only [reindexRange_repr_self] ** case h \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d\u00b9 : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i\u271d : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i\u271d, property := (_ : \u2203 y, \u2191b y = \u2191b i\u271d) }) i j : \u03b9 \u22a2 (\u2191fun\u2080 | { val := \u2191b i, property := (_ : \u2191b i \u2208 range \u2191b) } => 1) { val := \u2191b j, property := (_ : \u2203 y, \u2191b y = \u2191b j) } = (\u2191fun\u2080 | i => 1) j ** apply Finsupp.single_apply_left (f := fun i => (\u27e8b i, _\u27e9 : Set.range b)) ** case h.hf \u03b9 : Type u_1 \u03b9' : Type u_2 R : Type u_3 R\u2082 : Type u_4 K : Type u_5 M : Type u_6 M' : Type u_7 M'' : Type u_8 V : Type u V' : Type u_9 inst\u271d\u2074 : Semiring R inst\u271d\u00b3 : AddCommMonoid M inst\u271d\u00b2 : Module R M inst\u271d\u00b9 : AddCommMonoid M' inst\u271d : Module R M' b b\u2081 : Basis \u03b9 R M i\u271d\u00b9 : \u03b9 c : R x\u271d : M b' : Basis \u03b9' R M' e : \u03b9 \u2243 \u03b9' x : M i\u271d : \u03b9 \u271d : Nontrivial ((fun x => R) { val := \u2191b i\u271d, property := (_ : \u2203 y, \u2191b y = \u2191b i\u271d) }) i j : \u03b9 \u22a2 Injective fun i => { val := \u2191b i, property := (_ : \u2191b i \u2208 range \u2191b) } ** exact fun i j h => b.injective (Subtype.mk.inj h) ** Qed", + "informal": "" + }, + { + "formal": "Filter.mem_generate_iff ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 \u22a2 U \u2208 generate s \u2194 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 U ** constructor <;> intro h ** case mp \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 h : U \u2208 generate s \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 U ** induction h ** case mp.basic \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d : Set \u03b1 a\u271d : s\u271d \u2208 s \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d case mp.univ \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 univ case mp.superset \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : s\u271d \u2286 t\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d case mp.inter \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : GenerateSets s t\u271d a_ih\u271d\u00b9 : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u2229 t\u271d ** case basic V V_in =>\n exact \u27e8{V}, singleton_subset_iff.2 V_in, finite_singleton _, (sInter_singleton _).subset\u27e9 ** case mp.univ \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 univ case mp.superset \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : s\u271d \u2286 t\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d case mp.inter \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : GenerateSets s t\u271d a_ih\u271d\u00b9 : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u2229 t\u271d ** case univ => exact \u27e8\u2205, empty_subset _, finite_empty, subset_univ _\u27e9 ** case mp.superset \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : s\u271d \u2286 t\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d case mp.inter \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : GenerateSets s t\u271d a_ih\u271d\u00b9 : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u2229 t\u271d ** case superset V W _ hVW hV =>\n rcases hV with \u27e8t, hts, ht, htV\u27e9\n exact \u27e8t, hts, ht, htV.trans hVW\u27e9 ** case mp.inter \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d\u00b9 t : Set \u03b1 s : Set (Set \u03b1) U s\u271d t\u271d : Set \u03b1 a\u271d\u00b9 : GenerateSets s s\u271d a\u271d : GenerateSets s t\u271d a_ih\u271d\u00b9 : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d a_ih\u271d : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 t\u271d \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 s\u271d \u2229 t\u271d ** case inter V W _ _ hV hW =>\n rcases hV, hW with \u27e8\u27e8t, hts, ht, htV\u27e9, u, hus, hu, huW\u27e9\n exact\n \u27e8t \u222a u, union_subset hts hus, ht.union hu,\n (sInter_union _ _).subset.trans <| inter_subset_inter htV huW\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U V : Set \u03b1 V_in : V \u2208 s \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 V ** exact \u27e8{V}, singleton_subset_iff.2 V_in, finite_singleton _, (sInter_singleton _).subset\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 univ ** exact \u27e8\u2205, empty_subset _, finite_empty, subset_univ _\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U V W : Set \u03b1 a\u271d : GenerateSets s V hVW : V \u2286 W hV : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 V \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 W ** rcases hV with \u27e8t, hts, ht, htV\u27e9 ** case intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t\u271d : Set \u03b1 s : Set (Set \u03b1) U V W : Set \u03b1 a\u271d : GenerateSets s V hVW : V \u2286 W t : Set (Set \u03b1) hts : t \u2286 s ht : Set.Finite t htV : \u22c2\u2080 t \u2286 V \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 W ** exact \u27e8t, hts, ht, htV.trans hVW\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U V W : Set \u03b1 a\u271d\u00b9 : GenerateSets s V a\u271d : GenerateSets s W hV : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 V hW : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 W \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 V \u2229 W ** rcases hV, hW with \u27e8\u27e8t, hts, ht, htV\u27e9, u, hus, hu, huW\u27e9 ** case intro.intro.intro.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t\u271d : Set \u03b1 s : Set (Set \u03b1) U V W : Set \u03b1 a\u271d\u00b9 : GenerateSets s V a\u271d : GenerateSets s W t : Set (Set \u03b1) hts : t \u2286 s ht : Set.Finite t htV : \u22c2\u2080 t \u2286 V u : Set (Set \u03b1) hus : u \u2286 s hu : Set.Finite u huW : \u22c2\u2080 u \u2286 W \u22a2 \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 V \u2229 W ** exact\n \u27e8t \u222a u, union_subset hts hus, ht.union hu,\n (sInter_union _ _).subset.trans <| inter_subset_inter htV huW\u27e9 ** case mpr \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 h : \u2203 t x, Set.Finite t \u2227 \u22c2\u2080 t \u2286 U \u22a2 U \u2208 generate s ** rcases h with \u27e8t, hts, tfin, h\u27e9 ** case mpr.intro.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b4 : Type u_1 \u03b9 : Sort x f g : Filter \u03b1 s\u271d t\u271d : Set \u03b1 s : Set (Set \u03b1) U : Set \u03b1 t : Set (Set \u03b1) hts : t \u2286 s tfin : Set.Finite t h : \u22c2\u2080 t \u2286 U \u22a2 U \u2208 generate s ** exact mem_of_superset ((sInter_mem tfin).2 fun V hV => GenerateSets.basic <| hts hV) h ** Qed", + "informal": "" + }, + { + "formal": "InnerProductSpaceable.inner_.norm_sq ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E \u22a2 \u2016x\u2016 ^ 2 = \u2191re (inner_ \ud835\udd5c x x) ** simp only [inner_] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E \u22a2 \u2016x\u2016 ^ 2 = \u2191re (4\u207b\u00b9 * (\u2191\ud835\udcda \u2016x + x\u2016 * \u2191\ud835\udcda \u2016x + x\u2016 - \u2191\ud835\udcda \u2016x - x\u2016 * \u2191\ud835\udcda \u2016x - x\u2016 + I * \u2191\ud835\udcda \u2016I \u2022 x + x\u2016 * \u2191\ud835\udcda \u2016I \u2022 x + x\u2016 - I * \u2191\ud835\udcda \u2016I \u2022 x - x\u2016 * \u2191\ud835\udcda \u2016I \u2022 x - x\u2016)) ** have h\u2081 : IsROrC.normSq (4 : \ud835\udd5c) = 16 := by\n have : ((4 : \u211d) : \ud835\udd5c) = (4 : \ud835\udd5c) := by norm_cast\n rw [\u2190 this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : \u211d) \u2264 4)]\n norm_num ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E h\u2081 : \u2191normSq 4 = 16 \u22a2 \u2016x\u2016 ^ 2 = \u2191re (4\u207b\u00b9 * (\u2191\ud835\udcda \u2016x + x\u2016 * \u2191\ud835\udcda \u2016x + x\u2016 - \u2191\ud835\udcda \u2016x - x\u2016 * \u2191\ud835\udcda \u2016x - x\u2016 + I * \u2191\ud835\udcda \u2016I \u2022 x + x\u2016 * \u2191\ud835\udcda \u2016I \u2022 x + x\u2016 - I * \u2191\ud835\udcda \u2016I \u2022 x - x\u2016 * \u2191\ud835\udcda \u2016I \u2022 x - x\u2016)) ** have h\u2082 : \u2016x + x\u2016 = 2 * \u2016x\u2016 := by rw [\u2190 two_smul \ud835\udd5c, norm_smul, IsROrC.norm_two] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E h\u2081 : \u2191normSq 4 = 16 h\u2082 : \u2016x + x\u2016 = 2 * \u2016x\u2016 \u22a2 \u2016x\u2016 ^ 2 = \u2191re (4\u207b\u00b9 * (\u2191\ud835\udcda \u2016x + x\u2016 * \u2191\ud835\udcda \u2016x + x\u2016 - \u2191\ud835\udcda \u2016x - x\u2016 * \u2191\ud835\udcda \u2016x - x\u2016 + I * \u2191\ud835\udcda \u2016I \u2022 x + x\u2016 * \u2191\ud835\udcda \u2016I \u2022 x + x\u2016 - I * \u2191\ud835\udcda \u2016I \u2022 x - x\u2016 * \u2191\ud835\udcda \u2016I \u2022 x - x\u2016)) ** simp only [h\u2081, h\u2082, algebraMap_eq_ofReal, sub_self, norm_zero, mul_re, inv_re, ofNat_re, map_sub,\n map_add, ofReal_re, ofNat_im, ofReal_im, mul_im, I_re, inv_im] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E h\u2081 : \u2191normSq 4 = 16 h\u2082 : \u2016x + x\u2016 = 2 * \u2016x\u2016 \u22a2 \u2016x\u2016 ^ 2 = 4 / 16 * (2 * \u2016x\u2016 * (2 * \u2016x\u2016) - 0 * 0 - 0 + ((0 * \u2016I \u2022 x + x\u2016 - \u2191im I * 0) * \u2016I \u2022 x + x\u2016 - (0 * 0 + \u2191im I * \u2016I \u2022 x + x\u2016) * 0) - ((0 * \u2016I \u2022 x - x\u2016 - \u2191im I * 0) * \u2016I \u2022 x - x\u2016 - (0 * 0 + \u2191im I * \u2016I \u2022 x - x\u2016) * 0)) - -0 / 16 * (2 * \u2016x\u2016 * 0 + 0 * (2 * \u2016x\u2016) - (0 * 0 + 0 * 0) + ((0 * \u2016I \u2022 x + x\u2016 - \u2191im I * 0) * 0 + (0 * 0 + \u2191im I * \u2016I \u2022 x + x\u2016) * \u2016I \u2022 x + x\u2016) - ((0 * \u2016I \u2022 x - x\u2016 - \u2191im I * 0) * 0 + (0 * 0 + \u2191im I * \u2016I \u2022 x - x\u2016) * \u2016I \u2022 x - x\u2016)) ** ring ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E \u22a2 \u2191normSq 4 = 16 ** have : ((4 : \u211d) : \ud835\udd5c) = (4 : \ud835\udd5c) := by norm_cast ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E this : \u21914 = 4 \u22a2 \u2191normSq 4 = 16 ** rw [\u2190 this, normSq_eq_def', IsROrC.norm_of_nonneg (by norm_num : (0 : \u211d) \u2264 4)] ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E this : \u21914 = 4 \u22a2 4 ^ 2 = 16 ** norm_num ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E \u22a2 \u21914 = 4 ** norm_cast ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E this : \u21914 = 4 \u22a2 0 \u2264 4 ** norm_num ** \ud835\udd5c : Type u_1 inst\u271d\u00b2 : IsROrC \ud835\udd5c E : Type u_2 inst\u271d\u00b9 : NormedAddCommGroup E inst\u271d : NormedSpace \ud835\udd5c E x : E h\u2081 : \u2191normSq 4 = 16 \u22a2 \u2016x + x\u2016 = 2 * \u2016x\u2016 ** rw [\u2190 two_smul \ud835\udd5c, norm_smul, IsROrC.norm_two] ** Qed", + "informal": "" + }, + { + "formal": "Nat.coprime_add_mul_right_right ** m n k : \u2115 \u22a2 Coprime m (n + k * m) \u2194 Coprime m n ** rw [Coprime, Coprime, gcd_add_mul_right_right] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.MonoidalCategory.rightUnitor_tensor_inv ** C\u271d : Type u \ud835\udc9e : Category.{v, u} C\u271d inst\u271d\u00b2 : MonoidalCategory C\u271d C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : MonoidalCategory C U V W X\u271d Y\u271d Z X Y : C \u22a2 inv (\u03c1_ (X \u2297 Y)).inv = inv ((\ud835\udfd9 X \u2297 (\u03c1_ Y).inv) \u226b (\u03b1_ X Y (\ud835\udfd9_ C)).inv) ** simp ** Qed", + "informal": "" + }, + { + "formal": "FirstOrder.Language.model_infiniteTheory_iff ** L : Language L' : Language M : Type w N : Type u_1 P : Type u_2 inst\u271d\u00b2 : Structure L M inst\u271d\u00b9 : Structure L N inst\u271d : Structure L P \u03b1 : Type u' \u03b2 : Type v' \u03b3 : Type u_3 n : \u2115 T : Theory L \u22a2 M \u22a8 infiniteTheory L \u2194 Infinite M ** simp [infiniteTheory, infinite_iff, aleph0_le] ** Qed", + "informal": "" + }, + { + "formal": "mem_extremePoints_iff_extreme_singleton ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 \u03b9 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 inst\u271d\u00b2 : OrderedSemiring \ud835\udd5c inst\u271d\u00b9 : AddCommMonoid E inst\u271d : SMul \ud835\udd5c E A B C : Set E x : E \u22a2 x \u2208 extremePoints \ud835\udd5c A \u2194 IsExtreme \ud835\udd5c A {x} ** refine' \u27e8_, fun hx \u21a6 \u27e8singleton_subset_iff.1 hx.1, fun x\u2081 hx\u2081 x\u2082 hx\u2082 \u21a6 hx.2 hx\u2081 hx\u2082 rfl\u27e9\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 \u03b9 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 inst\u271d\u00b2 : OrderedSemiring \ud835\udd5c inst\u271d\u00b9 : AddCommMonoid E inst\u271d : SMul \ud835\udd5c E A B C : Set E x : E \u22a2 x \u2208 extremePoints \ud835\udd5c A \u2192 IsExtreme \ud835\udd5c A {x} ** rintro \u27e8hxA, hAx\u27e9 ** case intro \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 \u03b9 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 inst\u271d\u00b2 : OrderedSemiring \ud835\udd5c inst\u271d\u00b9 : AddCommMonoid E inst\u271d : SMul \ud835\udd5c E A B C : Set E x : E hxA : x \u2208 A hAx : \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x \u22a2 IsExtreme \ud835\udd5c A {x} ** use singleton_subset_iff.2 hxA ** case right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 \u03b9 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 inst\u271d\u00b2 : OrderedSemiring \ud835\udd5c inst\u271d\u00b9 : AddCommMonoid E inst\u271d : SMul \ud835\udd5c E A B C : Set E x : E hxA : x \u2208 A hAx : \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 x \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = x \u2227 x\u2082 = x \u22a2 \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 \u2200 \u2983x_1 : E\u2984, x_1 \u2208 {x} \u2192 x_1 \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 \u2208 {x} \u2227 x\u2082 \u2208 {x} ** rintro x\u2081 hx\u2081A x\u2082 hx\u2082A y (rfl : y = x) ** case right \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 \u03b9 : Type u_4 \u03c0 : \u03b9 \u2192 Type u_5 inst\u271d\u00b2 : OrderedSemiring \ud835\udd5c inst\u271d\u00b9 : AddCommMonoid E inst\u271d : SMul \ud835\udd5c E A B C : Set E x\u2081 : E hx\u2081A : x\u2081 \u2208 A x\u2082 : E hx\u2082A : x\u2082 \u2208 A y : E hxA : y \u2208 A hAx : \u2200 \u2983x\u2081 : E\u2984, x\u2081 \u2208 A \u2192 \u2200 \u2983x\u2082 : E\u2984, x\u2082 \u2208 A \u2192 y \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 = y \u2227 x\u2082 = y \u22a2 y \u2208 openSegment \ud835\udd5c x\u2081 x\u2082 \u2192 x\u2081 \u2208 {y} \u2227 x\u2082 \u2208 {y} ** exact hAx hx\u2081A hx\u2082A ** Qed", + "informal": "" + }, + { + "formal": "Seminorm.zero_or_exists_apply_eq_finset_sup ** R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E s : Finset \u03b9 x : E \u22a2 \u2191(Finset.sup s p) x = 0 \u2228 \u2203 i, i \u2208 s \u2227 \u2191(Finset.sup s p) x = \u2191(p i) x ** rcases Finset.eq_empty_or_nonempty s with (rfl|hs) ** case inl R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E x : E \u22a2 \u2191(Finset.sup \u2205 p) x = 0 \u2228 \u2203 i, i \u2208 \u2205 \u2227 \u2191(Finset.sup \u2205 p) x = \u2191(p i) x ** left ** case inl.h R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E x : E \u22a2 \u2191(Finset.sup \u2205 p) x = 0 ** rfl ** case inr R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E s : Finset \u03b9 x : E hs : Finset.Nonempty s \u22a2 \u2191(Finset.sup s p) x = 0 \u2228 \u2203 i, i \u2208 s \u2227 \u2191(Finset.sup s p) x = \u2191(p i) x ** right ** case inr.h R : Type u_1 R' : Type u_2 \ud835\udd5c : Type u_3 \ud835\udd5c\u2082 : Type u_4 \ud835\udd5c\u2083 : Type u_5 \ud835\udd5d : Type u_6 E : Type u_7 E\u2082 : Type u_8 E\u2083 : Type u_9 F : Type u_10 G : Type u_11 \u03b9 : Type u_12 inst\u271d\u00b9\u2078 : SeminormedRing \ud835\udd5c inst\u271d\u00b9\u2077 : SeminormedRing \ud835\udd5c\u2082 inst\u271d\u00b9\u2076 : SeminormedRing \ud835\udd5c\u2083 \u03c3\u2081\u2082 : \ud835\udd5c \u2192+* \ud835\udd5c\u2082 inst\u271d\u00b9\u2075 : RingHomIsometric \u03c3\u2081\u2082 \u03c3\u2082\u2083 : \ud835\udd5c\u2082 \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u2074 : RingHomIsometric \u03c3\u2082\u2083 \u03c3\u2081\u2083 : \ud835\udd5c \u2192+* \ud835\udd5c\u2083 inst\u271d\u00b9\u00b3 : RingHomIsometric \u03c3\u2081\u2083 inst\u271d\u00b9\u00b2 : AddCommGroup E inst\u271d\u00b9\u00b9 : AddCommGroup E\u2082 inst\u271d\u00b9\u2070 : AddCommGroup E\u2083 inst\u271d\u2079 : AddCommGroup F inst\u271d\u2078 : AddCommGroup G inst\u271d\u2077 : Module \ud835\udd5c E inst\u271d\u2076 : Module \ud835\udd5c\u2082 E\u2082 inst\u271d\u2075 : Module \ud835\udd5c\u2083 E\u2083 inst\u271d\u2074 : Module \ud835\udd5c F inst\u271d\u00b3 : Module \ud835\udd5c G inst\u271d\u00b2 : SMul R \u211d inst\u271d\u00b9 : SMul R \u211d\u22650 inst\u271d : IsScalarTower R \u211d\u22650 \u211d p : \u03b9 \u2192 Seminorm \ud835\udd5c E s : Finset \u03b9 x : E hs : Finset.Nonempty s \u22a2 \u2203 i, i \u2208 s \u2227 \u2191(Finset.sup s p) x = \u2191(p i) x ** exact exists_apply_eq_finset_sup p hs x ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.condDistrib_ae_eq_condexp ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) =\u1d50[\u03bc] \u03bc[indicator (Y \u207b\u00b9' s) fun \u03c9 => 1|MeasurableSpace.comap X m\u03b2] ** refine' ae_eq_condexp_of_forall_set_integral_eq hX.comap_le _ _ _ _ ** case refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 Integrable (indicator (Y \u207b\u00b9' s) fun \u03c9 => 1) ** exact (integrable_const _).indicator (hY hs) ** case refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 < \u22a4 \u2192 IntegrableOn (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) s_1 ** exact fun t _ _ => (integrable_toReal_condDistrib hX.aemeasurable hs).integrableOn ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 \u2200 (s_1 : Set \u03b1), MeasurableSet s_1 \u2192 \u2191\u2191\u03bc s_1 < \u22a4 \u2192 \u222b (x : \u03b1) in s_1, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc = \u222b (x : \u03b1) in s_1, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc ** intro t ht _ ** case refine'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t\u271d : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s t : Set \u03b1 ht : MeasurableSet t a\u271d : \u2191\u2191\u03bc t < \u22a4 \u22a2 \u222b (x : \u03b1) in t, ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X x)) s) \u2202\u03bc = \u222b (x : \u03b1) in t, indicator (Y \u207b\u00b9' s) (fun \u03c9 => 1) x \u2202\u03bc ** rw [integral_toReal ((measurable_condDistrib hs).mono hX.comap_le le_rfl).aemeasurable\n (eventually_of_forall fun \u03c9 => measure_lt_top (condDistrib Y X \u03bc (X \u03c9)) _),\n integral_indicator_const _ (hY hs), Measure.restrict_apply (hY hs), smul_eq_mul, mul_one,\n inter_comm, set_lintegral_condDistrib_of_measurableSet hX hY.aemeasurable hs ht] ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 AEStronglyMeasurable' (MeasurableSpace.comap X m\u03b2) (fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s)) \u03bc ** refine' (Measurable.stronglyMeasurable _).aeStronglyMeasurable' ** case refine'_4 \u03b1 : Type u_1 \u03b2 : Type u_2 \u03a9 : Type u_3 F : Type u_4 inst\u271d\u2076 : TopologicalSpace \u03a9 inst\u271d\u2075 : MeasurableSpace \u03a9 inst\u271d\u2074 : PolishSpace \u03a9 inst\u271d\u00b3 : BorelSpace \u03a9 inst\u271d\u00b2 : Nonempty \u03a9 inst\u271d\u00b9 : NormedAddCommGroup F m\u03b1 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d : IsFiniteMeasure \u03bc X : \u03b1 \u2192 \u03b2 Y : \u03b1 \u2192 \u03a9 m\u03b2 : MeasurableSpace \u03b2 s : Set \u03a9 t : Set \u03b2 f : \u03b2 \u00d7 \u03a9 \u2192 F hX : Measurable X hY : Measurable Y hs : MeasurableSet s \u22a2 Measurable fun a => ENNReal.toReal (\u2191\u2191(\u2191(condDistrib Y X \u03bc) (X a)) s) ** exact @Measurable.ennreal_toReal _ (m\u03b2.comap X) _ (measurable_condDistrib hs) ** Qed", + "informal": "" + }, + { + "formal": "Set.Finite.convexHull_eq ** R : Type u_1 R' : Type u_2 E : Type u_3 F : Type u_4 \u03b9 : Type u_5 \u03b9' : Type u_6 \u03b1 : Type u_7 inst\u271d\u2078 : LinearOrderedField R inst\u271d\u2077 : LinearOrderedField R' inst\u271d\u2076 : AddCommGroup E inst\u271d\u2075 : AddCommGroup F inst\u271d\u2074 : LinearOrderedAddCommGroup \u03b1 inst\u271d\u00b3 : Module R E inst\u271d\u00b2 : Module R F inst\u271d\u00b9 : Module R \u03b1 inst\u271d : OrderedSMul R \u03b1 s\u271d : Set E i j : \u03b9 c : R t : Finset \u03b9 w : \u03b9 \u2192 R z : \u03b9 \u2192 E s : Set E hs : Set.Finite s \u22a2 \u2191(convexHull R) s = {x | \u2203 w x_1 x_2, centerMass (Finite.toFinset hs) w id = x} ** simpa only [Set.Finite.coe_toFinset, Set.Finite.mem_toFinset, exists_prop] using\n hs.toFinset.convexHull_eq ** Qed", + "informal": "" + }, + { + "formal": "Finsupp.neLocus_eq_empty ** \u03b1 : Type u_1 M : Type u_2 N : Type u_3 P : Type u_4 inst\u271d\u00b2 : DecidableEq \u03b1 inst\u271d\u00b9 : DecidableEq N inst\u271d : Zero N f\u271d g\u271d f g : \u03b1 \u2192\u2080 N h : f = g \u22a2 neLocus f f = \u2205 ** simp only [neLocus, Ne.def, eq_self_iff_true, not_true, Finset.filter_False] ** Qed", + "informal": "" + }, + { + "formal": "Finset.coe_eq_univ ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : Fintype \u03b1 s t : Finset \u03b1 \u22a2 \u2191s = Set.univ \u2194 s = univ ** rw [\u2190 coe_univ, coe_inj] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_lt_prod_of_nonempty' ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : OrderedCancelCommMonoid M f g : \u03b9 \u2192 M s t : Finset \u03b9 hs : Finset.Nonempty s Hlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i \u22a2 \u220f i in s, f i < \u220f i in s, g i ** apply prod_lt_prod' ** case Hlt \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : OrderedCancelCommMonoid M f g : \u03b9 \u2192 M s t : Finset \u03b9 hs : Finset.Nonempty s Hlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i \u22a2 \u2203 i, i \u2208 s \u2227 f i < g i ** cases' hs with i hi ** case Hlt.intro \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : OrderedCancelCommMonoid M f g : \u03b9 \u2192 M s t : Finset \u03b9 Hlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i i : \u03b9 hi : i \u2208 s \u22a2 \u2203 i, i \u2208 s \u2227 f i < g i ** exact \u27e8i, hi, Hlt i hi\u27e9 ** case Hle \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : OrderedCancelCommMonoid M f g : \u03b9 \u2192 M s t : Finset \u03b9 hs : Finset.Nonempty s Hlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i \u22a2 \u2200 (i : \u03b9), i \u2208 s \u2192 f i \u2264 g i ** intro i hi ** case Hle \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : OrderedCancelCommMonoid M f g : \u03b9 \u2192 M s t : Finset \u03b9 hs : Finset.Nonempty s Hlt : \u2200 (i : \u03b9), i \u2208 s \u2192 f i < g i i : \u03b9 hi : i \u2208 s \u22a2 f i \u2264 g i ** apply le_of_lt (Hlt i hi) ** Qed", + "informal": "" + }, + { + "formal": "Set.encard_insert_le ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 x : \u03b1 \u22a2 encard (insert x s) \u2264 encard s + 1 ** rw [\u2190union_singleton, \u2190encard_singleton x] ** \u03b1 : Type u_1 s\u271d t s : Set \u03b1 x : \u03b1 \u22a2 encard (s \u222a {x}) \u2264 encard s + encard {x} ** apply encard_union_le ** Qed", + "informal": "" + }, + { + "formal": "Finset.Nontrivial.sdiff_singleton_nonempty ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 Finset.Nonempty (s \\ {c}) ** rw [Finset.sdiff_nonempty, Finset.subset_singleton_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 \u00ac(s = \u2205 \u2228 s = {c}) ** push_neg ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 s \u2260 \u2205 \u2227 s \u2260 {c} ** exact \u27e8by rintro rfl; exact Finset.not_nontrivial_empty hS, hS.ne_singleton\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s\u271d t u v : Finset \u03b1 a b c : \u03b1 s : Finset \u03b1 hS : Finset.Nontrivial s \u22a2 s \u2260 \u2205 ** rintro rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d : DecidableEq \u03b1 s t u v : Finset \u03b1 a b c : \u03b1 hS : Finset.Nontrivial \u2205 \u22a2 False ** exact Finset.not_nontrivial_empty hS ** Qed", + "informal": "" + }, + { + "formal": "Module.End.generalizedEigenspace_zero ** K R : Type v V M : Type w inst\u271d\u2075 : CommRing R inst\u271d\u2074 : AddCommGroup M inst\u271d\u00b3 : Module R M inst\u271d\u00b2 : Field K inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module K V f : End R M k : \u2115 \u22a2 \u2191(generalizedEigenspace f 0) k = LinearMap.ker (f ^ k) ** simp [Module.End.generalizedEigenspace] ** Qed", + "informal": "" + }, + { + "formal": "PrimeSpectrum.exists_primeSpectrum_prod_le ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I : Ideal R \u22a2 \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I ** refine' IsNoetherian.induction\n (P := fun I => \u2203 Z : Multiset (PrimeSpectrum R), Multiset.prod (Z.map asIdeal) \u2264 I)\n (fun (M : Ideal R) hgt => _) I ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** by_cases h_prM : M.IsPrime ** case neg R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** by_cases htop : M = \u22a4 ** case neg R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** have lt_add : \u2200 (z) (_ : z \u2209 M), M < M + span R {z} := by\n intro z hz\n refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _\n rw [m_eq]\n exact Ideal.mem_sup_right (mem_span_singleton_self z) ** case neg R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** obtain \u27e8x, hx, y, hy, hxy\u27e9 := (Ideal.not_isPrime_iff.mp h_prM).resolve_left htop ** case neg.intro.intro.intro.intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** obtain \u27e8Wx, h_Wx\u27e9 := hgt (M + span R {x}) (lt_add _ hx) ** case neg.intro.intro.intro.intro.intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** obtain \u27e8Wy, h_Wy\u27e9 := hgt (M + span R {y}) (lt_add _ hy) ** case neg.intro.intro.intro.intro.intro.intro R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** use Wx + Wy ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 Multiset.prod (Multiset.map asIdeal (Wx + Wy)) \u2264 M ** rw [Multiset.map_add, Multiset.prod_add] ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 Multiset.prod (Multiset.map asIdeal Wx) * Multiset.prod (Multiset.map asIdeal Wy) \u2264 M ** apply le_trans (Submodule.mul_le_mul h_Wx h_Wy) ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 (M + span R {x}) * (M + span R {y}) \u2264 M ** rw [add_mul] ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 M * (M + span R {y}) + span R {x} * (M + span R {y}) \u2264 M ** apply sup_le (show M * (M + span R {y}) \u2264 M from Ideal.mul_le_right) ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 span R {x} * (M + span R {y}) \u2264 M ** rw [mul_add] ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 span R {x} * M + span R {x} * span R {y} \u2264 M ** apply sup_le (show span R {x} * M \u2264 M from Ideal.mul_le_left) ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 lt_add : \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} x : R hx : \u00acx \u2208 M y : R hy : \u00acy \u2208 M hxy : x * y \u2208 M Wx : Multiset (PrimeSpectrum R) h_Wx : Multiset.prod (Multiset.map asIdeal Wx) \u2264 M + span R {x} Wy : Multiset (PrimeSpectrum R) h_Wy : Multiset.prod (Multiset.map asIdeal Wy) \u2264 M + span R {y} \u22a2 span R {x} * span R {y} \u2264 M ** rwa [span_mul_span, Set.singleton_mul_singleton, span_singleton_le_iff_mem] ** case pos R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : Ideal.IsPrime M \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** use {\u27e8M, h_prM\u27e9} ** case h R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : Ideal.IsPrime M \u22a2 Multiset.prod (Multiset.map asIdeal {{ asIdeal := M, IsPrime := h_prM }}) \u2264 M ** rw [Multiset.map_singleton, Multiset.prod_singleton] ** case pos R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : M = \u22a4 \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) M ** rw [htop] ** case pos R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : M = \u22a4 \u22a2 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) \u22a4 ** exact \u27e80, le_top\u27e9 ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 \u22a2 \u2200 (z : R), \u00acz \u2208 M \u2192 M < M + span R {z} ** intro z hz ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 z : R hz : \u00acz \u2208 M \u22a2 M < M + span R {z} ** refine' lt_of_le_of_ne le_sup_left fun m_eq => hz _ ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 z : R hz : \u00acz \u2208 M m_eq : M = M + span R {z} \u22a2 z \u2208 M ** rw [m_eq] ** R : Type u inst\u271d\u2074 : CommRing R inst\u271d\u00b3 : IsNoetherianRing R A : Type u inst\u271d\u00b2 : CommRing A inst\u271d\u00b9 : IsDomain A inst\u271d : IsNoetherianRing A I M : Ideal R hgt : \u2200 (J : Submodule R R), J > M \u2192 (fun I => \u2203 Z, Multiset.prod (Multiset.map asIdeal Z) \u2264 I) J h_prM : \u00acIdeal.IsPrime M htop : \u00acM = \u22a4 z : R hz : \u00acz \u2208 M m_eq : M = M + span R {z} \u22a2 z \u2208 M + span R {z} ** exact Ideal.mem_sup_right (mem_span_singleton_self z) ** Qed", + "informal": "" + }, + { + "formal": "edist_div_right ** M : Type u G : Type v X : Type w inst\u271d\u2076 : PseudoEMetricSpace X inst\u271d\u2075 : Group G inst\u271d\u2074 : MulAction G X inst\u271d\u00b3 : IsometricSMul G X inst\u271d\u00b2 : DivInvMonoid M inst\u271d\u00b9 : PseudoEMetricSpace M inst\u271d : IsometricSMul M\u1d50\u1d52\u1d56 M a b c : M \u22a2 edist (a / c) (b / c) = edist a b ** simp only [div_eq_mul_inv, edist_mul_right] ** Qed", + "informal": "" + }, + { + "formal": "DFinsupp.coe_tsub ** \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : (i : \u03b9) \u2192 CanonicallyOrderedAddCommMonoid (\u03b1 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Sub (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), OrderedSub (\u03b1 i) f\u271d g\u271d : \u03a0\u2080 (i : \u03b9), \u03b1 i i : \u03b9 a b : \u03b1 i f g : \u03a0\u2080 (i : \u03b9), \u03b1 i \u22a2 \u2191(f - g) = \u2191f - \u2191g ** ext i ** case h \u03b9 : Type u_1 \u03b1 : \u03b9 \u2192 Type u_2 inst\u271d\u00b2 : (i : \u03b9) \u2192 CanonicallyOrderedAddCommMonoid (\u03b1 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Sub (\u03b1 i) inst\u271d : \u2200 (i : \u03b9), OrderedSub (\u03b1 i) f\u271d g\u271d : \u03a0\u2080 (i : \u03b9), \u03b1 i i\u271d : \u03b9 a b : \u03b1 i\u271d f g : \u03a0\u2080 (i : \u03b9), \u03b1 i i : \u03b9 \u22a2 \u2191(f - g) i = (\u2191f - \u2191g) i ** exact tsub_apply f g i ** Qed", + "informal": "" + }, + { + "formal": "Partrec.merge ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 h : x \u2208 f a \u2228 x \u2208 g a \u22a2 x \u2208 k a ** have : (k a).Dom := (K _).2.2 (h.imp Exists.fst Exists.fst) ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 h : x \u2208 f a \u2228 x \u2208 g a this : (k a).Dom \u22a2 x \u2208 k a ** refine' \u27e8this, _\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 h : x \u2208 f a \u2228 x \u2208 g a this : (k a).Dom \u22a2 Part.get (k a) this = x ** cases' h with h h <;> cases' (K _).1 _ \u27e8this, rfl\u27e9 with h' h' ** case inl.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 f a h' : Part.get (k a) this \u2208 f a \u22a2 Part.get (k a) this = x ** exact mem_unique h' h ** case inl.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 f a h' : Part.get (k a) this \u2208 g a \u22a2 Part.get (k a) this = x ** exact (H _ _ h _ h').symm ** case inr.inl \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 g a h' : Part.get (k a) this \u2208 f a \u22a2 Part.get (k a) this = x ** exact H _ _ h' _ h ** case inr.inr \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03c3 : Type u_4 inst\u271d\u00b3 : Primcodable \u03b1 inst\u271d\u00b2 : Primcodable \u03b2 inst\u271d\u00b9 : Primcodable \u03b3 inst\u271d : Primcodable \u03c3 f g : \u03b1 \u2192. \u03c3 hf : Partrec f hg : Partrec g H : \u2200 (a : \u03b1) (x : \u03c3), x \u2208 f a \u2192 \u2200 (y : \u03c3), y \u2208 g a \u2192 x = y k : \u03b1 \u2192. \u03c3 hk : Partrec k K : \u2200 (a : \u03b1), (\u2200 (x : \u03c3), x \u2208 k a \u2192 x \u2208 f a \u2228 x \u2208 g a) \u2227 ((k a).Dom \u2194 (f a).Dom \u2228 (g a).Dom) a : \u03b1 x : \u03c3 this : (k a).Dom h : x \u2208 g a h' : Part.get (k a) this \u2208 g a \u22a2 Part.get (k a) this = x ** exact mem_unique h' h ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.predictablePart_add_ae_eq ** \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 \u22a2 predictablePart (f + g) \u2131 \u03bc n =\u1d50[\u03bc] g n ** filter_upwards [martingalePart_add_ae_eq hf hg hg0 hgint n] with \u03c9 h\u03c9 ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 \u03c9 : \u03a9 h\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 \u22a2 predictablePart (f + g) \u2131 \u03bc n \u03c9 = g n \u03c9 ** rw [\u2190 add_right_inj (f n \u03c9)] ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 \u03c9 : \u03a9 h\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 \u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 + g n \u03c9 ** conv_rhs => rw [\u2190 Pi.add_apply, \u2190 Pi.add_apply, \u2190 martingalePart_add_predictablePart \u2131 \u03bc (f + g)] ** case h \u03a9 : Type u_1 E : Type u_2 m0 : MeasurableSpace \u03a9 \u03bc : Measure \u03a9 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \u211d E inst\u271d\u00b9 : CompleteSpace E f\u271d : \u2115 \u2192 \u03a9 \u2192 E \u2131 : Filtration \u2115 m0 n\u271d : \u2115 inst\u271d : SigmaFiniteFiltration \u03bc \u2131 f g : \u2115 \u2192 \u03a9 \u2192 E hf : Martingale f \u2131 \u03bc hg : Adapted \u2131 fun n => g (n + 1) hg0 : g 0 = 0 hgint : \u2200 (n : \u2115), Integrable (g n) n : \u2115 \u03c9 : \u03a9 h\u03c9 : martingalePart (f + g) \u2131 \u03bc n \u03c9 = f n \u03c9 \u22a2 f n \u03c9 + predictablePart (f + g) \u2131 \u03bc n \u03c9 = (martingalePart (f + g) \u2131 \u03bc + predictablePart (f + g) \u2131 \u03bc) n \u03c9 ** rw [Pi.add_apply, Pi.add_apply, h\u03c9] ** Qed", + "informal": "" + }, + { + "formal": "SeparatingDual.exists_separating_of_ne ** R : Type u_1 V : Type u_2 inst\u271d\u2075 : Ring R inst\u271d\u2074 : AddCommGroup V inst\u271d\u00b3 : TopologicalSpace V inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : Module R V inst\u271d : SeparatingDual R V x y : V h : x \u2260 y \u22a2 \u2203 f, \u2191f x \u2260 \u2191f y ** rcases exists_ne_zero (R := R) (sub_ne_zero_of_ne h) with \u27e8f, hf\u27e9 ** case intro R : Type u_1 V : Type u_2 inst\u271d\u2075 : Ring R inst\u271d\u2074 : AddCommGroup V inst\u271d\u00b3 : TopologicalSpace V inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : Module R V inst\u271d : SeparatingDual R V x y : V h : x \u2260 y f : V \u2192L[R] R hf : \u2191f (x - y) \u2260 0 \u22a2 \u2203 f, \u2191f x \u2260 \u2191f y ** exact \u27e8f, by simpa [sub_ne_zero] using hf\u27e9 ** R : Type u_1 V : Type u_2 inst\u271d\u2075 : Ring R inst\u271d\u2074 : AddCommGroup V inst\u271d\u00b3 : TopologicalSpace V inst\u271d\u00b2 : TopologicalSpace R inst\u271d\u00b9 : Module R V inst\u271d : SeparatingDual R V x y : V h : x \u2260 y f : V \u2192L[R] R hf : \u2191f (x - y) \u2260 0 \u22a2 \u2191f x \u2260 \u2191f y ** simpa [sub_ne_zero] using hf ** Qed", + "informal": "" + }, + { + "formal": "Real.rpow_le_one ** x\u271d y z\u271d x z : \u211d hx1 : 0 \u2264 x hx2 : x \u2264 1 hz : 0 \u2264 z \u22a2 x ^ z \u2264 1 ** rw [\u2190 one_rpow z] ** x\u271d y z\u271d x z : \u211d hx1 : 0 \u2264 x hx2 : x \u2264 1 hz : 0 \u2264 z \u22a2 x ^ z \u2264 1 ^ z ** exact rpow_le_rpow hx1 hx2 hz ** Qed", + "informal": "" + }, + { + "formal": "ConvolutionExistsAt.integrable_swap ** \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b9\u2074 : NormedAddCommGroup E inst\u271d\u00b9\u00b3 : NormedAddCommGroup E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup E'' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : NormedSpace \ud835\udd5c E' inst\u271d\u2077 : NormedSpace \ud835\udd5c E'' inst\u271d\u2076 : NormedSpace \ud835\udd5c F L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u2075 : MeasurableSpace G \u03bc \u03bd : Measure G inst\u271d\u2074 : AddCommGroup G inst\u271d\u00b3 : MeasurableNeg G inst\u271d\u00b2 : IsAddLeftInvariant \u03bc inst\u271d\u00b9 : MeasurableAdd G inst\u271d : IsNegInvariant \u03bc h : ConvolutionExistsAt f g x L \u22a2 Integrable fun t => \u2191(\u2191L (f (x - t))) (g t) ** convert h.comp_sub_left x ** case h.e'_5.h.h.e'_6.h.e'_1 \ud835\udd5c : Type u\ud835\udd5c G : Type uG E : Type uE E' : Type uE' E'' : Type uE'' F : Type uF F' : Type uF' F'' : Type uF'' P : Type uP inst\u271d\u00b9\u2074 : NormedAddCommGroup E inst\u271d\u00b9\u00b3 : NormedAddCommGroup E' inst\u271d\u00b9\u00b2 : NormedAddCommGroup E'' inst\u271d\u00b9\u00b9 : NormedAddCommGroup F f f' : G \u2192 E g g' : G \u2192 E' x x' : G y y' : E inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedSpace \ud835\udd5c E inst\u271d\u2078 : NormedSpace \ud835\udd5c E' inst\u271d\u2077 : NormedSpace \ud835\udd5c E'' inst\u271d\u2076 : NormedSpace \ud835\udd5c F L : E \u2192L[\ud835\udd5c] E' \u2192L[\ud835\udd5c] F inst\u271d\u2075 : MeasurableSpace G \u03bc \u03bd : Measure G inst\u271d\u2074 : AddCommGroup G inst\u271d\u00b3 : MeasurableNeg G inst\u271d\u00b2 : IsAddLeftInvariant \u03bc inst\u271d\u00b9 : MeasurableAdd G inst\u271d : IsNegInvariant \u03bc h : ConvolutionExistsAt f g x L x\u271d : G \u22a2 x\u271d = x - (x - x\u271d) ** simp_rw [sub_sub_self] ** Qed", + "informal": "" + }, + { + "formal": "FormalMultilinearSeries.leftInv_eq_rightInv ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i \u22a2 leftInv p i = leftInv (removeZero p) i ** rw [leftInv_removeZero] ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i \u22a2 leftInv (removeZero p) i = rightInv (removeZero p) i ** apply leftInv_eq_rightInv_aux <;> simp ** case h \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i \u22a2 p (0 + 1) = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i ** exact h ** \ud835\udd5c : Type u_1 inst\u271d\u2074 : NontriviallyNormedField \ud835\udd5c E : Type u_2 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedSpace \ud835\udd5c E F : Type u_3 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \ud835\udd5c F p : FormalMultilinearSeries \ud835\udd5c E F i : E \u2243L[\ud835\udd5c] F h : p 1 = \u2191(LinearIsometryEquiv.symm (continuousMultilinearCurryFin1 \ud835\udd5c E F)) \u2191i \u22a2 rightInv (removeZero p) i = rightInv p i ** rw [rightInv_removeZero] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearMap.integral_comp_comm ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 \u222b (a : \u03b1), \u2191L (\u03c6 a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u03c6 a \u2202\u03bc) ** apply Integrable.induction (P := fun \u03c6 => (\u222b a, L (\u03c6 a) \u2202\u03bc) = L (\u222b a, \u03c6 a \u2202\u03bc)) ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 Integrable fun a => \u03c6 a ** all_goals assumption ** case h_ind \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 \u2200 (c : E) \u2983s : Set \u03b1\u2984, MeasurableSet s \u2192 \u2191\u2191?m.414671 s < \u22a4 \u2192 \u222b (a : \u03b1), \u2191L (indicator s (fun x => c) a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), indicator s (fun x => c) a \u2202\u03bc) ** intro e s s_meas _ ** case h_ind \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 e : E s : Set \u03b1 s_meas : MeasurableSet s a\u271d : \u2191\u2191?m.414671 s < \u22a4 \u22a2 \u222b (a : \u03b1), \u2191L (indicator s (fun x => e) a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), indicator s (fun x => e) a \u2202\u03bc) ** rw [integral_indicator_const e s_meas, \u2190 @smul_one_smul E \u211d \ud835\udd5c _ _ _ _ _ (\u03bc s).toReal e,\n ContinuousLinearMap.map_smul, @smul_one_smul F \u211d \ud835\udd5c _ _ _ _ _ (\u03bc s).toReal (L e), \u2190\n integral_indicator_const (L e) s_meas] ** case h_ind \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 e : E s : Set \u03b1 s_meas : MeasurableSet s a\u271d : \u2191\u2191?m.414671 s < \u22a4 \u22a2 \u222b (a : \u03b1), \u2191L (indicator s (fun x => e) a) \u2202\u03bc = \u222b (a : \u03b1), indicator s (fun x => \u2191L e) a \u2202\u03bc ** congr 1 with a ** case h_ind.e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 e : E s : Set \u03b1 s_meas : MeasurableSet s a\u271d : \u2191\u2191?m.414671 s < \u22a4 a : \u03b1 \u22a2 \u2191L (indicator s (fun x => e) a) = indicator s (fun x => \u2191L e) a ** erw [Set.indicator_comp_of_zero L.map_zero] ** case h_ind.e_f.h \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 e : E s : Set \u03b1 s_meas : MeasurableSet s a\u271d : \u2191\u2191?m.414671 s < \u22a4 a : \u03b1 \u22a2 \u2191L (indicator s (fun x => e) a) = (\u2191L \u2218 indicator s fun x => e) a ** rfl ** case h_add \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984, Disjoint (support f) (support g) \u2192 Integrable f \u2192 Integrable g \u2192 \u222b (a : \u03b1), \u2191L (f a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), f a \u2202\u03bc) \u2192 \u222b (a : \u03b1), \u2191L (g a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), g a \u2202\u03bc) \u2192 \u222b (a : \u03b1), \u2191L ((f + g) a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), (f + g) a \u2202\u03bc) ** intro f g _ f_int g_int hf hg ** case h_add \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 f g : \u03b1 \u2192 E a\u271d : Disjoint (support f) (support g) f_int : Integrable f g_int : Integrable g hf : \u222b (a : \u03b1), \u2191L (f a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), f a \u2202\u03bc) hg : \u222b (a : \u03b1), \u2191L (g a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), g a \u2202\u03bc) \u22a2 \u222b (a : \u03b1), \u2191L ((f + g) a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), (f + g) a \u2202\u03bc) ** simp [L.map_add, integral_add (\u03bc := \u03bc) f_int g_int,\n integral_add (\u03bc := \u03bc) (L.integrable_comp f_int) (L.integrable_comp g_int), hf, hg] ** case h_closed \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 IsClosed {f | \u222b (a : \u03b1), \u2191L (\u2191\u2191f a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), \u2191\u2191f a \u2202\u03bc)} ** exact isClosed_eq L.continuous_integral_comp_L1 (L.continuous.comp continuous_integral) ** case h_ae \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 \u2200 \u2983f g : \u03b1 \u2192 E\u2984, f =\u1d50[\u03bc] g \u2192 Integrable f \u2192 \u222b (a : \u03b1), \u2191L (f a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), f a \u2202\u03bc) \u2192 \u222b (a : \u03b1), \u2191L (g a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), g a \u2202\u03bc) ** intro f g hfg _ hf ** case h_ae \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 f g : \u03b1 \u2192 E hfg : f =\u1d50[\u03bc] g a\u271d : Integrable f hf : \u222b (a : \u03b1), \u2191L (f a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), f a \u2202\u03bc) \u22a2 \u222b (a : \u03b1), \u2191L (g a) \u2202\u03bc = \u2191L (\u222b (a : \u03b1), g a \u2202\u03bc) ** convert hf using 1 <;> clear hf ** case h.e'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 f g : \u03b1 \u2192 E hfg : f =\u1d50[\u03bc] g a\u271d : Integrable f \u22a2 \u222b (a : \u03b1), \u2191L (g a) \u2202\u03bc = \u222b (a : \u03b1), \u2191L (f a) \u2202\u03bc ** exact integral_congr_ae (hfg.fun_comp L).symm ** case h.e'_3 \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 f g : \u03b1 \u2192 E hfg : f =\u1d50[\u03bc] g a\u271d : Integrable f \u22a2 \u2191L (\u222b (a : \u03b1), g a \u2202\u03bc) = \u2191L (\u222b (a : \u03b1), f a \u2202\u03bc) ** rw [integral_congr_ae hfg.symm] ** case a \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2079 : MeasurableSpace \u03b1 \u03b9 : Type u_5 inst\u271d\u2078 : NormedAddCommGroup E \u03bc : Measure \u03b1 \ud835\udd5c : Type u_6 inst\u271d\u2077 : IsROrC \ud835\udd5c inst\u271d\u2076 : NormedSpace \ud835\udd5c E inst\u271d\u2075 : NormedAddCommGroup F inst\u271d\u2074 : NormedSpace \ud835\udd5c F p : \u211d\u22650\u221e inst\u271d\u00b3 : NormedSpace \u211d F inst\u271d\u00b2 : CompleteSpace E inst\u271d\u00b9 : CompleteSpace F inst\u271d : NormedSpace \u211d E L : E \u2192L[\ud835\udd5c] F \u03c6 : \u03b1 \u2192 E \u03c6_int : Integrable \u03c6 \u22a2 Integrable fun a => \u03c6 a ** assumption ** Qed", + "informal": "" + }, + { + "formal": "Complex.ofReal_mul ** r s : \u211d \u22a2 (\u2191(r * s)).re = (\u2191r * \u2191s).re \u2227 (\u2191(r * s)).im = (\u2191r * \u2191s).im ** simp [ofReal'] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.ne_of_oangle_sign_eq_neg_one ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) x y : V h : Real.Angle.sign (oangle o x y) = -1 \u22a2 -1 \u2260 0 ** decide ** Qed", + "informal": "" + }, + { + "formal": "Orientation.angle_eq_iff_oangle_eq_of_sign_eq ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) w x y z : V hw : w \u2260 0 hx : x \u2260 0 hy : y \u2260 0 hz : z \u2260 0 hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z) \u22a2 InnerProductGeometry.angle w x = InnerProductGeometry.angle y z \u2194 oangle o w x = oangle o y z ** refine' \u27e8fun h => o.oangle_eq_of_angle_eq_of_sign_eq h hs, fun h => _\u27e9 ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) w x y z : V hw : w \u2260 0 hx : x \u2260 0 hy : y \u2260 0 hz : z \u2260 0 hs : Real.Angle.sign (oangle o w x) = Real.Angle.sign (oangle o y z) h : oangle o w x = oangle o y z \u22a2 InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ** rw [o.angle_eq_abs_oangle_toReal hw hx, o.angle_eq_abs_oangle_toReal hy hz, h] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.addHaar_image_homothety ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d : Set E x : E r : \u211d s : Set E \u22a2 \u2191\u2191\u03bc (\u2191(AffineMap.homothety x r) '' s) = \u2191\u2191\u03bc ((fun y => y + x) '' (r \u2022 (fun y => y + -x) '' s)) ** simp only [\u2190 image_smul, image_image, \u2190 sub_eq_add_neg] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d : Set E x : E r : \u211d s : Set E \u22a2 \u2191\u2191\u03bc ((fun a => \u2191(AffineMap.homothety x r) a) '' s) = \u2191\u2191\u03bc ((fun x_1 => r \u2022 (x_1 - x) + x) '' s) ** rfl ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d : Set E x : E r : \u211d s : Set E \u22a2 \u2191\u2191\u03bc ((fun y => y + x) '' (r \u2022 (fun y => y + -x) '' s)) = ENNReal.ofReal |r ^ finrank \u211d E| * \u2191\u2191\u03bc s ** simp only [image_add_right, measure_preimage_add_right, addHaar_smul] ** Qed", + "informal": "" + }, + { + "formal": "wbtw_iff_sameRay_vsub ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P \u22a2 Wbtw R x y z \u2194 SameRay R (y -\u1d65 x) (z -\u1d65 y) ** refine' \u27e8Wbtw.sameRay_vsub, fun h => _\u27e9 ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P h : SameRay R (y -\u1d65 x) (z -\u1d65 y) \u22a2 Wbtw R x y z ** rcases h with (h | h | \u27e8r\u2081, r\u2082, hr\u2081, hr\u2082, h\u27e9) ** case inl R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P h : y -\u1d65 x = 0 \u22a2 Wbtw R x y z ** rw [vsub_eq_zero_iff_eq] at h ** case inl R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P h : y = x \u22a2 Wbtw R x y z ** simp [h] ** case inr.inl R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P h : z -\u1d65 y = 0 \u22a2 Wbtw R x y z ** rw [vsub_eq_zero_iff_eq] at h ** case inr.inl R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P h : z = y \u22a2 Wbtw R x y z ** simp [h] ** case inr.inr.intro.intro.intro.intro R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) \u22a2 Wbtw R x y z ** refine'\n \u27e8r\u2082 / (r\u2081 + r\u2082),\n \u27e8div_nonneg hr\u2082.le (add_nonneg hr\u2081.le hr\u2082.le),\n div_le_one_of_le (le_add_of_nonneg_left hr\u2081.le) (add_nonneg hr\u2081.le hr\u2082.le)\u27e9,\n _\u27e9 ** case inr.inr.intro.intro.intro.intro R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) \u22a2 \u2191(lineMap x z) (r\u2082 / (r\u2081 + r\u2082)) = y ** have h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y := by simp [h, hr\u2082.ne'] ** case inr.inr.intro.intro.intro.intro R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y \u22a2 \u2191(lineMap x z) (r\u2082 / (r\u2081 + r\u2082)) = y ** rw [eq_comm] ** case inr.inr.intro.intro.intro.intro R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y \u22a2 y = \u2191(lineMap x z) (r\u2082 / (r\u2081 + r\u2082)) ** simp only [lineMap_apply, h', vadd_vsub_assoc, smul_smul, \u2190 add_smul, eq_vadd_iff_vsub_eq,\n smul_add] ** case inr.inr.intro.intro.intro.intro R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y \u22a2 y -\u1d65 x = (r\u2082 / (r\u2081 + r\u2082) * (r\u2082\u207b\u00b9 * r\u2081) + r\u2082 / (r\u2081 + r\u2082)) \u2022 (y -\u1d65 x) ** convert (one_smul R (y -\u1d65 x)).symm ** case h.e'_3.h.e'_5 R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y \u22a2 r\u2082 / (r\u2081 + r\u2082) * (r\u2082\u207b\u00b9 * r\u2081) + r\u2082 / (r\u2081 + r\u2082) = 1 ** field_simp [(add_pos hr\u2081 hr\u2082).ne', hr\u2082.ne'] ** case h.e'_3.h.e'_5 R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) h' : z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y \u22a2 r\u2082 * r\u2081 * (r\u2081 + r\u2082) + r\u2082 * ((r\u2081 + r\u2082) * r\u2082) = (r\u2081 + r\u2082) * r\u2082 * (r\u2081 + r\u2082) ** ring ** R : Type u_1 V : Type u_2 V' : Type u_3 P : Type u_4 P' : Type u_5 inst\u271d\u00b3 : LinearOrderedField R inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module R V inst\u271d : AddTorsor V P x y z : P r\u2081 r\u2082 : R hr\u2081 : 0 < r\u2081 hr\u2082 : 0 < r\u2082 h : r\u2081 \u2022 (y -\u1d65 x) = r\u2082 \u2022 (z -\u1d65 y) \u22a2 z = r\u2082\u207b\u00b9 \u2022 r\u2081 \u2022 (y -\u1d65 x) +\u1d65 y ** simp [h, hr\u2082.ne'] ** Qed", + "informal": "" + }, + { + "formal": "ZMod.valMinAbs_natCast_eq_self ** n a : \u2115 inst\u271d : NeZero n \u22a2 valMinAbs \u2191a = \u2191a \u2194 a \u2264 n / 2 ** refine' \u27e8fun ha => _, valMinAbs_natCast_of_le_half\u27e9 ** n a : \u2115 inst\u271d : NeZero n ha : valMinAbs \u2191a = \u2191a \u22a2 a \u2264 n / 2 ** rw [\u2190 Int.natAbs_ofNat a, \u2190 ha] ** n a : \u2115 inst\u271d : NeZero n ha : valMinAbs \u2191a = \u2191a \u22a2 Int.natAbs (valMinAbs \u2191a) \u2264 n / 2 ** exact natAbs_valMinAbs_le a ** Qed", + "informal": "" + }, + { + "formal": "Subgroup.index_inf_ne_zero ** G : Type u_1 inst\u271d : Group G H K L : Subgroup G hH : index H \u2260 0 hK : index K \u2260 0 \u22a2 index (H \u2293 K) \u2260 0 ** rw [\u2190 relindex_top_right] at hH hK \u22a2 ** G : Type u_1 inst\u271d : Group G H K L : Subgroup G hH : relindex H \u22a4 \u2260 0 hK : relindex K \u22a4 \u2260 0 \u22a2 relindex (H \u2293 K) \u22a4 \u2260 0 ** exact relindex_inf_ne_zero hH hK ** Qed", + "informal": "" + }, + { + "formal": "NumberField.Units.dirichletUnitTheorem.seq_norm_le ** K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : minkowskiBound K < convexBodyLtFactor K * \u2191B n : \u2115 \u22a2 Int.natAbs (\u2191(Algebra.norm \u2124) \u2191(seq K w\u2081 hB n)) \u2264 B ** cases n with\n| zero =>\n have : 1 \u2264 B := by\n contrapose! hB\n simp only [Nat.lt_one_iff.mp hB, CharP.cast_eq_zero, mul_zero, zero_le]\n simp only [ne_eq, seq, map_one, Int.natAbs_one, this]\n| succ n =>\n rw [\u2190 Nat.cast_le (\u03b1 := \u211a), Int.cast_natAbs, Int.cast_abs, Algebra.coe_norm_int]\n exact (seq_next K w\u2081 hB (seq K w\u2081 hB n).prop).choose_spec.2.2 ** case zero K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : minkowskiBound K < convexBodyLtFactor K * \u2191B \u22a2 Int.natAbs (\u2191(Algebra.norm \u2124) \u2191(seq K w\u2081 hB Nat.zero)) \u2264 B ** have : 1 \u2264 B := by\n contrapose! hB\n simp only [Nat.lt_one_iff.mp hB, CharP.cast_eq_zero, mul_zero, zero_le] ** case zero K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : minkowskiBound K < convexBodyLtFactor K * \u2191B this : 1 \u2264 B \u22a2 Int.natAbs (\u2191(Algebra.norm \u2124) \u2191(seq K w\u2081 hB Nat.zero)) \u2264 B ** simp only [ne_eq, seq, map_one, Int.natAbs_one, this] ** K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : minkowskiBound K < convexBodyLtFactor K * \u2191B \u22a2 1 \u2264 B ** contrapose! hB ** K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : B < 1 \u22a2 convexBodyLtFactor K * \u2191B \u2264 minkowskiBound K ** simp only [Nat.lt_one_iff.mp hB, CharP.cast_eq_zero, mul_zero, zero_le] ** case succ K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : minkowskiBound K < convexBodyLtFactor K * \u2191B n : \u2115 \u22a2 Int.natAbs (\u2191(Algebra.norm \u2124) \u2191(seq K w\u2081 hB (Nat.succ n))) \u2264 B ** rw [\u2190 Nat.cast_le (\u03b1 := \u211a), Int.cast_natAbs, Int.cast_abs, Algebra.coe_norm_int] ** case succ K : Type u_1 inst\u271d\u00b9 : Field K inst\u271d : NumberField K w\u2081 : InfinitePlace K B : \u2115 hB : minkowskiBound K < convexBodyLtFactor K * \u2191B n : \u2115 \u22a2 |\u2191(Algebra.norm \u211a) \u2191\u2191(seq K w\u2081 hB (Nat.succ n))| \u2264 \u2191B ** exact (seq_next K w\u2081 hB (seq K w\u2081 hB n).prop).choose_spec.2.2 ** Qed", + "informal": "" + }, + { + "formal": "Fin.cycleRange_of_lt ** n : \u2115 i j : Fin (Nat.succ n) h : j < i \u22a2 \u2191(cycleRange i) j = j + 1 ** rw [cycleRange_of_le h.le, if_neg h.ne] ** Qed", + "informal": "" + }, + { + "formal": "finrank_span_finset_eq_card ** K : Type u V : Type v inst\u271d\u00b2 : DivisionRing K inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module K V s : Finset V hs : LinearIndependent K Subtype.val \u22a2 finrank K { x // x \u2208 span K \u2191s } = Finset.card s ** convert finrank_span_set_eq_card (s : Set V) hs ** case h.e'_3.h.e'_2 K : Type u V : Type v inst\u271d\u00b2 : DivisionRing K inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module K V s : Finset V hs : LinearIndependent K Subtype.val \u22a2 s = Set.toFinset \u2191s ** ext ** case h.e'_3.h.e'_2.a K : Type u V : Type v inst\u271d\u00b2 : DivisionRing K inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module K V s : Finset V hs : LinearIndependent K Subtype.val a\u271d : V \u22a2 a\u271d \u2208 s \u2194 a\u271d \u2208 Set.toFinset \u2191s ** simp ** Qed", + "informal": "" + }, + { + "formal": "Option.map_map\u2082_antidistrib_left ** \u03b1 : Type u_4 \u03b2 : Type u_5 \u03b3 : Type u_3 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 a : Option \u03b1 b : Option \u03b2 c : Option \u03b3 \u03b4 : Type u_1 \u03b2' : Type u_2 g : \u03b3 \u2192 \u03b4 f' : \u03b2' \u2192 \u03b1 \u2192 \u03b4 g' : \u03b2 \u2192 \u03b2' h_antidistrib : \u2200 (a : \u03b1) (b : \u03b2), g (f a b) = f' (g' b) a \u22a2 Option.map g (map\u2082 f a b) = map\u2082 f' (Option.map g' b) a ** cases a <;> cases b <;> simp [h_antidistrib] ** Qed", + "informal": "" + }, + { + "formal": "TendstoLocallyUniformlyOn.deriv ** E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U \u22a2 TendstoLocallyUniformlyOn (_root_.deriv \u2218 F) (_root_.deriv f) \u03c6 U ** rw [tendstoLocallyUniformlyOn_iff_forall_isCompact hU] ** E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U \u22a2 \u2200 (K : Set \u2102), K \u2286 U \u2192 IsCompact K \u2192 TendstoUniformlyOn (_root_.deriv \u2218 F) (_root_.deriv f) \u03c6 K ** rcases \u03c6.eq_or_neBot with rfl | hne ** case inr E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U hne : NeBot \u03c6 \u22a2 \u2200 (K : Set \u2102), K \u2286 U \u2192 IsCompact K \u2192 TendstoUniformlyOn (_root_.deriv \u2218 F) (_root_.deriv f) \u03c6 K ** rintro K hKU hK ** case inr E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4 : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U hne : NeBot \u03c6 K : Set \u2102 hKU : K \u2286 U hK : IsCompact K \u22a2 TendstoUniformlyOn (_root_.deriv \u2218 F) (_root_.deriv f) \u03c6 K ** obtain \u27e8\u03b4, h\u03b4, hK4, h\u27e9 := exists_cthickening_tendstoUniformlyOn hf hF hK hU hKU ** case inr.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K\u271d : Set \u2102 z : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U hne : NeBot \u03c6 K : Set \u2102 hKU : K \u2286 U hK : IsCompact K \u03b4 : \u211d h\u03b4 : \u03b4 > 0 hK4 : cthickening \u03b4 K \u2286 U h : TendstoUniformlyOn (_root_.deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K \u22a2 TendstoUniformlyOn (_root_.deriv \u2218 F) (_root_.deriv f) \u03c6 K ** refine' h.congr_right fun z hz => cderiv_eq_deriv hU (hf.differentiableOn hF hU) h\u03b4 _ ** case inr.intro.intro.intro E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K\u271d : Set \u2102 z\u271d : \u2102 M r \u03b4\u271d : \u211d \u03c6 : Filter \u03b9 F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hf : TendstoLocallyUniformlyOn F f \u03c6 U hF : \u2200\u1da0 (n : \u03b9) in \u03c6, DifferentiableOn \u2102 (F n) U hU : IsOpen U hne : NeBot \u03c6 K : Set \u2102 hKU : K \u2286 U hK : IsCompact K \u03b4 : \u211d h\u03b4 : \u03b4 > 0 hK4 : cthickening \u03b4 K \u2286 U h : TendstoUniformlyOn (_root_.deriv \u2218 F) (cderiv \u03b4 f) \u03c6 K z : \u2102 hz : z \u2208 K \u22a2 closedBall z \u03b4 \u2286 U ** exact (closedBall_subset_cthickening hz \u03b4).trans hK4 ** case inl E : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : NormedSpace \u2102 E inst\u271d : CompleteSpace E U K : Set \u2102 z : \u2102 M r \u03b4 : \u211d F : \u03b9 \u2192 \u2102 \u2192 E f g : \u2102 \u2192 E hU : IsOpen U hf : TendstoLocallyUniformlyOn F f \u22a5 U hF : \u2200\u1da0 (n : \u03b9) in \u22a5, DifferentiableOn \u2102 (F n) U \u22a2 \u2200 (K : Set \u2102), K \u2286 U \u2192 IsCompact K \u2192 TendstoUniformlyOn (deriv \u2218 F) (deriv f) \u22a5 K ** simp only [TendstoUniformlyOn, eventually_bot, imp_true_iff] ** Qed", + "informal": "" + }, + { + "formal": "Finset.mem_range_iff_mem_finset_range_of_mod_eq ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b2 f\u271d g : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a\u271d : \u03b1 b c : \u03b2 inst\u271d : DecidableEq \u03b1 f : \u2124 \u2192 \u03b1 a : \u03b1 n : \u2115 hn : 0 < n h : \u2200 (i : \u2124), f (i % \u2191n) = f i this : (\u2203 i, f (i % \u2191n) = a) \u2194 \u2203 i, i < n \u2227 f \u2191i = a \u22a2 a \u2208 Set.range f \u2194 a \u2208 image (fun i => f \u2191i) (range n) ** simpa [h] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b2 f\u271d g : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a\u271d : \u03b1 b c : \u03b2 inst\u271d : DecidableEq \u03b1 f : \u2124 \u2192 \u03b1 a : \u03b1 n : \u2115 hn : 0 < n h : \u2200 (i : \u2124), f (i % \u2191n) = f i hn' : 0 < \u2191n x\u271d : \u2203 i, f (i % \u2191n) = a i : \u2124 hi : f (i % \u2191n) = a this : 0 \u2264 i % \u2191n \u22a2 Int.toNat (i % \u2191n) < n \u2227 f \u2191(Int.toNat (i % \u2191n)) = a ** rw [\u2190 Int.ofNat_lt, Int.toNat_of_nonneg this] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b2 f\u271d g : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a\u271d : \u03b1 b c : \u03b2 inst\u271d : DecidableEq \u03b1 f : \u2124 \u2192 \u03b1 a : \u03b1 n : \u2115 hn : 0 < n h : \u2200 (i : \u2124), f (i % \u2191n) = f i hn' : 0 < \u2191n x\u271d : \u2203 i, f (i % \u2191n) = a i : \u2124 hi : f (i % \u2191n) = a this : 0 \u2264 i % \u2191n \u22a2 i % \u2191n < \u2191n \u2227 f (i % \u2191n) = a ** exact \u27e8Int.emod_lt_of_pos i hn', hi\u27e9 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 inst\u271d\u00b9 : DecidableEq \u03b2 f\u271d g : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 a\u271d : \u03b1 b c : \u03b2 inst\u271d : DecidableEq \u03b1 f : \u2124 \u2192 \u03b1 a : \u03b1 n : \u2115 hn : 0 < n h : \u2200 (i : \u2124), f (i % \u2191n) = f i hn' : 0 < \u2191n x\u271d : \u2203 i, i < n \u2227 f \u2191i = a i : \u2115 hi : i < n ha : f \u2191i = a \u22a2 f (\u2191i % \u2191n) = a ** rw [Int.emod_eq_of_lt (Int.ofNat_zero_le _) (Int.ofNat_lt_ofNat_of_lt hi), ha] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.eval_zero_map ** R : Type u S : Type v T : Type w \u03b9 : Type y a b : R m n : \u2115 inst\u271d\u00b9 : Semiring R p\u271d q r : R[X] inst\u271d : Semiring S f\u271d f : R \u2192+* S p : R[X] \u22a2 eval 0 (map f p) = \u2191f (eval 0 p) ** simp [\u2190 coeff_zero_eq_eval_zero] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.AEStronglyMeasurable.nullMeasurableSet_le ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : Preorder \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f g : \u03b1 \u2192 \u03b2 hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc \u22a2 NullMeasurableSet {a | f a \u2264 g a} ** apply\n (hf.stronglyMeasurable_mk.measurableSet_le hg.stronglyMeasurable_mk).nullMeasurableSet.congr ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : Preorder \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f g : \u03b1 \u2192 \u03b2 hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc \u22a2 {a | AEStronglyMeasurable.mk f hf a \u2264 AEStronglyMeasurable.mk g hg a} =\u1da0[ae \u03bc] {a | f a \u2264 g a} ** filter_upwards [hf.ae_eq_mk, hg.ae_eq_mk] with x hfx hgx ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : Preorder \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f g : \u03b1 \u2192 \u03b2 hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc x : \u03b1 hfx : f x = AEStronglyMeasurable.mk f hf x hgx : g x = AEStronglyMeasurable.mk g hg x \u22a2 setOf (fun a => AEStronglyMeasurable.mk f hf a \u2264 AEStronglyMeasurable.mk g hg a) x = setOf (fun a => f a \u2264 g a) x ** change (hf.mk f x \u2264 hg.mk g x) = (f x \u2264 g x) ** case h \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b9 : Type u_4 inst\u271d\u2075 : Countable \u03b9 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d\u00b2 : Preorder \u03b2 inst\u271d\u00b9 : OrderClosedTopology \u03b2 inst\u271d : PseudoMetrizableSpace \u03b2 f g : \u03b1 \u2192 \u03b2 hf : AEStronglyMeasurable f \u03bc hg : AEStronglyMeasurable g \u03bc x : \u03b1 hfx : f x = AEStronglyMeasurable.mk f hf x hgx : g x = AEStronglyMeasurable.mk g hg x \u22a2 (AEStronglyMeasurable.mk f hf x \u2264 AEStronglyMeasurable.mk g hg x) = (f x \u2264 g x) ** simp only [hfx, hgx] ** Qed", + "informal": "" + }, + { + "formal": "Finset.prod_dite_of_true ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 p : \u03b1 \u2192 Prop hp : DecidablePred p h : \u2200 (x : \u03b1), x \u2208 s \u2192 p x f : (x : \u03b1) \u2192 p x \u2192 \u03b2 g : (x : \u03b1) \u2192 \u00acp x \u2192 \u03b2 x : \u03b1 hx : x \u2208 s \u22a2 (fun x hx => { val := x, property := hx }) x hx \u2208 univ ** simp ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a\u271d : \u03b1 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 p : \u03b1 \u2192 Prop hp : DecidablePred p h : \u2200 (x : \u03b1), x \u2208 s \u2192 p x f : (x : \u03b1) \u2192 p x \u2192 \u03b2 g : (x : \u03b1) \u2192 \u00acp x \u2192 \u03b2 a : \u03b1 ha : a \u2208 s \u22a2 (if hx : p a then f a hx else g a hx) = f \u2191((fun x hx => { val := x, property := hx }) a ha) (_ : p \u2191((fun x hx => { val := x, property := hx }) a ha)) ** dsimp ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a\u271d : \u03b1 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 p : \u03b1 \u2192 Prop hp : DecidablePred p h : \u2200 (x : \u03b1), x \u2208 s \u2192 p x f : (x : \u03b1) \u2192 p x \u2192 \u03b2 g : (x : \u03b1) \u2192 \u00acp x \u2192 \u03b2 a : \u03b1 ha : a \u2208 s \u22a2 (if hx : p a then f a hx else g a hx) = f a (_ : p a) ** rw [dif_pos] ** \u03b9 : Type u_1 \u03b2 : Type u \u03b1 : Type v \u03b3 : Type w s s\u2081 s\u2082 : Finset \u03b1 a : \u03b1 f\u271d g\u271d : \u03b1 \u2192 \u03b2 inst\u271d : CommMonoid \u03b2 p : \u03b1 \u2192 Prop hp : DecidablePred p h : \u2200 (x : \u03b1), x \u2208 s \u2192 p x f : (x : \u03b1) \u2192 p x \u2192 \u03b2 g : (x : \u03b1) \u2192 \u00acp x \u2192 \u03b2 b : { x // x \u2208 s } _hb : b \u2208 univ \u22a2 b = (fun x hx => { val := x, property := hx }) \u2191b (_ : \u2191b \u2208 s) ** simp ** Qed", + "informal": "" + }, + { + "formal": "interior_prod_eq ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type u_1 \u03b4 : Type u_2 \u03b5 : Type u_3 \u03b6 : Type u_4 inst\u271d\u2075 : TopologicalSpace \u03b1 inst\u271d\u2074 : TopologicalSpace \u03b2 inst\u271d\u00b3 : TopologicalSpace \u03b3 inst\u271d\u00b2 : TopologicalSpace \u03b4 inst\u271d\u00b9 : TopologicalSpace \u03b5 inst\u271d : TopologicalSpace \u03b6 s : Set \u03b1 t : Set \u03b2 x\u271d : \u03b1 \u00d7 \u03b2 a : \u03b1 b : \u03b2 \u22a2 (a, b) \u2208 interior (s \u00d7\u02e2 t) \u2194 (a, b) \u2208 interior s \u00d7\u02e2 interior t ** simp only [mem_interior_iff_mem_nhds, mem_prod, prod_mem_nhds_iff] ** Qed", + "informal": "" + }, + { + "formal": "TopCat.Presheaf.SheafConditionEqualizerProducts.w ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasProducts C X : TopCat F : Presheaf C X \u03b9 : Type v' U : \u03b9 \u2192 Opens \u2191X \u22a2 res F U \u226b leftRes F U = res F U \u226b rightRes F U ** dsimp [res, leftRes, rightRes] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasProducts C X : TopCat F : Presheaf C X \u03b9 : Type v' U : \u03b9 \u2192 Opens \u2191X \u22a2 ((Pi.lift fun i => F.map (leSupr U i).op) \u226b Pi.lift fun p => Pi.\u03c0 (fun i => F.obj (op (U i))) p.1 \u226b F.map (infLELeft (U p.1) (U p.2)).op) = (Pi.lift fun i => F.map (leSupr U i).op) \u226b Pi.lift fun p => Pi.\u03c0 (fun i => F.obj (op (U i))) p.2 \u226b F.map (infLERight (U p.1) (U p.2)).op ** refine limit.hom_ext (fun _ => ?_) ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasProducts C X : TopCat F : Presheaf C X \u03b9 : Type v' U : \u03b9 \u2192 Opens \u2191X x\u271d : Discrete (\u03b9 \u00d7 \u03b9) \u22a2 ((Pi.lift fun i => F.map (leSupr U i).op) \u226b Pi.lift fun p => Pi.\u03c0 (fun i => F.obj (op (U i))) p.1 \u226b F.map (infLELeft (U p.1) (U p.2)).op) \u226b limit.\u03c0 (Discrete.functor fun p => F.obj (op (U p.1 \u2293 U p.2))) x\u271d = ((Pi.lift fun i => F.map (leSupr U i).op) \u226b Pi.lift fun p => Pi.\u03c0 (fun i => F.obj (op (U i))) p.2 \u226b F.map (infLERight (U p.1) (U p.2)).op) \u226b limit.\u03c0 (Discrete.functor fun p => F.obj (op (U p.1 \u2293 U p.2))) x\u271d ** simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, Fan.mk_\u03c0_app, Category.assoc] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasProducts C X : TopCat F : Presheaf C X \u03b9 : Type v' U : \u03b9 \u2192 Opens \u2191X x\u271d : Discrete (\u03b9 \u00d7 \u03b9) \u22a2 F.map (leSupr U x\u271d.as.1).op \u226b F.map (infLELeft (U x\u271d.as.1) (U x\u271d.as.2)).op = F.map (leSupr U x\u271d.as.2).op \u226b F.map (infLERight (U x\u271d.as.1) (U x\u271d.as.2)).op ** rw [\u2190 F.map_comp] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasProducts C X : TopCat F : Presheaf C X \u03b9 : Type v' U : \u03b9 \u2192 Opens \u2191X x\u271d : Discrete (\u03b9 \u00d7 \u03b9) \u22a2 F.map ((leSupr U x\u271d.as.1).op \u226b (infLELeft (U x\u271d.as.1) (U x\u271d.as.2)).op) = F.map (leSupr U x\u271d.as.2).op \u226b F.map (infLERight (U x\u271d.as.1) (U x\u271d.as.2)).op ** rw [\u2190 F.map_comp] ** C : Type u inst\u271d\u00b9 : Category.{v, u} C inst\u271d : HasProducts C X : TopCat F : Presheaf C X \u03b9 : Type v' U : \u03b9 \u2192 Opens \u2191X x\u271d : Discrete (\u03b9 \u00d7 \u03b9) \u22a2 F.map ((leSupr U x\u271d.as.1).op \u226b (infLELeft (U x\u271d.as.1) (U x\u271d.as.2)).op) = F.map ((leSupr U x\u271d.as.2).op \u226b (infLERight (U x\u271d.as.1) (U x\u271d.as.2)).op) ** congr 1 ** Qed", + "informal": "" + }, + { + "formal": "integrableOn_Icc_iff_integrableOn_Ico ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : PartialOrder \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 \u2191\u2191\u03bc {b} \u2260 \u22a4 ** rw [measure_singleton] ** \u03b1 : Type u_1 \u03b2 : Type u_2 E : Type u_3 F : Type u_4 inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : PartialOrder \u03b1 inst\u271d\u00b9 : MeasurableSingletonClass \u03b1 f : \u03b1 \u2192 E \u03bc : Measure \u03b1 a b : \u03b1 inst\u271d : NoAtoms \u03bc \u22a2 0 \u2260 \u22a4 ** exact ENNReal.zero_ne_top ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.Sphere.oangle_eq_pi_sub_two_zsmul_oangle_center_right ** V : Type u_1 P : Type u_2 inst\u271d\u2074 : NormedAddCommGroup V inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : MetricSpace P inst\u271d\u00b9 : NormedAddTorsor V P hd2 : Fact (finrank \u211d V = 2) inst\u271d : Module.Oriented \u211d V (Fin 2) s : Sphere P p\u2081 p\u2082 : P hp\u2081 : p\u2081 \u2208 s hp\u2082 : p\u2082 \u2208 s h : p\u2081 \u2260 p\u2082 \u22a2 \u2221 p\u2081 s.center p\u2082 = \u2191\u03c0 - 2 \u2022 \u2221 p\u2082 p\u2081 s.center ** rw [oangle_eq_pi_sub_two_zsmul_oangle_center_left hp\u2081 hp\u2082 h,\n oangle_eq_oangle_of_dist_eq (dist_center_eq_dist_center_of_mem_sphere' hp\u2082 hp\u2081)] ** Qed", + "informal": "" + }, + { + "formal": "ONote.fundamentalSequence_has_prop ** o : ONote \u22a2 FundamentalSequenceProp o (fundamentalSequence o) ** induction' o with a m b iha ihb ** case oadd a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) ihb : FundamentalSequenceProp b (fundamentalSequence b) \u22a2 FundamentalSequenceProp (oadd a m b) (fundamentalSequence (oadd a m b)) ** rw [fundamentalSequence] ** case oadd a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) ihb : FundamentalSequenceProp b (fundamentalSequence b) \u22a2 FundamentalSequenceProp (oadd a m b) (match fundamentalSequence b with | Sum.inr f => Sum.inr fun i => oadd a m (f i) | Sum.inl (some b') => Sum.inl (some (oadd a m b')) | Sum.inl none => match fundamentalSequence a, PNat.natPred m with | Sum.inl none, 0 => Sum.inl (some zero) | Sum.inl none, Nat.succ m => Sum.inl (some (oadd zero (Nat.succPNat m) zero)) | Sum.inl (some a'), 0 => Sum.inr fun i => oadd a' (Nat.succPNat i) zero | Sum.inl (some a'), Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd a' (Nat.succPNat i) zero) | Sum.inr f, 0 => Sum.inr fun i => oadd (f i) 1 zero | Sum.inr f, Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd (f i) 1 zero)) ** rcases e : b.fundamentalSequence with (\u27e8_ | b'\u27e9 | f) <;>\n simp only [FundamentalSequenceProp] <;>\n rw [e, FundamentalSequenceProp] at ihb ** case zero \u22a2 FundamentalSequenceProp zero (fundamentalSequence zero) ** exact rfl ** case oadd.inl.none a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) ihb : b = 0 e : fundamentalSequence b = Sum.inl none \u22a2 FundamentalSequenceProp (oadd a m b) (match fundamentalSequence a, PNat.natPred m with | Sum.inl none, 0 => Sum.inl (some zero) | Sum.inl none, Nat.succ m => Sum.inl (some (oadd zero (Nat.succPNat m) zero)) | Sum.inl (some a'), 0 => Sum.inr fun i => oadd a' (Nat.succPNat i) zero | Sum.inl (some a'), Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd a' (Nat.succPNat i) zero) | Sum.inr f, 0 => Sum.inr fun i => oadd (f i) 1 zero | Sum.inr f, Nat.succ m => Sum.inr fun i => oadd a (Nat.succPNat m) (oadd (f i) 1 zero)) ** rcases e : a.fundamentalSequence with (\u27e8_ | a'\u27e9 | f) <;> cases' e' : m.natPred with m' <;>\n simp only [FundamentalSequenceProp] <;>\n rw [e, FundamentalSequenceProp] at iha <;>\n (try rw [show m = 1 by\n have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm]) <;>\n (try rw [show m = m'.succ.succPNat by\n rw [\u2190 e', \u2190 PNat.coe_inj, Nat.succPNat_coe, \u2190 Nat.add_one, PNat.natPred_add_one]]) <;>\n simp only [repr, iha, ihb, opow_lt_opow_iff_right one_lt_omega, add_lt_add_iff_left, add_zero,\n eq_self_iff_true, lt_add_iff_pos_right, lt_def, mul_one, Nat.cast_zero, Nat.cast_succ,\n Nat.succPNat_coe, opow_succ, opow_zero, mul_add_one, PNat.one_coe, succ_zero,\n true_and_iff, _root_.zero_add, zero_def] ** case oadd.inl.none.inr.zero a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero \u22a2 IsLimit (repr (oadd a m b)) \u2227 (\u2200 (i : \u2115), oadd (f i) 1 zero < oadd (f (i + 1)) 1 zero \u2227 oadd (f i) 1 zero < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd (f i) 1 zero))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) \u2192 \u2203 i, a_1 < repr (oadd (f i) 1 zero) ** try rw [show m = 1 by\n have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm] ** case oadd.inl.none.inr.succ a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 IsLimit (repr (oadd a m b)) \u2227 (\u2200 (i : \u2115), oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a (Nat.succPNat m') (oadd (f (i + 1)) 1 zero) \u2227 oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) \u2192 \u2203 i, a_1 < repr (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)) ** rw [show m = 1 by\n have := PNat.natPred_add_one m; rw [e'] at this; exact PNat.coe_inj.1 this.symm] ** a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 m = 1 ** have := PNat.natPred_add_one m ** a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : PNat.natPred m + 1 = \u2191m \u22a2 m = 1 ** rw [e'] at this ** a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero this : Nat.zero + 1 = \u2191m \u22a2 m = 1 ** exact PNat.coe_inj.1 this.symm ** case oadd.inl.none.inr.succ a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 IsLimit (repr (oadd a m b)) \u2227 (\u2200 (i : \u2115), oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a (Nat.succPNat m') (oadd (f (i + 1)) 1 zero) \u2227 oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) \u2192 \u2203 i, a_1 < repr (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)) ** try rw [show m = m'.succ.succPNat by\n rw [\u2190 e', \u2190 PNat.coe_inj, Nat.succPNat_coe, \u2190 Nat.add_one, PNat.natPred_add_one]] ** case oadd.inl.none.inr.succ a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 IsLimit (repr (oadd a m b)) \u2227 (\u2200 (i : \u2115), oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a (Nat.succPNat m') (oadd (f (i + 1)) 1 zero) \u2227 oadd a (Nat.succPNat m') (oadd (f i) 1 zero) < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) \u2192 \u2203 i, a_1 < repr (oadd a (Nat.succPNat m') (oadd (f i) 1 zero)) ** rw [show m = m'.succ.succPNat by\n rw [\u2190 e', \u2190 PNat.coe_inj, Nat.succPNat_coe, \u2190 Nat.add_one, PNat.natPred_add_one]] ** a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 m = Nat.succPNat (Nat.succ m') ** rw [\u2190 e', \u2190 PNat.coe_inj, Nat.succPNat_coe, \u2190 Nat.add_one, PNat.natPred_add_one] ** case oadd.inl.none.inl.none.succ a : ONote m : \u2115+ b : ONote iha : a = 0 ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none e : fundamentalSequence a = Sum.inl none m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 1 * \u2191m' + 1 + 1 = succ (1 * \u2191m' + 1) \u2227 (NF (oadd 0 (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd 0 (Nat.succPNat m') 0)) ** exact \u27e8rfl, inferInstance\u27e9 ** case oadd.inl.none.inl.some.zero a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero \u22a2 IsLimit (\u03c9 ^ repr a' * \u03c9) \u2227 (\u2200 (i : \u2115), 0 < \u03c9 ^ repr a' \u2227 \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' < \u03c9 ^ repr a' * \u03c9 \u2227 (NF (oadd a 1 0) \u2192 NF (oadd a' (Nat.succPNat i) 0))) \u2227 \u2200 (a : Ordinal.{0}), a < \u03c9 ^ repr a' * \u03c9 \u2192 \u2203 i, a < \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' ** have := opow_pos (repr a') omega_pos ** case oadd.inl.none.inl.some.zero a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero this : 0 < \u03c9 ^ repr a' \u22a2 IsLimit (\u03c9 ^ repr a' * \u03c9) \u2227 (\u2200 (i : \u2115), 0 < \u03c9 ^ repr a' \u2227 \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' < \u03c9 ^ repr a' * \u03c9 \u2227 (NF (oadd a 1 0) \u2192 NF (oadd a' (Nat.succPNat i) 0))) \u2227 \u2200 (a : Ordinal.{0}), a < \u03c9 ^ repr a' * \u03c9 \u2192 \u2203 i, a < \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' ** refine'\n \u27e8mul_isLimit this omega_isLimit, fun i =>\n \u27e8this, _, fun H => @NF.oadd_zero _ _ (iha.2 H.fst)\u27e9, exists_lt_mul_omega'\u27e9 ** case oadd.inl.none.inl.some.zero a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero this : 0 < \u03c9 ^ repr a' i : \u2115 \u22a2 \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' < \u03c9 ^ repr a' * \u03c9 ** rw [\u2190 mul_succ, \u2190 nat_cast_succ, Ordinal.mul_lt_mul_iff_left this] ** case oadd.inl.none.inl.some.zero a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') e' : PNat.natPred m = Nat.zero this : 0 < \u03c9 ^ repr a' i : \u2115 \u22a2 \u2191(Nat.succ i) < \u03c9 ** apply nat_lt_omega ** case oadd.inl.none.inl.some.succ a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 IsLimit (\u03c9 ^ repr a' * \u03c9 * \u2191m' + \u03c9 ^ repr a' * \u03c9 + \u03c9 ^ repr a' * \u03c9) \u2227 (\u2200 (i : \u2115), 0 < \u03c9 ^ repr a' \u2227 \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' < \u03c9 ^ repr a' * \u03c9 \u2227 (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd a' (Nat.succPNat i) 0)))) \u2227 \u2200 (a : Ordinal.{0}), a < \u03c9 ^ repr a' * \u03c9 * \u2191m' + \u03c9 ^ repr a' * \u03c9 + \u03c9 ^ repr a' * \u03c9 \u2192 \u2203 i, a < \u03c9 ^ repr a' * \u03c9 * \u2191m' + \u03c9 ^ repr a' * \u03c9 + (\u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a') ** have := opow_pos (repr a') omega_pos ** case oadd.inl.none.inl.some.succ a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : 0 < \u03c9 ^ repr a' \u22a2 IsLimit (\u03c9 ^ repr a' * \u03c9 * \u2191m' + \u03c9 ^ repr a' * \u03c9 + \u03c9 ^ repr a' * \u03c9) \u2227 (\u2200 (i : \u2115), 0 < \u03c9 ^ repr a' \u2227 \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' < \u03c9 ^ repr a' * \u03c9 \u2227 (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd a' (Nat.succPNat i) 0)))) \u2227 \u2200 (a : Ordinal.{0}), a < \u03c9 ^ repr a' * \u03c9 * \u2191m' + \u03c9 ^ repr a' * \u03c9 + \u03c9 ^ repr a' * \u03c9 \u2192 \u2203 i, a < \u03c9 ^ repr a' * \u03c9 * \u2191m' + \u03c9 ^ repr a' * \u03c9 + (\u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a') ** refine'\n \u27e8add_isLimit _ (mul_isLimit this omega_isLimit), fun i => \u27e8this, _, _\u27e9,\n exists_lt_add exists_lt_mul_omega'\u27e9 ** case oadd.inl.none.inl.some.succ.refine'_1 a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : 0 < \u03c9 ^ repr a' i : \u2115 \u22a2 \u03c9 ^ repr a' * \u2191i + \u03c9 ^ repr a' < \u03c9 ^ repr a' * \u03c9 ** rw [\u2190 mul_succ, \u2190 nat_cast_succ, Ordinal.mul_lt_mul_iff_left this] ** case oadd.inl.none.inl.some.succ.refine'_1 a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : 0 < \u03c9 ^ repr a' i : \u2115 \u22a2 \u2191(Nat.succ i) < \u03c9 ** apply nat_lt_omega ** case oadd.inl.none.inl.some.succ.refine'_2 a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : 0 < \u03c9 ^ repr a' i : \u2115 \u22a2 NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd a' (Nat.succPNat i) 0)) ** refine' fun H => H.fst.oadd _ (NF.below_of_lt' _ (@NF.oadd_zero _ _ (iha.2 H.fst))) ** case oadd.inl.none.inl.some.succ.refine'_2 a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : 0 < \u03c9 ^ repr a' i : \u2115 H : NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u22a2 repr (oadd a' (Nat.succPNat i) 0) < \u03c9 ^ repr a ** rw [repr, \u2190 zero_def, repr, add_zero, iha.1, opow_succ, Ordinal.mul_lt_mul_iff_left this] ** case oadd.inl.none.inl.some.succ.refine'_2 a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none a' : ONote iha : repr a = succ (repr a') \u2227 (NF a \u2192 NF a') e : fundamentalSequence a = Sum.inl (some a') m' : \u2115 e' : PNat.natPred m = Nat.succ m' this : 0 < \u03c9 ^ repr a' i : \u2115 H : NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u22a2 \u2191\u2191(Nat.succPNat i) < \u03c9 ** apply nat_lt_omega ** case oadd.inl.none.inr.zero a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero \u22a2 IsLimit (\u03c9 ^ repr a) \u2227 (\u2200 (i : \u2115), repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a 1 0) \u2192 NF (oadd (f i) 1 0))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < \u03c9 ^ repr a \u2192 \u2203 i, a_1 < \u03c9 ^ repr (f i) ** rcases iha with \u27e8h1, h2, h3\u27e9 ** case oadd.inl.none.inr.zero.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) \u22a2 IsLimit (\u03c9 ^ repr a) \u2227 (\u2200 (i : \u2115), repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a 1 0) \u2192 NF (oadd (f i) 1 0))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < \u03c9 ^ repr a \u2192 \u2203 i, a_1 < \u03c9 ^ repr (f i) ** refine' \u27e8opow_isLimit one_lt_omega h1, fun i => _, exists_lt_omega_opow' one_lt_omega h1 h3\u27e9 ** case oadd.inl.none.inr.zero.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) i : \u2115 \u22a2 repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a 1 0) \u2192 NF (oadd (f i) 1 0)) ** obtain \u27e8h4, h5, h6\u27e9 := h2 i ** case oadd.inl.none.inr.zero.intro.intro.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f e' : PNat.natPred m = Nat.zero h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) i : \u2115 h4 : f i < f (i + 1) h5 : f i < a h6 : NF a \u2192 NF (f i) \u22a2 repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a 1 0) \u2192 NF (oadd (f i) 1 0)) ** exact \u27e8h4, h5, fun H => @NF.oadd_zero _ _ (h6 H.fst)\u27e9 ** case oadd.inl.none.inr.succ a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote iha : IsLimit (repr a) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' \u22a2 IsLimit (\u03c9 ^ repr a * \u2191m' + \u03c9 ^ repr a + \u03c9 ^ repr a) \u2227 (\u2200 (i : \u2115), repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < \u03c9 ^ repr a * \u2191m' + \u03c9 ^ repr a + \u03c9 ^ repr a \u2192 \u2203 i, a_1 < \u03c9 ^ repr a * \u2191m' + \u03c9 ^ repr a + \u03c9 ^ repr (f i) ** rcases iha with \u27e8h1, h2, h3\u27e9 ** case oadd.inl.none.inr.succ.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) \u22a2 IsLimit (\u03c9 ^ repr a * \u2191m' + \u03c9 ^ repr a + \u03c9 ^ repr a) \u2227 (\u2200 (i : \u2115), repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < \u03c9 ^ repr a * \u2191m' + \u03c9 ^ repr a + \u03c9 ^ repr a \u2192 \u2203 i, a_1 < \u03c9 ^ repr a * \u2191m' + \u03c9 ^ repr a + \u03c9 ^ repr (f i) ** refine'\n \u27e8add_isLimit _ (opow_isLimit one_lt_omega h1), fun i => _,\n exists_lt_add (exists_lt_omega_opow' one_lt_omega h1 h3)\u27e9 ** case oadd.inl.none.inr.succ.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) i : \u2115 \u22a2 repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0))) ** obtain \u27e8h4, h5, h6\u27e9 := h2 i ** case oadd.inl.none.inr.succ.intro.intro.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) i : \u2115 h4 : f i < f (i + 1) h5 : f i < a h6 : NF a \u2192 NF (f i) \u22a2 repr (f i) < repr (f (i + 1)) \u2227 repr (f i) < repr a \u2227 (NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u2192 NF (oadd a (Nat.succPNat m') (oadd (f i) 1 0))) ** refine' \u27e8h4, h5, fun H => H.fst.oadd _ (NF.below_of_lt' _ (@NF.oadd_zero _ _ (h6 H.fst)))\u27e9 ** case oadd.inl.none.inr.succ.intro.intro.intro.intro a : ONote m : \u2115+ b : ONote ihb : b = 0 e\u271d : fundamentalSequence b = Sum.inl none f : \u2115 \u2192 ONote e : fundamentalSequence a = Sum.inr f m' : \u2115 e' : PNat.natPred m = Nat.succ m' h1 : IsLimit (repr a) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < a \u2227 (NF a \u2192 NF (f i)) h3 : \u2200 (a_1 : Ordinal.{0}), a_1 < repr a \u2192 \u2203 i, a_1 < repr (f i) i : \u2115 h4 : f i < f (i + 1) h5 : f i < a h6 : NF a \u2192 NF (f i) H : NF (oadd a (Nat.succPNat (Nat.succ m')) 0) \u22a2 repr (oadd (f i) 1 0) < \u03c9 ^ repr a ** rwa [repr, \u2190 zero_def, repr, add_zero, PNat.one_coe, Nat.cast_one, mul_one,\n opow_lt_opow_iff_right one_lt_omega] ** case oadd.inl.some a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') \u2227 (NF b \u2192 NF b') e : fundamentalSequence b = Sum.inl (some b') \u22a2 repr (oadd a m b) = succ (repr (oadd a m b')) \u2227 (NF (oadd a m b) \u2192 NF (oadd a m b')) ** refine'\n \u27e8by rw [repr, ihb.1, add_succ, repr], fun H => H.fst.oadd _ (NF.below_of_lt' _ (ihb.2 H.snd))\u27e9 ** case oadd.inl.some a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') \u2227 (NF b \u2192 NF b') e : fundamentalSequence b = Sum.inl (some b') H : NF (oadd a m b) \u22a2 repr b' < \u03c9 ^ repr a ** have := H.snd'.repr_lt ** case oadd.inl.some a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') \u2227 (NF b \u2192 NF b') e : fundamentalSequence b = Sum.inl (some b') H : NF (oadd a m b) this : repr b < \u03c9 ^ repr a \u22a2 repr b' < \u03c9 ^ repr a ** rw [ihb.1] at this ** case oadd.inl.some a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') \u2227 (NF b \u2192 NF b') e : fundamentalSequence b = Sum.inl (some b') H : NF (oadd a m b) this : succ (repr b') < \u03c9 ^ repr a \u22a2 repr b' < \u03c9 ^ repr a ** exact (lt_succ _).trans this ** a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) b' : ONote ihb : repr b = succ (repr b') \u2227 (NF b \u2192 NF b') e : fundamentalSequence b = Sum.inl (some b') \u22a2 repr (oadd a m b) = succ (repr (oadd a m b')) ** rw [repr, ihb.1, add_succ, repr] ** case oadd.inr a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) f : \u2115 \u2192 ONote ihb : IsLimit (repr b) \u2227 (\u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < b \u2227 (NF b \u2192 NF (f i))) \u2227 \u2200 (a : Ordinal.{0}), a < repr b \u2192 \u2203 i, a < repr (f i) e : fundamentalSequence b = Sum.inr f \u22a2 IsLimit (repr (oadd a m b)) \u2227 (\u2200 (i : \u2115), oadd a m (f i) < oadd a m (f (i + 1)) \u2227 oadd a m (f i) < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd a m (f i)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) \u2192 \u2203 i, a_1 < repr (oadd a m (f i)) ** rcases ihb with \u27e8h1, h2, h3\u27e9 ** case oadd.inr.intro.intro a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) f : \u2115 \u2192 ONote e : fundamentalSequence b = Sum.inr f h1 : IsLimit (repr b) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < b \u2227 (NF b \u2192 NF (f i)) h3 : \u2200 (a : Ordinal.{0}), a < repr b \u2192 \u2203 i, a < repr (f i) \u22a2 IsLimit (repr (oadd a m b)) \u2227 (\u2200 (i : \u2115), oadd a m (f i) < oadd a m (f (i + 1)) \u2227 oadd a m (f i) < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd a m (f i)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < repr (oadd a m b) \u2192 \u2203 i, a_1 < repr (oadd a m (f i)) ** simp only [repr] ** case oadd.inr.intro.intro a : ONote m : \u2115+ b : ONote iha : FundamentalSequenceProp a (fundamentalSequence a) f : \u2115 \u2192 ONote e : fundamentalSequence b = Sum.inr f h1 : IsLimit (repr b) h2 : \u2200 (i : \u2115), f i < f (i + 1) \u2227 f i < b \u2227 (NF b \u2192 NF (f i)) h3 : \u2200 (a : Ordinal.{0}), a < repr b \u2192 \u2203 i, a < repr (f i) \u22a2 IsLimit (\u03c9 ^ repr a * \u2191\u2191m + repr b) \u2227 (\u2200 (i : \u2115), oadd a m (f i) < oadd a m (f (i + 1)) \u2227 oadd a m (f i) < oadd a m b \u2227 (NF (oadd a m b) \u2192 NF (oadd a m (f i)))) \u2227 \u2200 (a_1 : Ordinal.{0}), a_1 < \u03c9 ^ repr a * \u2191\u2191m + repr b \u2192 \u2203 i, a_1 < \u03c9 ^ repr a * \u2191\u2191m + repr (f i) ** exact\n \u27e8Ordinal.add_isLimit _ h1, fun i =>\n \u27e8oadd_lt_oadd_3 (h2 i).1, oadd_lt_oadd_3 (h2 i).2.1, fun H =>\n H.fst.oadd _ (NF.below_of_lt' (lt_trans (h2 i).2.1 H.snd'.repr_lt) ((h2 i).2.2 H.snd))\u27e9,\n exists_lt_add h3\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Ideal.mem_span_singleton_mul ** R : Type u \u03b9 : Type u_1 inst\u271d : CommSemiring R I\u271d J K L : Ideal R x y : R I : Ideal R \u22a2 x \u2208 span {y} * I \u2194 \u2203 z, z \u2208 I \u2227 y * z = x ** simp only [mul_comm, mem_mul_span_singleton] ** Qed", + "informal": "" + }, + { + "formal": "rat_inv_continuous_lemma ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K \u22a2 \u2203 \u03b4, \u03b4 > 0 \u2227 \u2200 {a b : \u03b2}, K \u2264 abv a \u2192 K \u2264 abv b \u2192 abv (a - b) < \u03b4 \u2192 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5 ** refine' \u27e8K * \u03b5 * K, mul_pos (mul_pos K0 \u03b50) K0, fun {a b} ha hb h => _\u27e9 ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K \u22a2 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5 ** have a0 := K0.trans_le ha ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K a0 : 0 < abv a \u22a2 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5 ** have b0 := K0.trans_le hb ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K a0 : 0 < abv a b0 : 0 < abv b \u22a2 abv (a\u207b\u00b9 - b\u207b\u00b9) < \u03b5 ** rw [inv_sub_inv' ((abv_pos abv).1 a0) ((abv_pos abv).1 b0), abv_mul abv, abv_mul abv, abv_inv abv,\n abv_inv abv, abv_sub abv] ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K a0 : 0 < abv a b0 : 0 < abv b \u22a2 (abv a)\u207b\u00b9 * abv (a - b) * (abv b)\u207b\u00b9 < \u03b5 ** refine' lt_of_mul_lt_mul_left (lt_of_mul_lt_mul_right _ b0.le) a0.le ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K a0 : 0 < abv a b0 : 0 < abv b \u22a2 abv a * ((abv a)\u207b\u00b9 * abv (a - b) * (abv b)\u207b\u00b9) * abv b < abv a * \u03b5 * abv b ** rw [mul_assoc, inv_mul_cancel_right\u2080 b0.ne', \u2190 mul_assoc, mul_inv_cancel a0.ne', one_mul] ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K a0 : 0 < abv a b0 : 0 < abv b \u22a2 abv (a - b) < abv a * \u03b5 * abv b ** refine' h.trans_le _ ** \u03b1 : Type u_2 \u03b2\u271d : Type ?u.21263 inst\u271d\u2074 : LinearOrderedField \u03b1 inst\u271d\u00b3 : Ring \u03b2\u271d abv\u271d : \u03b2\u271d \u2192 \u03b1 inst\u271d\u00b2 : IsAbsoluteValue abv\u271d \u03b2 : Type u_1 inst\u271d\u00b9 : DivisionRing \u03b2 abv : \u03b2 \u2192 \u03b1 inst\u271d : IsAbsoluteValue abv \u03b5 K : \u03b1 \u03b50 : 0 < \u03b5 K0 : 0 < K a b : \u03b2 ha : K \u2264 abv a hb : K \u2264 abv b h : abv (a - b) < K * \u03b5 * K a0 : 0 < abv a b0 : 0 < abv b \u22a2 K * \u03b5 * K \u2264 abv a * \u03b5 * abv b ** gcongr ** Qed", + "informal": "" + }, + { + "formal": "List.countP_mono_left ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool l : List \u03b1 h : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true \u22a2 countP p l \u2264 countP q l ** induction l with\n| nil => apply Nat.le_refl\n| cons a l ihl =>\n rw [forall_mem_cons] at h\n have \u27e8ha, hl\u27e9 := h\n simp [countP_cons]\n cases h : p a\n . simp\n apply Nat.le_trans ?_ (Nat.le_add_right _ _)\n apply ihl hl\n . simp [ha h, Nat.add_one]\n apply Nat.succ_le_succ\n apply ihl hl ** case nil \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool h : \u2200 (x : \u03b1), x \u2208 [] \u2192 p x = true \u2192 q x = true \u22a2 countP p [] \u2264 countP q [] ** apply Nat.le_refl ** case cons \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h : \u2200 (x : \u03b1), x \u2208 a :: l \u2192 p x = true \u2192 q x = true \u22a2 countP p (a :: l) \u2264 countP q (a :: l) ** rw [forall_mem_cons] at h ** case cons \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true \u22a2 countP p (a :: l) \u2264 countP q (a :: l) ** have \u27e8ha, hl\u27e9 := h ** case cons \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true \u22a2 countP p (a :: l) \u2264 countP q (a :: l) ** simp [countP_cons] ** case cons \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true \u22a2 (countP p l + if p a = true then 1 else 0) \u2264 countP q l + if q a = true then 1 else 0 ** cases h : p a ** case cons.false \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = false \u22a2 (countP p l + if false = true then 1 else 0) \u2264 countP q l + if q a = true then 1 else 0 case cons.true \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = true \u22a2 (countP p l + if true = true then 1 else 0) \u2264 countP q l + if q a = true then 1 else 0 ** . simp\n apply Nat.le_trans ?_ (Nat.le_add_right _ _)\n apply ihl hl ** case cons.true \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = true \u22a2 (countP p l + if true = true then 1 else 0) \u2264 countP q l + if q a = true then 1 else 0 ** . simp [ha h, Nat.add_one]\n apply Nat.succ_le_succ\n apply ihl hl ** case cons.false \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = false \u22a2 (countP p l + if false = true then 1 else 0) \u2264 countP q l + if q a = true then 1 else 0 ** simp ** case cons.false \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = false \u22a2 countP p l \u2264 countP q l + if q a = true then 1 else 0 ** apply Nat.le_trans ?_ (Nat.le_add_right _ _) ** \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = false \u22a2 countP p l \u2264 countP q l ** apply ihl hl ** case cons.true \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = true \u22a2 (countP p l + if true = true then 1 else 0) \u2264 countP q l + if q a = true then 1 else 0 ** simp [ha h, Nat.add_one] ** case cons.true \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = true \u22a2 succ (countP p l) \u2264 succ (countP q l) ** apply Nat.succ_le_succ ** case cons.true.a \u03b1 : Type u_1 p q : \u03b1 \u2192 Bool a : \u03b1 l : List \u03b1 ihl : (\u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true) \u2192 countP p l \u2264 countP q l h\u271d : (p a = true \u2192 q a = true) \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true ha : p a = true \u2192 q a = true hl : \u2200 (x : \u03b1), x \u2208 l \u2192 p x = true \u2192 q x = true h : p a = true \u22a2 countP p l \u2264 countP q l ** apply ihl hl ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.biproduct.toSubtype_eq_desc ** J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p \u22a2 \u2200 (j : J), \u03b9 f j \u226b toSubtype f p = \u03b9 f j \u226b desc fun j => if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0 ** simp ** Qed", + "informal": "" + }, + { + "formal": "imp_and_neg_imp_iff ** p q : Prop \u22a2 (p \u2192 q) \u2227 (\u00acp \u2192 q) \u2194 q ** rw [imp_iff_or_not, imp_iff_or_not, not_not, \u2190 or_and_left, not_and_self_iff, or_false_iff] ** Qed", + "informal": "" + }, + { + "formal": "Ordinal.one_out_eq ** \u03b1 : Type u \u03b2 : Type u_1 \u03b3 : Type u_2 r : \u03b1 \u2192 \u03b1 \u2192 Prop s : \u03b2 \u2192 \u03b2 \u2192 Prop t : \u03b3 \u2192 \u03b3 \u2192 Prop x : (Quotient.out 1).\u03b1 \u22a2 0 < type fun x x_1 => x < x_1 ** simp ** Qed", + "informal": "" + }, + { + "formal": "PMF.bernoulli_expectation ** p : \u211d\u22650\u221e h : p \u2264 1 \u22a2 \u222b (b : Bool), bif b then 1 else 0 \u2202toMeasure (bernoulli p h) = ENNReal.toReal p ** simp [integral_eq_sum] ** Qed", + "informal": "" + }, + { + "formal": "List.pmap_congr ** \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p q : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 g : (a : \u03b1) \u2192 q a \u2192 \u03b2 l : List \u03b1 H\u2081 : \u2200 (a : \u03b1), a \u2208 l \u2192 p a H\u2082 : \u2200 (a : \u03b1), a \u2208 l \u2192 q a h : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082 \u22a2 pmap f l H\u2081 = pmap g l H\u2082 ** induction' l with _ _ ih ** case nil \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p q : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 g : (a : \u03b1) \u2192 q a \u2192 \u03b2 l : List \u03b1 H\u2081\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a H\u2082\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 q a h\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082 H\u2081 : \u2200 (a : \u03b1), a \u2208 [] \u2192 p a H\u2082 : \u2200 (a : \u03b1), a \u2208 [] \u2192 q a h : \u2200 (a : \u03b1), a \u2208 [] \u2192 \u2200 (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082 \u22a2 pmap f [] H\u2081 = pmap g [] H\u2082 ** rfl ** case cons \u03b9 : Type u_1 \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w l\u2081 l\u2082 : List \u03b1 p q : \u03b1 \u2192 Prop f : (a : \u03b1) \u2192 p a \u2192 \u03b2 g : (a : \u03b1) \u2192 q a \u2192 \u03b2 l : List \u03b1 H\u2081\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 p a H\u2082\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 q a h\u271d : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082 head\u271d : \u03b1 tail\u271d : List \u03b1 ih : \u2200 {H\u2081 : \u2200 (a : \u03b1), a \u2208 tail\u271d \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 tail\u271d \u2192 q a}, (\u2200 (a : \u03b1), a \u2208 tail\u271d \u2192 \u2200 (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082) \u2192 pmap f tail\u271d H\u2081 = pmap g tail\u271d H\u2082 H\u2081 : \u2200 (a : \u03b1), a \u2208 head\u271d :: tail\u271d \u2192 p a H\u2082 : \u2200 (a : \u03b1), a \u2208 head\u271d :: tail\u271d \u2192 q a h : \u2200 (a : \u03b1), a \u2208 head\u271d :: tail\u271d \u2192 \u2200 (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082 \u22a2 pmap f (head\u271d :: tail\u271d) H\u2081 = pmap g (head\u271d :: tail\u271d) H\u2082 ** rw [pmap, pmap, h _ (mem_cons_self _ _), ih fun a ha => h a (mem_cons_of_mem _ ha)] ** Qed", + "informal": "" + }, + { + "formal": "WithTop.coe_mul ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 a b : \u03b1 \u22a2 \u2191(a * b) = \u2191a * \u2191b ** by_cases ha : a = 0 ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 a b : \u03b1 ha : a = 0 \u22a2 \u2191(a * b) = \u2191a * \u2191b ** simp [ha] ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 a b : \u03b1 ha : \u00aca = 0 \u22a2 \u2191(a * b) = \u2191a * \u2191b ** by_cases hb : b = 0 ** case pos \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 a b : \u03b1 ha : \u00aca = 0 hb : b = 0 \u22a2 \u2191(a * b) = \u2191a * \u2191b ** simp [hb] ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 a b : \u03b1 ha : \u00aca = 0 hb : \u00acb = 0 \u22a2 \u2191(a * b) = \u2191a * \u2191b ** simp [*, mul_def] ** case neg \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : MulZeroClass \u03b1 a b : \u03b1 ha : \u00aca = 0 hb : \u00acb = 0 \u22a2 \u2191(a * b) = Option.map\u2082 (fun x x_1 => x * x_1) \u2191a \u2191b ** rfl ** Qed", + "informal": "" + }, + { + "formal": "DiscreteQuotient.LEComap.comp ** \u03b1 : Type u_1 X : Type u_2 Y : Type u_3 Z : Type u_4 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : TopologicalSpace Y inst\u271d : TopologicalSpace Z S : DiscreteQuotient X f : C(X, Y) A A' : DiscreteQuotient X B B' : DiscreteQuotient Y g : C(Y, Z) C : DiscreteQuotient Z \u22a2 LEComap g B C \u2192 LEComap f A B \u2192 LEComap (ContinuousMap.comp g f) A C ** tauto ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.pow_sub_pow_of_prime ** R : Type u_1 n\u271d : \u2115 inst\u271d\u00b2 : CommRing R a b x\u271d y\u271d : R p\u271d : \u2115 inst\u271d\u00b9 : IsDomain R inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 p : R hp : Prime p x y : R hxy : p \u2223 x - y hx : \u00acp \u2223 x n : \u2115 hn : \u00acp \u2223 \u2191n \u22a2 multiplicity p (x ^ n - y ^ n) = multiplicity p (x - y) ** rw [\u2190 geom_sum\u2082_mul, multiplicity.mul hp, multiplicity_eq_zero.2 (not_dvd_geom_sum\u2082 hp hxy hx hn),\n zero_add] ** Qed", + "informal": "" + }, + { + "formal": "UpperHalfPlane.c_mul_im_sq_le_normSq_denom ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) z : \u210d g : SL(2, \u211d) \u22a2 (\u2191\u2191(\u2191toGLPos g) 1 0 * im z) ^ 2 \u2264 \u2191Complex.normSq (denom (\u2191toGLPos g) z) ** let c := (\u2191\u2098g 1 0 : \u211d) ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) z : \u210d g : SL(2, \u211d) c : \u211d := \u2191\u2191(\u2191toGLPos g) 1 0 \u22a2 (\u2191\u2191(\u2191toGLPos g) 1 0 * im z) ^ 2 \u2264 \u2191Complex.normSq (denom (\u2191toGLPos g) z) ** let d := (\u2191\u2098g 1 1 : \u211d) ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) z : \u210d g : SL(2, \u211d) c : \u211d := \u2191\u2191(\u2191toGLPos g) 1 0 d : \u211d := \u2191\u2191(\u2191toGLPos g) 1 1 \u22a2 (\u2191\u2191(\u2191toGLPos g) 1 0 * im z) ^ 2 \u2264 \u2191Complex.normSq (denom (\u2191toGLPos g) z) ** calc\n (c * z.im) ^ 2 \u2264 (c * z.im) ^ 2 + (c * z.re + d) ^ 2 := by nlinarith\n _ = Complex.normSq (denom g z) := by dsimp [denom, Complex.normSq]; ring ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) z : \u210d g : SL(2, \u211d) c : \u211d := \u2191\u2191(\u2191toGLPos g) 1 0 d : \u211d := \u2191\u2191(\u2191toGLPos g) 1 1 \u22a2 (c * im z) ^ 2 \u2264 (c * im z) ^ 2 + (c * re z + d) ^ 2 ** nlinarith ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) z : \u210d g : SL(2, \u211d) c : \u211d := \u2191\u2191(\u2191toGLPos g) 1 0 d : \u211d := \u2191\u2191(\u2191toGLPos g) 1 1 \u22a2 (c * im z) ^ 2 + (c * re z + d) ^ 2 = \u2191Complex.normSq (denom (\u2191toGLPos g) z) ** dsimp [denom, Complex.normSq] ** g\u271d : SL(2, \u2124) z\u271d : \u210d \u0393 : Subgroup SL(2, \u2124) z : \u210d g : SL(2, \u211d) c : \u211d := \u2191\u2191(\u2191toGLPos g) 1 0 d : \u211d := \u2191\u2191(\u2191toGLPos g) 1 1 \u22a2 (\u2191g 1 0 * im z) ^ 2 + (\u2191g 1 0 * re z + \u2191g 1 1) ^ 2 = (\u2191g 1 0 * re z - 0 * im z + \u2191g 1 1) * (\u2191g 1 0 * re z - 0 * im z + \u2191g 1 1) + (\u2191g 1 0 * im z + 0 * re z + 0) * (\u2191g 1 0 * im z + 0 * re z + 0) ** ring ** Qed", + "informal": "" + }, + { + "formal": "Mathlib.Meta.NormNum.isInt_pow ** \u03b1 : Type u_1 inst\u271d : Ring \u03b1 n\u271d\u00b9 : \u2124 n\u271d : \u2115 \u22a2 \u2191n\u271d\u00b9 ^ \u2191n\u271d = \u2191(Int.pow n\u271d\u00b9 n\u271d) ** simp ** Qed", + "informal": "" + }, + { + "formal": "toIcoMod_add_right ** \u03b1 : Type u_1 inst\u271d : LinearOrderedAddCommGroup \u03b1 h\u03b1 : Archimedean \u03b1 p : \u03b1 hp : 0 < p a\u271d b\u271d c : \u03b1 n : \u2124 a b : \u03b1 \u22a2 toIcoMod hp a (b + p) = toIcoMod hp a b ** simpa only [one_zsmul] using toIcoMod_add_zsmul hp a b 1 ** Qed", + "informal": "" + }, + { + "formal": "Acc.ndrecOn_eq_ndrecOnC ** \u22a2 @ndrecOn = @Acc.ndrecOnC ** funext \u03b1 r motive intro a t ** case h.h.h.h.h.h \u03b1 : Sort u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop motive : \u03b1 \u2192 Sort u_2 intro : \u03b1 a : Acc r intro t : (x : \u03b1) \u2192 (\u2200 (y : \u03b1), r y x \u2192 Acc r y) \u2192 ((y : \u03b1) \u2192 r y x \u2192 motive y) \u2192 motive x \u22a2 ndrecOn a t = Acc.ndrecOnC a t ** rw [Acc.ndrecOn, rec_eq_recC, Acc.ndrecOnC] ** Qed", + "informal": "" + }, + { + "formal": "mul_ball ** E : Type u_1 inst\u271d : SeminormedCommGroup E \u03b5 \u03b4 : \u211d s t : Set E x y : E \u22a2 s * ball x \u03b4 = x \u2022 thickening \u03b4 s ** rw [\u2190 smul_ball_one, mul_smul_comm, mul_ball_one] ** Qed", + "informal": "" + }, + { + "formal": "hasMellin_one_Ioc ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re \u22a2 HasMellin (indicator (Ioc 0 1) fun x => 1) s (1 / s) ** have aux1 : -1 < (s - 1).re := by\n simpa only [sub_re, one_re, sub_eq_add_neg] using lt_add_of_pos_left _ hs ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re \u22a2 HasMellin (indicator (Ioc 0 1) fun x => 1) s (1 / s) ** have aux2 : s \u2260 0 := by contrapose! hs; rw [hs, zero_re] ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re aux2 : s \u2260 0 \u22a2 HasMellin (indicator (Ioc 0 1) fun x => 1) s (1 / s) ** have aux3 : MeasurableSet (Ioc (0 : \u211d) 1) := measurableSet_Ioc ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re aux2 : s \u2260 0 aux3 : MeasurableSet (Ioc 0 1) \u22a2 HasMellin (indicator (Ioc 0 1) fun x => 1) s (1 / s) ** simp_rw [HasMellin, mellin, MellinConvergent, \u2190 indicator_smul, IntegrableOn,\n integrable_indicator_iff aux3, smul_eq_mul, integral_indicator aux3, mul_one, IntegrableOn,\n Measure.restrict_restrict_of_subset Ioc_subset_Ioi_self] ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re aux2 : s \u2260 0 aux3 : MeasurableSet (Ioc 0 1) \u22a2 (Integrable fun x => \u2191x ^ (s - 1)) \u2227 \u222b (x : \u211d) in Ioc 0 1, \u2191x ^ (s - 1) = 1 / s ** rw [\u2190 IntegrableOn, \u2190 intervalIntegrable_iff_integrable_Ioc_of_le zero_le_one] ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re aux2 : s \u2260 0 aux3 : MeasurableSet (Ioc 0 1) \u22a2 IntervalIntegrable (fun x => \u2191x ^ (s - 1)) volume 0 1 \u2227 \u222b (x : \u211d) in Ioc 0 1, \u2191x ^ (s - 1) = 1 / s ** refine' \u27e8intervalIntegral.intervalIntegrable_cpow' aux1, _\u27e9 ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re aux2 : s \u2260 0 aux3 : MeasurableSet (Ioc 0 1) \u22a2 \u222b (x : \u211d) in Ioc 0 1, \u2191x ^ (s - 1) = 1 / s ** rw [\u2190 intervalIntegral.integral_of_le zero_le_one, integral_cpow (Or.inl aux1), sub_add_cancel,\n ofReal_zero, ofReal_one, one_cpow, zero_cpow aux2, sub_zero] ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re \u22a2 -1 < (s - 1).re ** simpa only [sub_re, one_re, sub_eq_add_neg] using lt_add_of_pos_left _ hs ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 hs : 0 < s.re aux1 : -1 < (s - 1).re \u22a2 s \u2260 0 ** contrapose! hs ** E : Type u_1 inst\u271d : NormedAddCommGroup E s : \u2102 aux1 : -1 < (s - 1).re hs : s = 0 \u22a2 s.re \u2264 0 ** rw [hs, zero_re] ** Qed", + "informal": "" + }, + { + "formal": "nonunits.subset_compl_ball ** R : Type u_1 inst\u271d\u00b9 : NormedRing R inst\u271d : CompleteSpace R x : R hx : x \u2208 nonunits R h\u2081 : x \u2208 Metric.ball 1 1 \u22a2 \u20161 - x\u2016 < 1 ** rwa [mem_ball_iff_norm'] at h\u2081 ** Qed", + "informal": "" + }, + { + "formal": "PolynomialModule.smul_apply ** R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M f : R[X] g : PolynomialModule R M n : \u2115 \u22a2 \u2191(f \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff f x.1 \u2022 \u2191g x.2 ** induction' f using Polynomial.induction_on' with p q hp hq f_n f_a ** case h_add R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n : \u2115 p q : R[X] hp : \u2191(p \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff p x.1 \u2022 \u2191g x.2 hq : \u2191(q \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff q x.1 \u2022 \u2191g x.2 \u22a2 \u2191((p + q) \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff (p + q) x.1 \u2022 \u2191g x.2 ** rw [add_smul, Finsupp.add_apply, hp, hq, \u2190 Finset.sum_add_distrib] ** case h_add R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n : \u2115 p q : R[X] hp : \u2191(p \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff p x.1 \u2022 \u2191g x.2 hq : \u2191(q \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff q x.1 \u2022 \u2191g x.2 \u22a2 \u2211 x in Finset.Nat.antidiagonal n, (coeff p x.1 \u2022 \u2191g x.2 + coeff q x.1 \u2022 \u2191g x.2) = \u2211 x in Finset.Nat.antidiagonal n, coeff (p + q) x.1 \u2022 \u2191g x.2 ** congr ** case h_add.e_f R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n : \u2115 p q : R[X] hp : \u2191(p \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff p x.1 \u2022 \u2191g x.2 hq : \u2191(q \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff q x.1 \u2022 \u2191g x.2 \u22a2 (fun x => coeff p x.1 \u2022 \u2191g x.2 + coeff q x.1 \u2022 \u2191g x.2) = fun x => coeff (p + q) x.1 \u2022 \u2191g x.2 ** ext ** case h_add.e_f.h R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n : \u2115 p q : R[X] hp : \u2191(p \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff p x.1 \u2022 \u2191g x.2 hq : \u2191(q \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff q x.1 \u2022 \u2191g x.2 x\u271d : \u2115 \u00d7 \u2115 \u22a2 coeff p x\u271d.1 \u2022 \u2191g x\u271d.2 + coeff q x\u271d.1 \u2022 \u2191g x\u271d.2 = coeff (p + q) x\u271d.1 \u2022 \u2191g x\u271d.2 ** rw [coeff_add, add_smul] ** case h_monomial R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R \u22a2 \u2191(\u2191(monomial f_n) f_a \u2022 g) n = \u2211 x in Finset.Nat.antidiagonal n, coeff (\u2191(monomial f_n) f_a) x.1 \u2022 \u2191g x.2 ** rw [Finset.Nat.sum_antidiagonal_eq_sum_range_succ fun i j => (monomial f_n f_a).coeff i \u2022 g j,\n monomial_smul_apply] ** case h_monomial R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R \u22a2 (if f_n \u2264 n then f_a \u2022 \u2191g (n - f_n) else 0) = \u2211 k in Finset.range (Nat.succ n), coeff (\u2191(monomial f_n) f_a) k \u2022 \u2191g (n - k) ** simp_rw [Polynomial.coeff_monomial, \u2190 Finset.mem_range_succ_iff] ** case h_monomial R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R \u22a2 (if f_n \u2208 Finset.range (Nat.succ n) then f_a \u2022 \u2191g (n - f_n) else 0) = \u2211 x in Finset.range (Nat.succ n), (if f_n = x then f_a else 0) \u2022 \u2191g (n - x) ** rw [\u2190 Finset.sum_ite_eq (Finset.range (Nat.succ n)) f_n (fun x => f_a \u2022 g (n - x))] ** case h_monomial R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R \u22a2 (\u2211 x in Finset.range (Nat.succ n), if f_n = x then f_a \u2022 \u2191g (n - x) else 0) = \u2211 x in Finset.range (Nat.succ n), (if f_n = x then f_a else 0) \u2022 \u2191g (n - x) ** congr ** case h_monomial.e_f R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R \u22a2 (fun x => if f_n = x then f_a \u2022 \u2191g (n - x) else 0) = fun x => (if f_n = x then f_a else 0) \u2022 \u2191g (n - x) ** ext x ** case h_monomial.e_f.h R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R x : \u2115 \u22a2 (if f_n = x then f_a \u2022 \u2191g (n - x) else 0) = (if f_n = x then f_a else 0) \u2022 \u2191g (n - x) ** split_ifs ** case pos R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R x : \u2115 h\u271d : f_n = x \u22a2 f_a \u2022 \u2191g (n - x) = f_a \u2022 \u2191g (n - x) case neg R : Type u_1 M : Type u_2 inst\u271d\u2076 : CommRing R inst\u271d\u2075 : AddCommGroup M inst\u271d\u2074 : Module R M I : Ideal R S : Type u_3 inst\u271d\u00b3 : CommSemiring S inst\u271d\u00b2 : Algebra S R inst\u271d\u00b9 : Module S M inst\u271d : IsScalarTower S R M g : PolynomialModule R M n f_n : \u2115 f_a : R x : \u2115 h\u271d : \u00acf_n = x \u22a2 0 = 0 \u2022 \u2191g (n - x) ** exacts [rfl, (zero_smul R _).symm] ** Qed", + "informal": "" + }, + { + "formal": "AlgebraicTopology.DoldKan.N\u2082\u0393\u2082_toKaroubi ** C : Type u_1 inst\u271d\u00b2 : Category.{u_2, u_1} C inst\u271d\u00b9 : Preadditive C inst\u271d : HasFiniteCoproducts C \u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081 ** have h := Functor.congr_obj (functorExtension\u2082_comp_whiskeringLeft_toKaroubi\n (ChainComplex C \u2115) (SimplicialObject C)) \u0393\u2080 ** C : Type u_1 inst\u271d\u00b2 : Category.{u_2, u_1} C inst\u271d\u00b9 : Preadditive C inst\u271d : HasFiniteCoproducts C h : (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C) \u22d9 (whiskeringLeft (ChainComplex C \u2115) (Karoubi (ChainComplex C \u2115)) (Karoubi (SimplicialObject C))).obj (toKaroubi (ChainComplex C \u2115))).obj \u0393\u2080 = ((whiskeringRight (ChainComplex C \u2115) (SimplicialObject C) (Karoubi (SimplicialObject C))).obj (toKaroubi (SimplicialObject C))).obj \u0393\u2080 \u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081 ** have h' := Functor.congr_obj (functorExtension\u2081_comp_whiskeringLeft_toKaroubi\n (SimplicialObject C) (ChainComplex C \u2115)) N\u2081 ** C : Type u_1 inst\u271d\u00b2 : Category.{u_2, u_1} C inst\u271d\u00b9 : Preadditive C inst\u271d : HasFiniteCoproducts C h : (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C) \u22d9 (whiskeringLeft (ChainComplex C \u2115) (Karoubi (ChainComplex C \u2115)) (Karoubi (SimplicialObject C))).obj (toKaroubi (ChainComplex C \u2115))).obj \u0393\u2080 = ((whiskeringRight (ChainComplex C \u2115) (SimplicialObject C) (Karoubi (SimplicialObject C))).obj (toKaroubi (SimplicialObject C))).obj \u0393\u2080 h' : (functorExtension\u2081 (SimplicialObject C) (ChainComplex C \u2115) \u22d9 (whiskeringLeft (SimplicialObject C) (Karoubi (SimplicialObject C)) (Karoubi (ChainComplex C \u2115))).obj (toKaroubi (SimplicialObject C))).obj N\u2081 = (\ud835\udfed (SimplicialObject C \u2964 Karoubi (ChainComplex C \u2115))).obj N\u2081 \u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081 ** dsimp [N\u2082, \u0393\u2082, functorExtension\u2081] at h h' \u22a2 ** C : Type u_1 inst\u271d\u00b2 : Category.{u_2, u_1} C inst\u271d\u00b9 : Preadditive C inst\u271d : HasFiniteCoproducts C h : toKaroubi (ChainComplex C \u2115) \u22d9 (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C)).obj \u0393\u2080 = \u0393\u2080 \u22d9 toKaroubi (SimplicialObject C) h' : toKaroubi (SimplicialObject C) \u22d9 FunctorExtension\u2081.obj N\u2081 = N\u2081 \u22a2 toKaroubi (ChainComplex C \u2115) \u22d9 (functorExtension\u2082 (ChainComplex C \u2115) (SimplicialObject C)).obj \u0393\u2080 \u22d9 FunctorExtension\u2081.obj N\u2081 = \u0393\u2080 \u22d9 N\u2081 ** rw [\u2190 Functor.assoc, h, Functor.assoc, h'] ** Qed", + "informal": "" + }, + { + "formal": "Nat.gcd_eq_gcd_ab ** x y : \u2115 \u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y ** have := @xgcdAux_P x y x y 1 0 0 1 (by simp [P]) (by simp [P]) ** x y : \u2115 this : Nat.P x y (xgcdAux x 1 0 y 0 1) \u22a2 \u2191(gcd x y) = \u2191x * gcdA x y + \u2191y * gcdB x y ** rwa [xgcdAux_val, xgcd_val] at this ** x y : \u2115 \u22a2 Nat.P x y (x, 1, 0) ** simp [P] ** x y : \u2115 \u22a2 Nat.P x y (y, 0, 1) ** simp [P] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.oangle_map_complex ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) f : V \u2243\u2097\u1d62[\u211d] \u2102 hf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation x y : V \u22a2 oangle o x y = \u2191(arg (\u2191(starRingEnd \u2102) (\u2191f x) * \u2191f y)) ** rw [\u2190 Complex.oangle, \u2190 hf, o.oangle_map] ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) f : V \u2243\u2097\u1d62[\u211d] \u2102 hf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation x y : V \u22a2 oangle o x y = oangle o (\u2191(LinearIsometryEquiv.symm f) (\u2191f x)) (\u2191(LinearIsometryEquiv.symm f) (\u2191f y)) ** iterate 2 rw [LinearIsometryEquiv.symm_apply_apply] ** V : Type u_1 V' : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : NormedAddCommGroup V' inst\u271d\u00b3 : InnerProductSpace \u211d V inst\u271d\u00b2 : InnerProductSpace \u211d V' inst\u271d\u00b9 : Fact (finrank \u211d V = 2) inst\u271d : Fact (finrank \u211d V' = 2) o : Orientation \u211d V (Fin 2) f : V \u2243\u2097\u1d62[\u211d] \u2102 hf : \u2191(map (Fin 2) f.toLinearEquiv) o = Complex.orientation x y : V \u22a2 oangle o x y = oangle o x (\u2191(LinearIsometryEquiv.symm f) (\u2191f y)) ** rw [LinearIsometryEquiv.symm_apply_apply] ** Qed", + "informal": "" + }, + { + "formal": "Rat.nonneg_total ** a b c : \u211a \u22a2 Rat.Nonneg a \u2228 Rat.Nonneg (-a) ** cases' a with n ** case mk' b c : \u211a n : \u2124 den\u271d : \u2115 den_nz\u271d : den\u271d \u2260 0 reduced\u271d : Nat.Coprime (Int.natAbs n) den\u271d \u22a2 Rat.Nonneg (mk' n den\u271d) \u2228 Rat.Nonneg (-mk' n den\u271d) ** exact Or.imp_right neg_nonneg_of_nonpos (le_total 0 n) ** Qed", + "informal": "" + }, + { + "formal": "Option.map\u2082_eq_none_iff ** \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_1 f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 a : Option \u03b1 b : Option \u03b2 c : Option \u03b3 \u22a2 map\u2082 f a b = none \u2194 a = none \u2228 b = none ** cases a <;> cases b <;> simp ** Qed", + "informal": "" + }, + { + "formal": "DiscreteQuotient.ofLE_refl_apply ** \u03b1 : Type u_1 X : Type u_2 Y : Type u_3 Z : Type u_4 inst\u271d\u00b2 : TopologicalSpace X inst\u271d\u00b9 : TopologicalSpace Y inst\u271d : TopologicalSpace Z S A B C : DiscreteQuotient X a : Quotient A.toSetoid \u22a2 ofLE (_ : A \u2264 A) a = a ** simp ** Qed", + "informal": "" + }, + { + "formal": "NNReal.pow_nat_rpow_nat_inv ** x : \u211d\u22650 n : \u2115 hn : n \u2260 0 \u22a2 (x ^ n) ^ (\u2191n)\u207b\u00b9 = x ** rw [\u2190 NNReal.coe_eq, coe_rpow, NNReal.coe_pow] ** x : \u211d\u22650 n : \u2115 hn : n \u2260 0 \u22a2 (\u2191x ^ n) ^ (\u2191n)\u207b\u00b9 = \u2191x ** exact Real.pow_nat_rpow_nat_inv x.2 hn ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.kernel.lintegral_const ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } f : \u03b2 \u2192 \u211d\u22650\u221e \u03bc : Measure \u03b2 a : \u03b1 \u22a2 \u222b\u207b (x : \u03b2), f x \u2202\u2191(const \u03b1 \u03bc) a = \u222b\u207b (x : \u03b2), f x \u2202\u03bc ** rw [kernel.const_apply] ** Qed", + "informal": "" + }, + { + "formal": "dvd_sub_pow_of_dvd_sub ** R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 \u22a2 \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k ** induction' k with k ih ** case succ R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k \u22a2 \u2191(p ^ (Nat.succ k + 1)) \u2223 a ^ p ^ Nat.succ k - b ^ p ^ Nat.succ k ** rw [pow_succ' p k, pow_mul, pow_mul, \u2190 geom_sum\u2082_mul, pow_succ, Nat.cast_mul] ** case succ R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k \u22a2 \u2191p * \u2191(p ^ (k + 1)) \u2223 (Finset.sum (Finset.range p) fun i => (a ^ p ^ k) ^ i * (b ^ p ^ k) ^ (p - 1 - i)) * (a ^ p ^ k - b ^ p ^ k) ** refine' mul_dvd_mul _ ih ** case succ R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k \u22a2 \u2191p \u2223 Finset.sum (Finset.range p) fun i => (a ^ p ^ k) ^ i * (b ^ p ^ k) ^ (p - 1 - i) ** let f : R \u2192+* R \u29f8 span {(p : R)} := mk (span {(p : R)}) ** case succ R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k f : R \u2192+* R \u29f8 span {\u2191p} := mk (span {\u2191p}) \u22a2 \u2191p \u2223 Finset.sum (Finset.range p) fun i => (a ^ p ^ k) ^ i * (b ^ p ^ k) ^ (p - 1 - i) ** have hf : \u2200 r : R, (p : R) \u2223 r \u2194 f r = 0 := fun r \u21a6 by rw [eq_zero_iff_mem, mem_span_singleton] ** case succ R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k f : R \u2192+* R \u29f8 span {\u2191p} := mk (span {\u2191p}) hf : \u2200 (r : R), \u2191p \u2223 r \u2194 \u2191f r = 0 \u22a2 \u2191p \u2223 Finset.sum (Finset.range p) fun i => (a ^ p ^ k) ^ i * (b ^ p ^ k) ^ (p - 1 - i) ** rw [hf, map_sub, sub_eq_zero] at h ** case succ R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k f : R \u2192+* R \u29f8 span {\u2191p} := mk (span {\u2191p}) h : \u2191f a = \u2191f b hf : \u2200 (r : R), \u2191p \u2223 r \u2194 \u2191f r = 0 \u22a2 \u2191p \u2223 Finset.sum (Finset.range p) fun i => (a ^ p ^ k) ^ i * (b ^ p ^ k) ^ (p - 1 - i) ** rw [hf, RingHom.map_geom_sum\u2082, map_pow, map_pow, h, geom_sum\u2082_self, mul_eq_zero_of_left] ** case succ.h R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k f : R \u2192+* R \u29f8 span {\u2191p} := mk (span {\u2191p}) h : \u2191f a = \u2191f b hf : \u2200 (r : R), \u2191p \u2223 r \u2194 \u2191f r = 0 \u22a2 \u2191p = 0 ** rw [\u2190 map_natCast f, eq_zero_iff_mem, mem_span_singleton] ** case zero R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b \u22a2 \u2191(p ^ (Nat.zero + 1)) \u2223 a ^ p ^ Nat.zero - b ^ p ^ Nat.zero ** rwa [pow_one, pow_zero, pow_one, pow_one] ** R : Type u_1 inst\u271d : CommRing R p : \u2115 a b : R h : \u2191p \u2223 a - b k : \u2115 ih : \u2191(p ^ (k + 1)) \u2223 a ^ p ^ k - b ^ p ^ k f : R \u2192+* R \u29f8 span {\u2191p} := mk (span {\u2191p}) r : R \u22a2 \u2191p \u2223 r \u2194 \u2191f r = 0 ** rw [eq_zero_iff_mem, mem_span_singleton] ** Qed", + "informal": "" + }, + { + "formal": "Orthonormal.inner_finsupp_eq_sum_right ** \ud835\udd5c : Type u_1 E : Type u_2 F : Type u_3 inst\u271d\u2074 : IsROrC \ud835\udd5c inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : InnerProductSpace \ud835\udd5c E inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : InnerProductSpace \u211d F dec_E : DecidableEq E \u03b9 : Type u_4 dec_\u03b9 : DecidableEq \u03b9 v : \u03b9 \u2192 E hv : Orthonormal \ud835\udd5c v l\u2081 l\u2082 : \u03b9 \u2192\u2080 \ud835\udd5c \u22a2 inner (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l\u2081) (\u2191(Finsupp.total \u03b9 E \ud835\udd5c v) l\u2082) = Finsupp.sum l\u2082 fun i y => \u2191(starRingEnd ((fun x => \ud835\udd5c) i)) (\u2191l\u2081 i) * y ** simp only [l\u2082.total_apply _, Finsupp.inner_sum, hv.inner_left_finsupp, mul_comm, smul_eq_mul] ** Qed", + "informal": "" + }, + { + "formal": "AddCircle.addOrderOf_coe_rat ** \ud835\udd5c : Type u_1 B : Type u_2 inst\u271d\u00b2 : LinearOrderedField \ud835\udd5c inst\u271d\u00b9 : TopologicalSpace \ud835\udd5c inst\u271d : OrderTopology \ud835\udd5c p q\u271d : \ud835\udd5c hp : Fact (0 < p) q : \u211a \u22a2 addOrderOf \u2191(\u2191q * p) = q.den ** have : (\u2191(q.den : \u2124) : \ud835\udd5c) \u2260 0 := by\n norm_cast\n exact q.pos.ne.symm ** \ud835\udd5c : Type u_1 B : Type u_2 inst\u271d\u00b2 : LinearOrderedField \ud835\udd5c inst\u271d\u00b9 : TopologicalSpace \ud835\udd5c inst\u271d : OrderTopology \ud835\udd5c p q\u271d : \ud835\udd5c hp : Fact (0 < p) q : \u211a this : \u2191\u2191q.den \u2260 0 \u22a2 addOrderOf \u2191(\u2191q * p) = q.den ** rw [\u2190 @Rat.num_den q, Rat.cast_mk_of_ne_zero _ _ this, Int.cast_ofNat, Rat.num_den,\n addOrderOf_div_of_gcd_eq_one' q.pos q.reduced] ** \ud835\udd5c : Type u_1 B : Type u_2 inst\u271d\u00b2 : LinearOrderedField \ud835\udd5c inst\u271d\u00b9 : TopologicalSpace \ud835\udd5c inst\u271d : OrderTopology \ud835\udd5c p q\u271d : \ud835\udd5c hp : Fact (0 < p) q : \u211a \u22a2 \u2191\u2191q.den \u2260 0 ** norm_cast ** \ud835\udd5c : Type u_1 B : Type u_2 inst\u271d\u00b2 : LinearOrderedField \ud835\udd5c inst\u271d\u00b9 : TopologicalSpace \ud835\udd5c inst\u271d : OrderTopology \ud835\udd5c p q\u271d : \ud835\udd5c hp : Fact (0 < p) q : \u211a \u22a2 \u00acq.den = 0 ** exact q.pos.ne.symm ** Qed", + "informal": "" + }, + { + "formal": "Finset.mulEtransformRight.card ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 inst\u271d : Group \u03b1 e : \u03b1 x : Finset \u03b1 \u00d7 Finset \u03b1 \u22a2 Finset.card x.2 + Finset.card (e\u207b\u00b9 \u2022 x.2) = 2 * Finset.card x.2 ** rw [card_smul_finset, two_mul] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.hasseDeriv_coeff ** R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n : \u2115 \u22a2 coeff (\u2191(hasseDeriv k) f) n = \u2191(choose (n + k) k) * coeff f (n + k) ** rw [hasseDeriv_apply, coeff_sum, sum_def, Finset.sum_eq_single (n + k), coeff_monomial] ** R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n : \u2115 \u22a2 (if n + k - k = n then \u2191(choose (n + k) k) * coeff f (n + k) else 0) = \u2191(choose (n + k) k) * coeff f (n + k) ** simp only [if_true, add_tsub_cancel_right, eq_self_iff_true] ** case h\u2080 R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n : \u2115 \u22a2 \u2200 (b : \u2115), b \u2208 support f \u2192 b \u2260 n + k \u2192 coeff (\u2191(monomial (b - k)) (\u2191(choose b k) * coeff f b)) n = 0 ** intro i _hi hink ** case h\u2080 R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hink : i \u2260 n + k \u22a2 coeff (\u2191(monomial (i - k)) (\u2191(choose i k) * coeff f i)) n = 0 ** rw [coeff_monomial] ** case h\u2080 R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hink : i \u2260 n + k \u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0 ** by_cases hik : i < k ** case pos R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hink : i \u2260 n + k hik : i < k \u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0 ** simp only [Nat.choose_eq_zero_of_lt hik, ite_self, Nat.cast_zero, zero_mul] ** case neg R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hink : i \u2260 n + k hik : \u00aci < k \u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0 ** push_neg at hik ** case neg R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hink : i \u2260 n + k hik : k \u2264 i \u22a2 (if i - k = n then \u2191(choose i k) * coeff f i else 0) = 0 ** rw [if_neg] ** case neg.hnc R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hink : i \u2260 n + k hik : k \u2264 i \u22a2 \u00aci - k = n ** contrapose! hink ** case neg.hnc R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n i : \u2115 _hi : i \u2208 support f hik : k \u2264 i hink : i - k = n \u22a2 i = n + k ** exact (tsub_eq_iff_eq_add_of_le hik).mp hink ** case h\u2081 R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n : \u2115 \u22a2 \u00acn + k \u2208 support f \u2192 coeff (\u2191(monomial (n + k - k)) (\u2191(choose (n + k) k) * coeff f (n + k))) n = 0 ** intro h ** case h\u2081 R : Type u_1 inst\u271d : Semiring R k : \u2115 f : R[X] n : \u2115 h : \u00acn + k \u2208 support f \u22a2 coeff (\u2191(monomial (n + k - k)) (\u2191(choose (n + k) k) * coeff f (n + k))) n = 0 ** simp only [not_mem_support_iff.mp h, monomial_zero_right, mul_zero, coeff_zero] ** Qed", + "informal": "" + }, + { + "formal": "TensorPower.one_mul ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 a : (\u2a02[R]^n) M \u22a2 \u2191(cast R M (_ : 0 + n = n)) (GradedMonoid.GMul.mul GradedMonoid.GOne.one a) = a ** rw [gMul_def, gOne_def] ** R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 a : (\u2a02[R]^n) M \u22a2 \u2191(cast R M (_ : 0 + n = n)) (\u2191mulEquiv (\u2191(tprod R) Fin.elim0' \u2297\u209c[R] a)) = a ** induction' a using PiTensorProduct.induction_on with r a x y hx hy ** case C1 R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 r : R a : Fin n \u2192 M \u22a2 \u2191(cast R M (_ : 0 + n = n)) (\u2191mulEquiv (\u2191(tprod R) Fin.elim0' \u2297\u209c[R] (r \u2022 \u2191(tprod R) a))) = r \u2022 \u2191(tprod R) a ** rw [TensorProduct.tmul_smul, LinearEquiv.map_smul, LinearEquiv.map_smul, \u2190 gMul_def,\n tprod_mul_tprod, cast_tprod] ** case C1 R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 r : R a : Fin n \u2192 M \u22a2 r \u2022 \u2191(tprod R) (Fin.append Fin.elim0' a \u2218 Fin.cast (_ : n = 0 + n)) = r \u2022 \u2191(tprod R) a ** congr 2 with i ** case C1.e_a.h.e_6.h.h R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 r : R a : Fin n \u2192 M i : Fin n \u22a2 (Fin.append Fin.elim0' a \u2218 Fin.cast (_ : n = 0 + n)) i = a i ** rw [Fin.elim0'_append] ** case C1.e_a.h.e_6.h.h R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 r : R a : Fin n \u2192 M i : Fin n \u22a2 ((a \u2218 Fin.cast (_ : 0 + n = n)) \u2218 Fin.cast (_ : n = 0 + n)) i = a i ** refine' congr_arg a (Fin.ext _) ** case C1.e_a.h.e_6.h.h R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 r : R a : Fin n \u2192 M i : Fin n \u22a2 \u2191(Fin.cast (_ : 0 + n = n) (Fin.cast (_ : n = 0 + n) i)) = \u2191i ** simp ** case Cp R : Type u_1 M : Type u_2 inst\u271d\u00b2 : CommSemiring R inst\u271d\u00b9 : AddCommMonoid M inst\u271d : Module R M n : \u2115 x y : \u2a02[R] (i : Fin n), M hx : \u2191(cast R M (_ : 0 + n = n)) (\u2191mulEquiv (\u2191(tprod R) Fin.elim0' \u2297\u209c[R] x)) = x hy : \u2191(cast R M (_ : 0 + n = n)) (\u2191mulEquiv (\u2191(tprod R) Fin.elim0' \u2297\u209c[R] y)) = y \u22a2 \u2191(cast R M (_ : 0 + n = n)) (\u2191mulEquiv (\u2191(tprod R) Fin.elim0' \u2297\u209c[R] (x + y))) = x + y ** rw [TensorProduct.tmul_add, map_add, map_add, hx, hy] ** Qed", + "informal": "" + }, + { + "formal": "Matrix.scalar.commute ** l : Type u_1 m : Type u_2 n : Type u_3 o : Type u_4 m' : o \u2192 Type u_5 n' : o \u2192 Type u_6 R : Type u_7 S : Type u_8 \u03b1 : Type v \u03b2 : Type w \u03b3 : Type u_9 inst\u271d\u00b2 : CommSemiring \u03b1 inst\u271d\u00b9 : Fintype n inst\u271d : DecidableEq n r : \u03b1 M : Matrix n n \u03b1 \u22a2 Commute (\u2191(scalar n) r) M ** simp [Commute, SemiconjBy] ** Qed", + "informal": "" + }, + { + "formal": "isRegular_mul_and_mul_iff ** R : Type u_1 inst\u271d : Semigroup R a b : R \u22a2 IsRegular (a * b) \u2227 IsRegular (b * a) \u2194 IsRegular a \u2227 IsRegular b ** refine' \u27e8_, _\u27e9 ** case refine'_1 R : Type u_1 inst\u271d : Semigroup R a b : R \u22a2 IsRegular (a * b) \u2227 IsRegular (b * a) \u2192 IsRegular a \u2227 IsRegular b ** rintro \u27e8ab, ba\u27e9 ** case refine'_1.intro R : Type u_1 inst\u271d : Semigroup R a b : R ab : IsRegular (a * b) ba : IsRegular (b * a) \u22a2 IsRegular a \u2227 IsRegular b ** exact\n \u27e8\u27e8IsLeftRegular.of_mul ba.left, IsRightRegular.of_mul ab.right\u27e9,\n \u27e8IsLeftRegular.of_mul ab.left, IsRightRegular.of_mul ba.right\u27e9\u27e9 ** case refine'_2 R : Type u_1 inst\u271d : Semigroup R a b : R \u22a2 IsRegular a \u2227 IsRegular b \u2192 IsRegular (a * b) \u2227 IsRegular (b * a) ** rintro \u27e8ha, hb\u27e9 ** case refine'_2.intro R : Type u_1 inst\u271d : Semigroup R a b : R ha : IsRegular a hb : IsRegular b \u22a2 IsRegular (a * b) \u2227 IsRegular (b * a) ** exact\n \u27e8\u27e8(mul_isLeftRegular_iff _ ha.left).mpr hb.left,\n (mul_isRightRegular_iff _ hb.right).mpr ha.right\u27e9,\n \u27e8(mul_isLeftRegular_iff _ hb.left).mpr ha.left,\n (mul_isRightRegular_iff _ ha.right).mpr hb.right\u27e9\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.Colorable.chromaticNumber_le_of_forall_imp ** V : Type u G : SimpleGraph V \u03b1 : Type v C : Coloring G \u03b1 V' : Type u_1 G' : SimpleGraph V' m : \u2115 hc : Colorable G' m h : \u2200 (n : \u2115), Colorable G' n \u2192 Colorable G n \u22a2 chromaticNumber G \u2264 chromaticNumber G' ** apply csInf_le chromaticNumber_bddBelow ** V : Type u G : SimpleGraph V \u03b1 : Type v C : Coloring G \u03b1 V' : Type u_1 G' : SimpleGraph V' m : \u2115 hc : Colorable G' m h : \u2200 (n : \u2115), Colorable G' n \u2192 Colorable G n \u22a2 chromaticNumber G' \u2208 {n | Colorable G n} ** apply h ** case a V : Type u G : SimpleGraph V \u03b1 : Type v C : Coloring G \u03b1 V' : Type u_1 G' : SimpleGraph V' m : \u2115 hc : Colorable G' m h : \u2200 (n : \u2115), Colorable G' n \u2192 Colorable G n \u22a2 Colorable G' (chromaticNumber G') ** apply colorable_chromaticNumber hc ** Qed", + "informal": "" + }, + { + "formal": "TrivSqZeroExt.liftAux_apply_inr ** S : Type u_1 R R' : Type u M : Type v inst\u271d\u00b9\u2077 : CommSemiring S inst\u271d\u00b9\u2076 : Semiring R inst\u271d\u00b9\u2075 : CommSemiring R' inst\u271d\u00b9\u2074 : AddCommMonoid M inst\u271d\u00b9\u00b3 : Algebra S R inst\u271d\u00b9\u00b2 : Algebra S R' inst\u271d\u00b9\u00b9 : Module S M inst\u271d\u00b9\u2070 : Module R M inst\u271d\u2079 : Module R\u1d50\u1d52\u1d56 M inst\u271d\u2078 : SMulCommClass R R\u1d50\u1d52\u1d56 M inst\u271d\u2077 : IsScalarTower S R M inst\u271d\u2076 : IsScalarTower S R\u1d50\u1d52\u1d56 M inst\u271d\u2075 : Module R' M inst\u271d\u2074 : Module R'\u1d50\u1d52\u1d56 M inst\u271d\u00b3 : IsCentralScalar R' M inst\u271d\u00b2 : IsScalarTower S R' M A : Type u_2 inst\u271d\u00b9 : Semiring A inst\u271d : Algebra R' A f : M \u2192\u2097[R'] A hf : \u2200 (x y : M), \u2191f x * \u2191f y = 0 m : M \u22a2 \u2191(algebraMap R' A) 0 + \u2191f m = \u2191f m ** rw [RingHom.map_zero, zero_add] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.tendsto_addHaar_inter_smul_zero_of_density_zero ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u22a2 Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) ** refine' tendsto_order.2 \u27e8fun a' ha' => (ENNReal.not_lt_zero ha').elim, fun \u03b5 (\u03b5pos : 0 < \u03b5) => _\u27e9 ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** rcases eq_or_ne (\u03bc t) 0 with (h't | h't) ** case inr E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** obtain \u27e8n, npos, hn\u27e9 : \u2203 n : \u2115, 0 < n \u2227 \u03bc (t \\ closedBall 0 n) < \u03b5 / 2 * \u03bc t := by\n have A :\n Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop\n (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) := by\n have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn))\n have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n iUnion_closedBall_nat, compl_univ, inter_empty]\n simp only [this, measure_empty] at A\n have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't\n exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists ** case inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** have L :\n Tendsto (fun r : \u211d => \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) / \u03bc ({x} + r \u2022 t)) (\ud835\udcdd[>] 0)\n (\ud835\udcdd 0) :=\n tendsto_addHaar_inter_smul_zero_of_density_zero_aux2 \u03bc s x h _ t h't n (Nat.cast_pos.2 npos)\n (inter_subset_right _ _) ** case inr.intro.intro E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** filter_upwards [(tendsto_order.1 L).2 _ (ENNReal.half_pos \u03b5pos.ne'), self_mem_nhdsWithin] ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) \u22a2 \u2200 (a : \u211d), \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 / 2 \u2192 a \u2208 Ioi 0 \u2192 \u2191\u2191\u03bc (s \u2229 ({x} + a \u2022 t)) / \u2191\u2191\u03bc ({x} + a \u2022 t) < \u03b5 ** rintro r hr (rpos : 0 < r) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** have I :\n \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264\n \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n)) :=\n calc\n \u03bc (s \u2229 ({x} + r \u2022 t)) =\n \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 n))) :=\n by rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff]\n _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc (s \u2229 ({x} + r \u2022 (t \\ closedBall 0 n))) :=\n (measure_union_le _ _)\n _ \u2264 \u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n)) :=\n add_le_add le_rfl (measure_mono (inter_subset_right _ _)) ** case h E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** calc\n \u03bc (s \u2229 ({x} + r \u2022 t)) / \u03bc ({x} + r \u2022 t) \u2264\n (\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 n))) + \u03bc ({x} + r \u2022 (t \\ closedBall 0 n))) /\n \u03bc ({x} + r \u2022 t) :=\n mul_le_mul_right' I _\n _ < \u03b5 / 2 + \u03b5 / 2 := by\n rw [ENNReal.add_div]\n apply ENNReal.add_lt_add hr _\n rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne',\n ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)]\n _ = \u03b5 := ENNReal.add_halves _ ** case inl E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 \u22a2 \u2200\u1da0 (b : \u211d) in \ud835\udcdd[Ioi 0] 0, \u2191\u2191\u03bc (s \u2229 ({x} + b \u2022 t)) / \u2191\u2191\u03bc ({x} + b \u2022 t) < \u03b5 ** apply eventually_of_forall fun r => ?_ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** suffices H : \u03bc (s \u2229 ({x} + r \u2022 t)) = 0 ** case H E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0 ** apply le_antisymm _ (zero_le _) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 0 ** calc\n \u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u03bc ({x} + r \u2022 t) := measure_mono (inter_subset_right _ _)\n _ = 0 := by\n simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,\n mul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d H : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0 \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** rw [H] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d H : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = 0 \u22a2 0 / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 ** simpa only [ENNReal.zero_div] using \u03b5pos ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t = 0 r : \u211d \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 t) = 0 ** simp only [h't, addHaar_smul, image_add_left, measure_preimage_add, singleton_add,\n mul_zero] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** have A :\n Tendsto (fun n : \u2115 => \u03bc (t \\ closedBall 0 n)) atTop\n (\ud835\udcdd (\u03bc (\u22c2 n : \u2115, t \\ closedBall 0 n))) := by\n have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9\n refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N\n exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn)) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** have : \u22c2 n : \u2115, t \\ closedBall 0 n = \u2205 := by\n simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n iUnion_closedBall_nat, compl_univ, inter_empty] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) this : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** simp only [this, measure_empty] at A ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 this : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0) \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** have I : 0 < \u03b5 / 2 * \u03bc t := ENNReal.mul_pos (ENNReal.half_pos \u03b5pos.ne').ne' h't ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 this : \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd 0) I : 0 < \u03b5 / 2 * \u2191\u2191\u03bc t \u22a2 \u2203 n, 0 < n \u2227 \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t ** exact (Eventually.and (Ioi_mem_atTop 0) ((tendsto_order.1 A).2 _ I)).exists ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 \u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) ** have N : \u2203 n : \u2115, \u03bc (t \\ closedBall 0 n) \u2260 \u221e :=\n \u27e80, ((measure_mono (diff_subset t _)).trans_lt h''t.lt_top).ne\u27e9 ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 N : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4 \u22a2 Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) ** refine' tendsto_measure_iInter (fun n \u21a6 ht.diff measurableSet_closedBall) (fun m n hmn \u21a6 _) N ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 N : \u2203 n, \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) \u2260 \u22a4 m n : \u2115 hmn : m \u2264 n \u22a2 t \\ closedBall 0 \u2191n \u2264 t \\ closedBall 0 \u2191m ** exact diff_subset_diff Subset.rfl (closedBall_subset_closedBall (Nat.cast_le.2 hmn)) ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 A : Tendsto (fun n => \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n)) atTop (\ud835\udcdd (\u2191\u2191\u03bc (\u22c2 n, t \\ closedBall 0 \u2191n))) \u22a2 \u22c2 n, t \\ closedBall 0 \u2191n = \u2205 ** simp_rw [diff_eq, \u2190 inter_iInter, iInter_eq_compl_iUnion_compl, compl_compl,\n iUnion_closedBall_nat, compl_univ, inter_empty] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) = \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n)) \u222a s \u2229 ({x} + r \u2022 (t \\ closedBall 0 \u2191n))) ** rw [\u2190 inter_union_distrib_left, \u2190 add_union, \u2190 smul_set_union, inter_union_diff] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 (\u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 + \u03b5 / 2 ** rw [ENNReal.add_div] ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 + \u03b5 / 2 ** apply ENNReal.add_lt_add hr _ ** E : Type u_1 inst\u271d\u2078 : NormedAddCommGroup E inst\u271d\u2077 : NormedSpace \u211d E inst\u271d\u2076 : MeasurableSpace E inst\u271d\u2075 : BorelSpace E inst\u271d\u2074 : FiniteDimensional \u211d E \u03bc : Measure E inst\u271d\u00b3 : IsAddHaarMeasure \u03bc F : Type u_2 inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedSpace \u211d F inst\u271d : CompleteSpace F s\u271d s : Set E x : E h : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 closedBall x r) / \u2191\u2191\u03bc (closedBall x r)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) t : Set E ht : MeasurableSet t h''t : \u2191\u2191\u03bc t \u2260 \u22a4 \u03b5 : \u211d\u22650\u221e \u03b5pos : 0 < \u03b5 h't : \u2191\u2191\u03bc t \u2260 0 n : \u2115 npos : 0 < n hn : \u2191\u2191\u03bc (t \\ closedBall 0 \u2191n) < \u03b5 / 2 * \u2191\u2191\u03bc t L : Tendsto (fun r => \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd 0) r : \u211d hr : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 rpos : 0 < r I : \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 t)) \u2264 \u2191\u2191\u03bc (s \u2229 ({x} + r \u2022 (t \u2229 closedBall 0 \u2191n))) + \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) \u22a2 \u2191\u2191\u03bc ({x} + r \u2022 (t \\ closedBall 0 \u2191n)) / \u2191\u2191\u03bc ({x} + r \u2022 t) < \u03b5 / 2 ** rwa [addHaar_singleton_add_smul_div_singleton_add_smul \u03bc rpos.ne',\n ENNReal.div_lt_iff (Or.inl h't) (Or.inl h''t)] ** Qed", + "informal": "" + }, + { + "formal": "geometric_hahn_banach_open ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** obtain rfl | \u27e8a\u2080, ha\u2080\u27e9 := s.eq_empty_or_nonempty ** case inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** obtain rfl | \u27e8b\u2080, hb\u2080\u27e9 := t.eq_empty_or_nonempty ** case inr.intro.inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** let x\u2080 := b\u2080 - a\u2080 ** case inr.intro.inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** let C := x\u2080 +\u1d65 (s - t) ** case inr.intro.inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** have : (0 : E) \u2208 C :=\n \u27e8a\u2080 - b\u2080, sub_mem_sub ha\u2080 hb\u2080, by simp_rw [vadd_eq_add, sub_add_sub_cancel', sub_self]\u27e9 ** case inr.intro.inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this : 0 \u2208 C \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** have : Convex \u211d C := (hs\u2081.sub ht).vadd _ ** case inr.intro.inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d : 0 \u2208 C this : Convex \u211d C \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** have : x\u2080 \u2209 C := by\n intro hx\u2080\n rw [\u2190 add_zero x\u2080] at hx\u2080\n exact disj.zero_not_mem_sub_set (vadd_mem_vadd_set_iff.1 hx\u2080) ** case inr.intro.inr.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b9 : 0 \u2208 C this\u271d : Convex \u211d C this : \u00acx\u2080 \u2208 C \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** obtain \u27e8f, hf\u2081, hf\u2082\u27e9 := separate_convex_open_set \u20390 \u2208 C\u203a \u2039_\u203a (hs\u2082.sub_right.vadd _) \u2039x\u2080 \u2209 C\u203a ** case inr.intro.inr.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b9 : 0 \u2208 C this\u271d : Convex \u211d C this : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** have : f b\u2080 = f a\u2080 + 1 := by simp [\u2190 hf\u2081] ** case inr.intro.inr.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** have forall_le : \u2200 a \u2208 s, \u2200 b \u2208 t, f a \u2264 f b := by\n intro a ha b hb\n have := hf\u2082 (x\u2080 + (a - b)) (vadd_mem_vadd_set <| sub_mem_sub ha hb)\n simp only [f.map_add, f.map_sub, hf\u2081] at this\n linarith ** case inr.intro.inr.intro.intro.intro \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** refine' \u27e8f, sInf (f '' t), image_subset_iff.1 (_ : f '' s \u2286 Iio (sInf (f '' t))), fun b hb => _\u27e9 ** case inl \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E t : Set E x y : E ht : Convex \u211d t hs\u2081 : Convex \u211d \u2205 hs\u2082 : IsOpen \u2205 disj : Disjoint \u2205 t \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 \u2205 \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 t \u2192 u \u2264 \u2191f b ** exact \u27e80, 0, by simp, fun b _hb => le_rfl\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E t : Set E x y : E ht : Convex \u211d t hs\u2081 : Convex \u211d \u2205 hs\u2082 : IsOpen \u2205 disj : Disjoint \u2205 t \u22a2 \u2200 (a : E), a \u2208 \u2205 \u2192 \u21910 a < 0 ** simp ** case inr.intro.inl \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s a\u2080 : E ha\u2080 : a\u2080 \u2208 s ht : Convex \u211d \u2205 disj : Disjoint s \u2205 \u22a2 \u2203 f u, (\u2200 (a : E), a \u2208 s \u2192 \u2191f a < u) \u2227 \u2200 (b : E), b \u2208 \u2205 \u2192 u \u2264 \u2191f b ** exact \u27e80, 1, fun a _ha => zero_lt_one, by simp\u27e9 ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s a\u2080 : E ha\u2080 : a\u2080 \u2208 s ht : Convex \u211d \u2205 disj : Disjoint s \u2205 \u22a2 \u2200 (b : E), b \u2208 \u2205 \u2192 1 \u2264 \u21910 b ** simp ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) \u22a2 (fun x => x\u2080 +\u1d65 x) (a\u2080 - b\u2080) = 0 ** simp_rw [vadd_eq_add, sub_add_sub_cancel', sub_self] ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d : 0 \u2208 C this : Convex \u211d C \u22a2 \u00acx\u2080 \u2208 C ** intro hx\u2080 ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d : 0 \u2208 C this : Convex \u211d C hx\u2080 : x\u2080 \u2208 C \u22a2 False ** rw [\u2190 add_zero x\u2080] at hx\u2080 ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d : 0 \u2208 C this : Convex \u211d C hx\u2080 : x\u2080 + 0 \u2208 C \u22a2 False ** exact disj.zero_not_mem_sub_set (vadd_mem_vadd_set_iff.1 hx\u2080) ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b9 : 0 \u2208 C this\u271d : Convex \u211d C this : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 \u22a2 \u2191f b\u2080 = \u2191f a\u2080 + 1 ** simp [\u2190 hf\u2081] ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 \u22a2 \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b ** intro a ha b hb ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 a : E ha : a \u2208 s b : E hb : b \u2208 t \u22a2 \u2191f a \u2264 \u2191f b ** have := hf\u2082 (x\u2080 + (a - b)) (vadd_mem_vadd_set <| sub_mem_sub ha hb) ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b3 : 0 \u2208 C this\u271d\u00b2 : Convex \u211d C this\u271d\u00b9 : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this\u271d : \u2191f b\u2080 = \u2191f a\u2080 + 1 a : E ha : a \u2208 s b : E hb : b \u2208 t this : \u2191f (x\u2080 + (a - b)) < 1 \u22a2 \u2191f a \u2264 \u2191f b ** simp only [f.map_add, f.map_sub, hf\u2081] at this ** \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b3 : 0 \u2208 C this\u271d\u00b2 : Convex \u211d C this\u271d\u00b9 : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this\u271d : \u2191f b\u2080 = \u2191f a\u2080 + 1 a : E ha : a \u2208 s b : E hb : b \u2208 t this : \u2191f b\u2080 - \u2191f a\u2080 + (\u2191f a - \u2191f b) < 1 \u22a2 \u2191f a \u2264 \u2191f b ** linarith ** case inr.intro.inr.intro.intro.intro.refine'_1 \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b \u22a2 \u2191f '' s \u2286 Iio (sInf (\u2191f '' t)) ** rw [\u2190 interior_Iic] ** case inr.intro.inr.intro.intro.intro.refine'_1 \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b \u22a2 \u2191f '' s \u2286 interior (Iic (sInf (\u2191f '' t))) ** refine' interior_maximal (image_subset_iff.2 fun a ha => _) (f.isOpenMap_of_ne_zero _ _ hs\u2082) ** case inr.intro.inr.intro.intro.intro.refine'_1.refine'_1 \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b a : E ha : a \u2208 s \u22a2 a \u2208 \u2191f \u207b\u00b9' Iic (sInf (\u2191f '' t)) ** exact le_csInf (Nonempty.image _ \u27e8_, hb\u2080\u27e9) (ball_image_of_ball <| forall_le _ ha) ** case inr.intro.inr.intro.intro.intro.refine'_1.refine'_2 \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b \u22a2 f \u2260 0 ** rintro rfl ** case inr.intro.inr.intro.intro.intro.refine'_1.refine'_2 \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C hf\u2081 : \u21910 x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u21910 x < 1 this : \u21910 b\u2080 = \u21910 a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u21910 a \u2264 \u21910 b \u22a2 False ** simp at hf\u2081 ** case inr.intro.inr.intro.intro.intro.refine'_2 \ud835\udd5c : Type u_1 E : Type u_2 inst\u271d\u2074 : TopologicalSpace E inst\u271d\u00b3 : AddCommGroup E inst\u271d\u00b2 : TopologicalAddGroup E inst\u271d\u00b9 : Module \u211d E inst\u271d : ContinuousSMul \u211d E s t : Set E x y : E hs\u2081 : Convex \u211d s hs\u2082 : IsOpen s ht : Convex \u211d t disj : Disjoint s t a\u2080 : E ha\u2080 : a\u2080 \u2208 s b\u2080 : E hb\u2080 : b\u2080 \u2208 t x\u2080 : E := b\u2080 - a\u2080 C : Set E := x\u2080 +\u1d65 (s - t) this\u271d\u00b2 : 0 \u2208 C this\u271d\u00b9 : Convex \u211d C this\u271d : \u00acx\u2080 \u2208 C f : E \u2192L[\u211d] \u211d hf\u2081 : \u2191f x\u2080 = 1 hf\u2082 : \u2200 (x : E), x \u2208 C \u2192 \u2191f x < 1 this : \u2191f b\u2080 = \u2191f a\u2080 + 1 forall_le : \u2200 (a : E), a \u2208 s \u2192 \u2200 (b : E), b \u2208 t \u2192 \u2191f a \u2264 \u2191f b b : E hb : b \u2208 t \u22a2 sInf (\u2191f '' t) \u2264 \u2191f b ** exact csInf_le \u27e8f a\u2080, ball_image_of_ball <| forall_le _ ha\u2080\u27e9 (mem_image_of_mem _ hb) ** Qed", + "informal": "" + }, + { + "formal": "OpenSubgroup.isClosed ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G \u22a2 IsClosed \u2191U ** apply isOpen_compl_iff.1 ** G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G \u22a2 IsOpen (\u2191U)\u1d9c ** refine' isOpen_iff_forall_mem_open.2 fun x hx => \u27e8(fun y => y * x\u207b\u00b9) \u207b\u00b9' U, _, _, _\u27e9 ** case refine'_1 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G x : G hx : x \u2208 (\u2191U)\u1d9c \u22a2 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U \u2286 (\u2191U)\u1d9c ** refine' fun u hux hu => hx _ ** case refine'_1 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G x : G hx : x \u2208 (\u2191U)\u1d9c u : G hux : u \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U hu : u \u2208 \u2191U \u22a2 x \u2208 \u2191U ** simp only [Set.mem_preimage, SetLike.mem_coe] at hux hu \u22a2 ** case refine'_1 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G x : G hx : x \u2208 (\u2191U)\u1d9c u : G hux : u * x\u207b\u00b9 \u2208 U hu : u \u2208 U \u22a2 x \u2208 U ** convert U.mul_mem (U.inv_mem hux) hu ** case a G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G x : G hx : x \u2208 (\u2191U)\u1d9c u : G hux : u * x\u207b\u00b9 \u2208 U hu : u \u2208 U \u22a2 x \u2208 U \u2194 (u * x\u207b\u00b9)\u207b\u00b9 * u \u2208 \u2191U ** simp ** case refine'_2 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G x : G hx : x \u2208 (\u2191U)\u1d9c \u22a2 IsOpen ((fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U) ** exact U.isOpen.preimage (continuous_mul_right _) ** case refine'_3 G : Type u_1 inst\u271d\u00b2 : Group G inst\u271d\u00b9 : TopologicalSpace G U\u271d V : OpenSubgroup G g : G inst\u271d : ContinuousMul G U : OpenSubgroup G x : G hx : x \u2208 (\u2191U)\u1d9c \u22a2 x \u2208 (fun y => y * x\u207b\u00b9) \u207b\u00b9' \u2191U ** simp [one_mem] ** Qed", + "informal": "" + }, + { + "formal": "Finset.card_le_mul_card_image_of_maps_to ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : DecidableEq \u03b2 f : \u03b1 \u2192 \u03b2 s : Finset \u03b1 t : Finset \u03b2 Hf : \u2200 (a : \u03b1), a \u2208 s \u2192 f a \u2208 t n : \u2115 hn : \u2200 (a : \u03b2), a \u2208 t \u2192 card (filter (fun x => f x = a) s) \u2264 n \u22a2 \u2211 _a in t, n = n * card t ** simp [mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "Nat.digits_add_two_add_one ** n\u271d b n : \u2115 \u22a2 digits (b + 2) (n + 1) = (n + 1) % (b + 2) :: digits (b + 2) ((n + 1) / (b + 2)) ** simp [digits, digitsAux_def] ** Qed", + "informal": "" + }, + { + "formal": "AffineSubspace.mem_perpBisector_iff_dist_eq' ** V : Type u_2 P : Type u_1 inst\u271d\u00b3 : NormedAddCommGroup V inst\u271d\u00b2 : InnerProductSpace \u211d V inst\u271d\u00b9 : MetricSpace P inst\u271d : NormedAddTorsor V P c c\u2081 c\u2082 p\u2081 p\u2082 : P \u22a2 c \u2208 perpBisector p\u2081 p\u2082 \u2194 dist p\u2081 c = dist p\u2082 c ** simp only [mem_perpBisector_iff_dist_eq, dist_comm] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.ofFinsupp_single ** R : Type u a b : R m n\u271d : \u2115 inst\u271d : Semiring R p q : R[X] n : \u2115 r : R \u22a2 { toFinsupp := fun\u2080 | n => r } = \u2191(monomial n) r ** simp [monomial] ** Qed", + "informal": "" + }, + { + "formal": "Complex.Gamma_eq_GammaAux ** s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 Gamma s = GammaAux n s ** convert (u <| n - \u230a1 - s.re\u230b\u208a).symm ** case h.e'_3.h.e'_1 s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 n = \u230a1 - s.re\u230b\u208a + (n - \u230a1 - s.re\u230b\u208a) ** rw [Nat.add_sub_of_le] ** case h.e'_3.h.e'_1 s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 \u230a1 - s.re\u230b\u208a \u2264 n ** by_cases 0 \u2264 1 - s.re ** s : \u2102 n : \u2115 h1 : -s.re < \u2191n \u22a2 \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s ** intro k ** s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 \u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s ** induction' k with k hk ** case zero s : \u2102 n : \u2115 h1 : -s.re < \u2191n \u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + Nat.zero) s = Gamma s ** simp [Gamma] ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + Nat.succ k) s = Gamma s ** rw [\u2190 hk, Nat.succ_eq_add_one, \u2190 add_assoc] ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 GammaAux (\u230a1 - s.re\u230b\u208a + k + 1) s = GammaAux (\u230a1 - s.re\u230b\u208a + k) s ** refine' (GammaAux_recurrence2 s (\u230a1 - s.re\u230b\u208a + k) _).symm ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 -s.re < \u2191(\u230a1 - s.re\u230b\u208a + k) ** rw [Nat.cast_add] ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s \u22a2 -s.re < \u2191\u230a1 - s.re\u230b\u208a + \u2191k ** have i0 := Nat.sub_one_lt_floor (1 - s.re) ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s i0 : 1 - s.re - 1 < \u2191\u230a1 - s.re\u230b\u208a \u22a2 -s.re < \u2191\u230a1 - s.re\u230b\u208a + \u2191k ** simp only [sub_sub_cancel_left] at i0 ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s i0 : -s.re < \u2191\u230a1 - s.re\u230b\u208a \u22a2 -s.re < \u2191\u230a1 - s.re\u230b\u208a + \u2191k ** refine' lt_add_of_lt_of_nonneg i0 _ ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s i0 : -s.re < \u2191\u230a1 - s.re\u230b\u208a \u22a2 0 \u2264 \u2191k ** rw [\u2190 Nat.cast_zero, Nat.cast_le] ** case succ s : \u2102 n : \u2115 h1 : -s.re < \u2191n k : \u2115 hk : GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s i0 : -s.re < \u2191\u230a1 - s.re\u230b\u208a \u22a2 0 \u2264 k ** exact Nat.zero_le k ** case pos s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : 0 \u2264 1 - s.re \u22a2 \u230a1 - s.re\u230b\u208a \u2264 n ** apply Nat.le_of_lt_succ ** case pos.a s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : 0 \u2264 1 - s.re \u22a2 \u230a1 - s.re\u230b\u208a < Nat.succ n ** exact_mod_cast lt_of_le_of_lt (Nat.floor_le h) (by linarith : 1 - s.re < n + 1) ** s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : 0 \u2264 1 - s.re \u22a2 1 - s.re < \u2191n + 1 ** linarith ** case neg s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : \u00ac0 \u2264 1 - s.re \u22a2 \u230a1 - s.re\u230b\u208a \u2264 n ** rw [Nat.floor_of_nonpos] ** case neg s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : \u00ac0 \u2264 1 - s.re \u22a2 0 \u2264 n case neg s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : \u00ac0 \u2264 1 - s.re \u22a2 1 - s.re \u2264 0 ** linarith ** case neg s : \u2102 n : \u2115 h1 : -s.re < \u2191n u : \u2200 (k : \u2115), GammaAux (\u230a1 - s.re\u230b\u208a + k) s = Gamma s h : \u00ac0 \u2264 1 - s.re \u22a2 1 - s.re \u2264 0 ** linarith ** Qed", + "informal": "" + }, + { + "formal": "Int.ediv_neg ** m : Nat \u22a2 ofNat (m / 0) = -\u2191(m / 0) ** rw [Nat.div_zero] ** m : Nat \u22a2 ofNat 0 = -\u21910 ** rfl ** Qed", + "informal": "" + }, + { + "formal": "List.mem_of_mem_inter_right ** \u03b1 : Type u_1 l l\u2081 l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 inst\u271d : DecidableEq \u03b1 h : a \u2208 l\u2081 \u2229 l\u2082 \u22a2 a \u2208 l\u2082 ** simpa using of_mem_filter h ** Qed", + "informal": "" + }, + { + "formal": "InnerProductGeometry.cos_angle_add_angle_sub_add_angle_sub_eq_neg_one ** V : Type u_1 inst\u271d\u00b9 : NormedAddCommGroup V inst\u271d : InnerProductSpace \u211d V x y : V hx : x \u2260 0 hy : y \u2260 0 \u22a2 Real.cos (angle x y + angle x (x - y) + angle y (y - x)) = -1 ** rw [add_assoc, Real.cos_add, cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle hx hy,\n sin_angle_sub_add_angle_sub_rev_eq_sin_angle hx hy, mul_neg, \u2190 neg_add', add_comm, \u2190 sq, \u2190 sq,\n Real.sin_sq_add_cos_sq] ** Qed", + "informal": "" + }, + { + "formal": "TensorProduct.add_left_neg ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x : M \u2297[R] N \u22a2 -0 + 0 = 0 ** rw [add_zero] ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x : M \u2297[R] N \u22a2 -0 = 0 ** apply (Neg.aux R).map_zero ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d : M \u2297[R] N x : M y : N \u22a2 -x \u2297\u209c[R] y + x \u2297\u209c[R] y = 0 ** convert (add_tmul (R := R) (-x) x y).symm ** case h.e'_3 R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d : M \u2297[R] N x : M y : N \u22a2 0 = (-x + x) \u2297\u209c[R] y ** rw [add_left_neg, zero_tmul] ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 \u22a2 -(x + y) + (x + y) = 0 ** suffices : -x + x + (-y + y) = 0 ** case this R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 \u22a2 -x + x + (-y + y) = 0 ** rw [hx, hy, add_zero] ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 this : -x + x + (-y + y) = 0 \u22a2 -(x + y) + (x + y) = 0 ** rw [\u2190 this] ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 this : -x + x + (-y + y) = 0 \u22a2 -(x + y) + (x + y) = -x + x + (-y + y) ** unfold Neg.neg neg ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 this : -x + x + (-y + y) = 0 \u22a2 { neg := \u2191(Neg.aux R) }.1 (x + y) + (x + y) = { neg := \u2191(Neg.aux R) }.1 x + x + ({ neg := \u2191(Neg.aux R) }.1 y + y) ** simp only ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 this : -x + x + (-y + y) = 0 \u22a2 \u2191(Neg.aux R) (x + y) + (x + y) = \u2191(Neg.aux R) x + x + (\u2191(Neg.aux R) y + y) ** rw [map_add] ** R : Type u_1 inst\u271d\u00b9\u2070 : CommSemiring R M : Type u_2 N : Type u_3 P : Type u_4 Q : Type u_5 S : Type u_6 inst\u271d\u2079 : AddCommGroup M inst\u271d\u2078 : AddCommGroup N inst\u271d\u2077 : AddCommGroup P inst\u271d\u2076 : AddCommGroup Q inst\u271d\u2075 : AddCommGroup S inst\u271d\u2074 : Module R M inst\u271d\u00b3 : Module R N inst\u271d\u00b2 : Module R P inst\u271d\u00b9 : Module R Q inst\u271d : Module R S x\u271d x y : M \u2297[R] N hx : -x + x = 0 hy : -y + y = 0 this : -x + x + (-y + y) = 0 \u22a2 \u2191(Neg.aux R) x + \u2191(Neg.aux R) y + (x + y) = \u2191(Neg.aux R) x + x + (\u2191(Neg.aux R) y + y) ** abel ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.map_roots_le ** R : Type u S : Type v T : Type w a b : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : IsDomain B p : A[X] f : A \u2192+* B h : map f p \u2260 0 \u22a2 Multiset.map (\u2191f) (roots p) \u2264 roots (map f p) ** classical\nexact Multiset.le_iff_count.2 fun b => by\n rw [count_roots]\n apply count_map_roots h ** R : Type u S : Type v T : Type w a b : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : IsDomain B p : A[X] f : A \u2192+* B h : map f p \u2260 0 \u22a2 Multiset.map (\u2191f) (roots p) \u2264 roots (map f p) ** exact Multiset.le_iff_count.2 fun b => by\n rw [count_roots]\n apply count_map_roots h ** R : Type u S : Type v T : Type w a b\u271d : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : IsDomain B p : A[X] f : A \u2192+* B h : map f p \u2260 0 b : B \u22a2 Multiset.count b (Multiset.map (\u2191f) (roots p)) \u2264 Multiset.count b (roots (map f p)) ** rw [count_roots] ** R : Type u S : Type v T : Type w a b\u271d : R n : \u2115 A : Type u_1 B : Type u_2 inst\u271d\u00b3 : CommRing A inst\u271d\u00b2 : CommRing B inst\u271d\u00b9 : IsDomain A inst\u271d : IsDomain B p : A[X] f : A \u2192+* B h : map f p \u2260 0 b : B \u22a2 Multiset.count b (Multiset.map (\u2191f) (roots p)) \u2264 rootMultiplicity b (map f p) ** apply count_map_roots h ** Qed", + "informal": "" + }, + { + "formal": "WfDvdMonoid.of_wellFounded_associates ** \u03b1 : Type u_1 inst\u271d : CancelCommMonoidWithZero \u03b1 h : WellFounded fun x x_1 => x < x_1 \u22a2 WellFounded DvdNotUnit ** convert h ** case h.e'_2.h.h.h.e \u03b1 : Type u_1 inst\u271d : CancelCommMonoidWithZero \u03b1 h : WellFounded fun x x_1 => x < x_1 x\u271d\u00b9 x\u271d : Associates \u03b1 \u22a2 DvdNotUnit = LT.lt ** ext ** case h.e'_2.h.h.h.e.h.h.a \u03b1 : Type u_1 inst\u271d : CancelCommMonoidWithZero \u03b1 h : WellFounded fun x x_1 => x < x_1 x\u271d\u00b3 x\u271d\u00b2 x\u271d\u00b9 x\u271d : Associates \u03b1 \u22a2 DvdNotUnit x\u271d\u00b9 x\u271d \u2194 x\u271d\u00b9 < x\u271d ** exact Associates.dvdNotUnit_iff_lt ** Qed", + "informal": "" + }, + { + "formal": "IsROrC.ofReal_mul_re ** K : Type u_1 E : Type u_2 inst\u271d : IsROrC K r : \u211d z : K \u22a2 \u2191re (\u2191r * z) = r * \u2191re z ** simp only [mul_re, ofReal_im, zero_mul, ofReal_re, sub_zero] ** Qed", + "informal": "" + }, + { + "formal": "Nat.find_pos ** m n k l : \u2115 p q : \u2115 \u2192 Prop inst\u271d\u00b9 : DecidablePred p inst\u271d : DecidablePred q h : \u2203 n, p n \u22a2 0 < Nat.find h \u2194 \u00acp 0 ** rw [pos_iff_ne_zero, Ne, Nat.find_eq_zero] ** Qed", + "informal": "" + }, + { + "formal": "LieSubmodule.lie_le_inf ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 \u2045I, J\u2046 \u2264 I \u2293 J ** rw [le_inf_iff] ** R : Type u L : Type v M : Type w M\u2082 : Type w\u2081 inst\u271d\u00b9\u2070 : CommRing R inst\u271d\u2079 : LieRing L inst\u271d\u2078 : LieAlgebra R L inst\u271d\u2077 : AddCommGroup M inst\u271d\u2076 : Module R M inst\u271d\u2075 : LieRingModule L M inst\u271d\u2074 : LieModule R L M inst\u271d\u00b3 : AddCommGroup M\u2082 inst\u271d\u00b2 : Module R M\u2082 inst\u271d\u00b9 : LieRingModule L M\u2082 inst\u271d : LieModule R L M\u2082 N N' : LieSubmodule R L M I J : LieIdeal R L N\u2082 : LieSubmodule R L M\u2082 \u22a2 \u2045I, J\u2046 \u2264 I \u2227 \u2045I, J\u2046 \u2264 J ** exact \u27e8lie_le_left I J, lie_le_right J I\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Ideal.spanNorm_singleton ** R : Type u_1 inst\u271d\u00b2 : CommRing R S : Type u_2 inst\u271d\u00b9 : CommRing S inst\u271d : Algebra R S r : S x : R hx : x \u2208 \u2191(Algebra.norm R) '' \u2191(span {r}) \u22a2 \u2191(Algebra.norm R) r \u2223 x ** obtain \u27e8x, hx', rfl\u27e9 := (Set.mem_image _ _ _).mp hx ** case intro.intro R : Type u_1 inst\u271d\u00b2 : CommRing R S : Type u_2 inst\u271d\u00b9 : CommRing S inst\u271d : Algebra R S r x : S hx' : x \u2208 \u2191(span {r}) hx : \u2191(Algebra.norm R) x \u2208 \u2191(Algebra.norm R) '' \u2191(span {r}) \u22a2 \u2191(Algebra.norm R) r \u2223 \u2191(Algebra.norm R) x ** exact map_dvd _ (mem_span_singleton.mp hx') ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.weightedSMul_empty ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 \u22a2 weightedSMul \u03bc \u2205 = 0 ** ext1 x ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 x : F \u22a2 \u2191(weightedSMul \u03bc \u2205) x = \u21910 x ** rw [weightedSMul_apply] ** case h \u03b1 : Type u_1 E : Type u_2 F : Type u_3 \ud835\udd5c : Type u_4 inst\u271d\u00b9 : NormedAddCommGroup F inst\u271d : NormedSpace \u211d F m\u271d : MeasurableSpace \u03b1 \u03bc\u271d : Measure \u03b1 m : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 x : F \u22a2 ENNReal.toReal (\u2191\u2191\u03bc \u2205) \u2022 x = \u21910 x ** simp ** Qed", + "informal": "" + }, + { + "formal": "Rack.ad_conj ** R\u271d : Type u_1 inst\u271d\u00b9 : Rack R\u271d R : Type u_2 inst\u271d : Rack R x y : R \u22a2 act' (x \u25c3 y) = act' x * act' y * (act' x)\u207b\u00b9 ** rw [eq_mul_inv_iff_mul_eq] ** R\u271d : Type u_1 inst\u271d\u00b9 : Rack R\u271d R : Type u_2 inst\u271d : Rack R x y : R \u22a2 act' (x \u25c3 y) * act' x = act' x * act' y ** ext z ** case H R\u271d : Type u_1 inst\u271d\u00b9 : Rack R\u271d R : Type u_2 inst\u271d : Rack R x y z : R \u22a2 \u2191(act' (x \u25c3 y) * act' x) z = \u2191(act' x * act' y) z ** apply self_distrib.symm ** Qed", + "informal": "" + }, + { + "formal": "EuclideanGeometry.orthogonalProjection_orthogonalProjection ** V : Type u_1 P : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : InnerProductSpace \u211d V inst\u271d\u00b3 : MetricSpace P inst\u271d\u00b2 : NormedAddTorsor V P s : AffineSubspace \u211d P inst\u271d\u00b9 : Nonempty { x // x \u2208 s } inst\u271d : HasOrthogonalProjection (direction s) p : P \u22a2 \u2191(orthogonalProjection s) \u2191(\u2191(orthogonalProjection s) p) = \u2191(orthogonalProjection s) p ** ext ** case a V : Type u_1 P : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : InnerProductSpace \u211d V inst\u271d\u00b3 : MetricSpace P inst\u271d\u00b2 : NormedAddTorsor V P s : AffineSubspace \u211d P inst\u271d\u00b9 : Nonempty { x // x \u2208 s } inst\u271d : HasOrthogonalProjection (direction s) p : P \u22a2 \u2191(\u2191(orthogonalProjection s) \u2191(\u2191(orthogonalProjection s) p)) = \u2191(\u2191(orthogonalProjection s) p) ** rw [orthogonalProjection_eq_self_iff] ** case a V : Type u_1 P : Type u_2 inst\u271d\u2075 : NormedAddCommGroup V inst\u271d\u2074 : InnerProductSpace \u211d V inst\u271d\u00b3 : MetricSpace P inst\u271d\u00b2 : NormedAddTorsor V P s : AffineSubspace \u211d P inst\u271d\u00b9 : Nonempty { x // x \u2208 s } inst\u271d : HasOrthogonalProjection (direction s) p : P \u22a2 \u2191(\u2191(orthogonalProjection s) p) \u2208 s ** exact orthogonalProjection_mem p ** Qed", + "informal": "" + }, + { + "formal": "Rat.cast_mk_of_ne_zero ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 \u22a2 \u2191(a /. b) = \u2191a / \u2191b ** have b0' : b \u2260 0 := by\n refine' mt _ b0\n simp (config := { contextual := true }) ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 \u22a2 \u2191(a /. b) = \u2191a / \u2191b ** cases' e : a /. b with n d h c ** case mk' F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d \u22a2 \u2191(mk' n d) = \u2191a / \u2191b ** have d0 : (d : \u03b1) \u2260 0 := by\n intro d0\n have dd := den_dvd a b\n cases' show (d : \u2124) \u2223 b by rwa [e] at dd with k ke\n have : (b : \u03b1) = (d : \u03b1) * (k : \u03b1) := by rw [ke, Int.cast_mul, Int.cast_ofNat]\n rw [d0, zero_mul] at this\n contradiction ** case mk' F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d \u2260 0 \u22a2 \u2191(mk' n d) = \u2191a / \u2191b ** rw [num_den'] at e ** case mk' F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = n /. \u2191d d0 : \u2191d \u2260 0 \u22a2 \u2191(mk' n d) = \u2191a / \u2191b ** have := congr_arg ((\u2191) : \u2124 \u2192 \u03b1)\n ((divInt_eq_iff b0' <| ne_of_gt <| Int.coe_nat_pos.2 h.bot_lt).1 e) ** case mk' F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = n /. \u2191d d0 : \u2191d \u2260 0 this : \u2191(a * \u2191d) = \u2191(n * b) \u22a2 \u2191(mk' n d) = \u2191a / \u2191b ** rw [Int.cast_mul, Int.cast_mul, Int.cast_ofNat] at this ** case mk' F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = n /. \u2191d d0 : \u2191d \u2260 0 this : \u2191a * \u2191d = \u2191n * \u2191b \u22a2 \u2191(mk' n d) = \u2191a / \u2191b ** apply Eq.symm ** case mk'.h F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = n /. \u2191d d0 : \u2191d \u2260 0 this : \u2191a * \u2191d = \u2191n * \u2191b \u22a2 \u2191a / \u2191b = \u2191(mk' n d) ** rw [cast_def, div_eq_mul_inv, eq_div_iff_mul_eq d0, mul_assoc, (d.commute_cast _).eq, \u2190 mul_assoc,\n this, mul_assoc, mul_inv_cancel b0, mul_one] ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 \u22a2 b \u2260 0 ** refine' mt _ b0 ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 \u22a2 b = 0 \u2192 \u2191b = 0 ** simp (config := { contextual := true }) ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d \u22a2 \u2191d \u2260 0 ** intro d0 ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 \u22a2 False ** have dd := den_dvd a b ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 dd : \u2191(a /. b).den \u2223 b \u22a2 False ** cases' show (d : \u2124) \u2223 b by rwa [e] at dd with k ke ** case intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 dd : \u2191(a /. b).den \u2223 b k : \u2124 ke : b = \u2191d * k \u22a2 False ** have : (b : \u03b1) = (d : \u03b1) * (k : \u03b1) := by rw [ke, Int.cast_mul, Int.cast_ofNat] ** case intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 dd : \u2191(a /. b).den \u2223 b k : \u2124 ke : b = \u2191d * k this : \u2191b = \u2191d * \u2191k \u22a2 False ** rw [d0, zero_mul] at this ** case intro F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 dd : \u2191(a /. b).den \u2223 b k : \u2124 ke : b = \u2191d * k this : \u2191b = 0 \u22a2 False ** contradiction ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 dd : \u2191(a /. b).den \u2223 b \u22a2 \u2191d \u2223 b ** rwa [e] at dd ** F : Type u_1 \u03b9 : Type u_2 \u03b1 : Type u_3 \u03b2 : Type u_4 inst\u271d : DivisionRing \u03b1 a b : \u2124 b0 : \u2191b \u2260 0 b0' : b \u2260 0 n : \u2124 d : \u2115 h : d \u2260 0 c : Nat.Coprime (Int.natAbs n) d e : a /. b = mk' n d d0 : \u2191d = 0 dd : \u2191(a /. b).den \u2223 b k : \u2124 ke : b = \u2191d * k \u22a2 \u2191b = \u2191d * \u2191k ** rw [ke, Int.cast_mul, Int.cast_ofNat] ** Qed", + "informal": "" + }, + { + "formal": "Vector.mem_cons_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 n : \u2115 a a' : \u03b1 v : Vector \u03b1 n \u22a2 a' \u2208 toList (a ::\u1d65 v) \u2194 a' = a \u2228 a' \u2208 toList v ** rw [Vector.toList_cons, List.mem_cons] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.prod_X_sub_C_coeff ** R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 coeff (prod (map (fun t => X - \u2191C t) s)) k = (-1) ^ (\u2191card s - k) * esymm s (\u2191card s - k) ** conv_lhs =>\n congr\n congr\n congr\n ext x\n rw [sub_eq_add_neg]\n rw [\u2190 map_neg C x] ** R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 coeff (prod (map (fun x => X + \u2191C (-x)) s)) k = (-1) ^ (\u2191card s - k) * esymm s (\u2191card s - k) ** convert prod_X_add_C_coeff (map (fun t => -t) s) _ using 1 ** case h.e'_2 R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 coeff (prod (map (fun x => X + \u2191C (-x)) s)) k = coeff (prod (map (fun r => X + \u2191C r) (map (fun t => -t) s))) ?convert_1 ** rw [map_map] ** case h.e'_2 R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 coeff (prod (map (fun x => X + \u2191C (-x)) s)) k = coeff (prod (map ((fun r => X + \u2191C r) \u2218 fun t => -t) s)) ?convert_1 case convert_1 R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 \u2115 ** rfl ** case h.e'_3 R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 (-1) ^ (\u2191card s - k) * esymm s (\u2191card s - k) = esymm (map (fun t => -t) s) (\u2191card (map (fun t => -t) s) - k) ** rw [esymm_neg, card_map] ** case convert_2 R : Type u_1 inst\u271d : CommRing R s : Multiset R k : \u2115 h : k \u2264 \u2191card s \u22a2 k \u2264 \u2191card (map (fun t => -t) s) ** rwa [card_map] ** Qed", + "informal": "" + }, + { + "formal": "GaloisConnection.bddAbove_l_image ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x \u03ba : \u03b9 \u2192 Sort u_1 a a\u2081 a\u2082 : \u03b1 b b\u2081 b\u2082 : \u03b2 inst\u271d\u00b9 : Preorder \u03b1 inst\u271d : Preorder \u03b2 l : \u03b1 \u2192 \u03b2 u : \u03b2 \u2192 \u03b1 gc : GaloisConnection l u s : Set \u03b1 x\u271d : BddAbove (l '' s) x : \u03b2 hx : x \u2208 upperBounds (l '' s) \u22a2 u x \u2208 upperBounds s ** rwa [gc.upperBounds_l_image] at hx ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.isBridge_iff_adj_and_forall_cycle_not_mem ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V \u22a2 IsBridge G (Quotient.mk (Sym2.Rel.setoid V) (v, w)) \u2194 Adj G v w \u2227 \u2200 \u2983u : V\u2984 (p : Walk G u u), Walk.IsCycle p \u2192 \u00acQuotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 Walk.edges p ** rw [isBridge_iff, and_congr_right_iff] ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V \u22a2 Adj G v w \u2192 (\u00acReachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w \u2194 \u2200 \u2983u : V\u2984 (p : Walk G u u), Walk.IsCycle p \u2192 \u00acQuotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 Walk.edges p) ** intro h ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V h : Adj G v w \u22a2 \u00acReachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w \u2194 \u2200 \u2983u : V\u2984 (p : Walk G u u), Walk.IsCycle p \u2192 \u00acQuotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 Walk.edges p ** rw [\u2190 not_iff_not] ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V h : Adj G v w \u22a2 \u00ac\u00acReachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w \u2194 \u00ac\u2200 \u2983u : V\u2984 (p : Walk G u u), Walk.IsCycle p \u2192 \u00acQuotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 Walk.edges p ** push_neg ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V h : Adj G v w \u22a2 Reachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w \u2194 Exists fun \u2983u\u2984 => \u2203 p, Walk.IsCycle p \u2227 Quotient.mk (Sym2.Rel.setoid V) (v, w) \u2208 Walk.edges p ** rw [\u2190 adj_and_reachable_delete_edges_iff_exists_cycle] ** V : Type u V' : Type v V'' : Type w G : SimpleGraph V G' : SimpleGraph V' G'' : SimpleGraph V'' v w : V h : Adj G v w \u22a2 Reachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w \u2194 Adj G v w \u2227 Reachable (G \\ fromEdgeSet {Quotient.mk (Sym2.Rel.setoid V) (v, w)}) v w ** simp only [h, true_and_iff] ** Qed", + "informal": "" + }, + { + "formal": "AffineSubspace.Parallel.symm ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s\u2081 s\u2082 : AffineSubspace k P h : s\u2081 \u2225 s\u2082 \u22a2 s\u2082 \u2225 s\u2081 ** rcases h with \u27e8v, rfl\u27e9 ** case intro k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s\u2081 : AffineSubspace k P v : V \u22a2 map (\u2191(constVAdd k P v)) s\u2081 \u2225 s\u2081 ** refine' \u27e8-v, _\u27e9 ** case intro k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V inst\u271d : AffineSpace V P s\u2081 : AffineSubspace k P v : V \u22a2 s\u2081 = map (\u2191(constVAdd k P (-v))) (map (\u2191(constVAdd k P v)) s\u2081) ** rw [map_map, \u2190 coe_trans_to_affineMap, \u2190 constVAdd_add, neg_add_self, constVAdd_zero,\n coe_refl_to_affineMap, map_id] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.incMatrix_of_not_mem_incidenceSet ** R : Type u_1 \u03b1 : Type u_2 G : SimpleGraph \u03b1 inst\u271d : MulZeroOneClass R a b : \u03b1 e : Sym2 \u03b1 h : \u00ace \u2208 incidenceSet G a \u22a2 incMatrix R G a e = 0 ** rw [incMatrix_apply, Set.indicator_of_not_mem h] ** Qed", + "informal": "" + }, + { + "formal": "ultrafilter_extend_extends ** \u03b1 : Type u \u03b3 : Type u_1 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : T2Space \u03b3 f : \u03b1 \u2192 \u03b3 \u22a2 Ultrafilter.extend f \u2218 pure = f ** letI : TopologicalSpace \u03b1 := \u22a5 ** \u03b1 : Type u \u03b3 : Type u_1 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : T2Space \u03b3 f : \u03b1 \u2192 \u03b3 this : TopologicalSpace \u03b1 := \u22a5 \u22a2 Ultrafilter.extend f \u2218 pure = f ** haveI : DiscreteTopology \u03b1 := \u27e8rfl\u27e9 ** \u03b1 : Type u \u03b3 : Type u_1 inst\u271d\u00b9 : TopologicalSpace \u03b3 inst\u271d : T2Space \u03b3 f : \u03b1 \u2192 \u03b3 this\u271d : TopologicalSpace \u03b1 := \u22a5 this : DiscreteTopology \u03b1 \u22a2 Ultrafilter.extend f \u2218 pure = f ** exact funext (denseInducing_pure.extend_eq continuous_of_discreteTopology) ** Qed", + "informal": "" + }, + { + "formal": "VitaliFamily.mem_filterAt_iff ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x : \u03b1 s : Set (Set \u03b1) \u22a2 s \u2208 filterAt v x \u2194 \u2203 \u03b5, \u03b5 > 0 \u2227 \u2200 (a : Set \u03b1), a \u2208 setsAt v x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s ** simp only [filterAt, exists_prop, gt_iff_lt] ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x : \u03b1 s : Set (Set \u03b1) \u22a2 s \u2208 \u2a05 \u03b5 \u2208 Ioi 0, \ud835\udcdf {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5} \u2194 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (a : Set \u03b1), a \u2208 setsAt v x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s ** rw [mem_biInf_of_directed] ** \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x : \u03b1 s : Set (Set \u03b1) \u22a2 (\u2203 i, i \u2208 Ioi 0 \u2227 s \u2208 \ud835\udcdf {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x i}) \u2194 \u2203 \u03b5, 0 < \u03b5 \u2227 \u2200 (a : Set \u03b1), a \u2208 setsAt v x \u2192 a \u2286 closedBall x \u03b5 \u2192 a \u2208 s ** simp only [subset_def, and_imp, exists_prop, mem_sep_iff, mem_Ioi, mem_principal] ** case h \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x : \u03b1 s : Set (Set \u03b1) \u22a2 DirectedOn ((fun \u03b5 => \ud835\udcdf {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x \u03b5}) \u207b\u00b9'o fun x x_1 => x \u2265 x_1) (Ioi 0) ** simp only [DirectedOn, exists_prop, ge_iff_le, le_principal_iff, mem_Ioi, Order.Preimage,\n mem_principal] ** case h \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x : \u03b1 s : Set (Set \u03b1) \u22a2 \u2200 (x_1 : \u211d), 0 < x_1 \u2192 \u2200 (y : \u211d), 0 < y \u2192 \u2203 z, 0 < z \u2227 {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x z} \u2286 {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x x_1} \u2227 {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x z} \u2286 {a | a \u2208 setsAt v x \u2227 a \u2286 closedBall x y} ** intro x hx y hy ** case h \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x\u271d : \u03b1 s : Set (Set \u03b1) x : \u211d hx : 0 < x y : \u211d hy : 0 < y \u22a2 \u2203 z, 0 < z \u2227 {a | a \u2208 setsAt v x\u271d \u2227 a \u2286 closedBall x\u271d z} \u2286 {a | a \u2208 setsAt v x\u271d \u2227 a \u2286 closedBall x\u271d x} \u2227 {a | a \u2208 setsAt v x\u271d \u2227 a \u2286 closedBall x\u271d z} \u2286 {a | a \u2208 setsAt v x\u271d \u2227 a \u2286 closedBall x\u271d y} ** refine' \u27e8min x y, lt_min hx hy,\n fun a ha => \u27e8ha.1, ha.2.trans (closedBall_subset_closedBall (min_le_left _ _))\u27e9,\n fun a ha => \u27e8ha.1, ha.2.trans (closedBall_subset_closedBall (min_le_right _ _))\u27e9\u27e9 ** case ne \u03b1 : Type u_1 inst\u271d : MetricSpace \u03b1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 v : VitaliFamily \u03bc x : \u03b1 s : Set (Set \u03b1) \u22a2 Set.Nonempty (Ioi 0) ** exact \u27e8(1 : \u211d), mem_Ioi.2 zero_lt_one\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Antivary.sum_smul_lt_sum_smul_comp_perm_iff ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 inst\u271d\u2074 : LinearOrderedRing \u03b1 inst\u271d\u00b3 : LinearOrderedAddCommGroup \u03b2 inst\u271d\u00b2 : Module \u03b1 \u03b2 inst\u271d\u00b9 : OrderedSMul \u03b1 \u03b2 s : Finset \u03b9 \u03c3 : Perm \u03b9 f : \u03b9 \u2192 \u03b1 g : \u03b9 \u2192 \u03b2 inst\u271d : Fintype \u03b9 hfg : Antivary f g \u22a2 \u2211 i : \u03b9, f i \u2022 g i < \u2211 i : \u03b9, f i \u2022 g (\u2191\u03c3 i) \u2194 \u00acAntivary f (g \u2218 \u2191\u03c3) ** simp [(hfg.antivaryOn _).sum_smul_lt_sum_smul_comp_perm_iff fun _ _ \u21a6 mem_univ _] ** Qed", + "informal": "" + }, + { + "formal": "IsLeast.union ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w \u03b9 : Sort x inst\u271d\u00b2 : Preorder \u03b1 inst\u271d\u00b9 : Preorder \u03b2 s\u271d t\u271d : Set \u03b1 a\u271d b\u271d : \u03b1 inst\u271d : LinearOrder \u03b3 a b : \u03b3 s t : Set \u03b3 ha : IsLeast s a hb : IsLeast t b \u22a2 min a b \u2208 s \u222a t ** cases' le_total a b with h h <;> simp [h, ha.1, hb.1] ** Qed", + "informal": "" + }, + { + "formal": "Multiset.prod_eq_prod_toEnumFinset ** \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 m\u271d : Multiset \u03b1 inst\u271d : CommMonoid \u03b1 m : Multiset \u03b1 \u22a2 prod m = \u220f x in toEnumFinset m, x.1 ** congr ** case e_a \u03b1 : Type u_1 inst\u271d\u00b9 : DecidableEq \u03b1 m\u271d : Multiset \u03b1 inst\u271d : CommMonoid \u03b1 m : Multiset \u03b1 \u22a2 m = map (fun x => x.1) (toEnumFinset m).val ** simp ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.condCount_singleton ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : MeasurableSingletonClass \u03a9 \u03c9 : \u03a9 t : Set \u03a9 inst\u271d : Decidable (\u03c9 \u2208 t) \u22a2 \u2191\u2191(condCount {\u03c9}) t = if \u03c9 \u2208 t then 1 else 0 ** rw [condCount, cond_apply _ (measurableSet_singleton \u03c9), Measure.count_singleton, inv_one,\n one_mul] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : MeasurableSingletonClass \u03a9 \u03c9 : \u03a9 t : Set \u03a9 inst\u271d : Decidable (\u03c9 \u2208 t) \u22a2 \u2191\u2191Measure.count ({\u03c9} \u2229 t) = if \u03c9 \u2208 t then 1 else 0 ** split_ifs ** case pos \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : MeasurableSingletonClass \u03a9 \u03c9 : \u03a9 t : Set \u03a9 inst\u271d : Decidable (\u03c9 \u2208 t) h\u271d : \u03c9 \u2208 t \u22a2 \u2191\u2191Measure.count ({\u03c9} \u2229 t) = 1 ** rw [(by simpa : ({\u03c9} : Set \u03a9) \u2229 t = {\u03c9}), Measure.count_singleton] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : MeasurableSingletonClass \u03a9 \u03c9 : \u03a9 t : Set \u03a9 inst\u271d : Decidable (\u03c9 \u2208 t) h\u271d : \u03c9 \u2208 t \u22a2 {\u03c9} \u2229 t = {\u03c9} ** simpa ** case neg \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : MeasurableSingletonClass \u03a9 \u03c9 : \u03a9 t : Set \u03a9 inst\u271d : Decidable (\u03c9 \u2208 t) h\u271d : \u00ac\u03c9 \u2208 t \u22a2 \u2191\u2191Measure.count ({\u03c9} \u2229 t) = 0 ** rw [(by simpa : ({\u03c9} : Set \u03a9) \u2229 t = \u2205), Measure.count_empty] ** \u03a9 : Type u_1 inst\u271d\u00b2 : MeasurableSpace \u03a9 inst\u271d\u00b9 : MeasurableSingletonClass \u03a9 \u03c9 : \u03a9 t : Set \u03a9 inst\u271d : Decidable (\u03c9 \u2208 t) h\u271d : \u00ac\u03c9 \u2208 t \u22a2 {\u03c9} \u2229 t = \u2205 ** simpa ** Qed", + "informal": "" + }, + { + "formal": "pythagoreanTriple_comm ** x y z : \u2124 \u22a2 PythagoreanTriple x y z \u2194 PythagoreanTriple y x z ** delta PythagoreanTriple ** x y z : \u2124 \u22a2 x * x + y * y = z * z \u2194 y * y + x * x = z * z ** rw [add_comm] ** Qed", + "informal": "" + }, + { + "formal": "Set.biUnion_Ioc_eq_Ioi_self_iff ** \u03b9 : Sort u \u03b1 : Type v \u03b2 : Type w inst\u271d : LinearOrder \u03b1 a\u2081 a\u2082 b\u2081 b\u2082 : \u03b1 p : \u03b9 \u2192 Prop f : (i : \u03b9) \u2192 p i \u2192 \u03b1 a : \u03b1 \u22a2 \u22c3 i, \u22c3 (hi : p i), Ioc a (f i hi) = Ioi a \u2194 \u2200 (x : \u03b1), a < x \u2192 \u2203 i hi, x \u2264 f i hi ** simp [\u2190 Ioi_inter_Iic, \u2190 inter_iUnion, subset_def] ** Qed", + "informal": "" + }, + { + "formal": "List.cons_bagInter_of_neg ** \u03b1 : Type u_1 l l\u2081\u271d l\u2082 : List \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 inst\u271d : DecidableEq \u03b1 l\u2081 : List \u03b1 h : \u00aca \u2208 l\u2082 \u22a2 List.bagInter (a :: l\u2081) l\u2082 = List.bagInter l\u2081 l\u2082 ** cases l\u2082 ** case cons \u03b1 : Type u_1 l l\u2081\u271d : List \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 inst\u271d : DecidableEq \u03b1 l\u2081 : List \u03b1 head\u271d : \u03b1 tail\u271d : List \u03b1 h : \u00aca \u2208 head\u271d :: tail\u271d \u22a2 List.bagInter (a :: l\u2081) (head\u271d :: tail\u271d) = List.bagInter l\u2081 (head\u271d :: tail\u271d) ** simp only [erase_of_not_mem h, List.bagInter, if_neg (mt mem_of_elem_eq_true h)] ** case nil \u03b1 : Type u_1 l l\u2081\u271d : List \u03b1 p : \u03b1 \u2192 Prop a : \u03b1 inst\u271d : DecidableEq \u03b1 l\u2081 : List \u03b1 h : \u00aca \u2208 [] \u22a2 List.bagInter (a :: l\u2081) [] = List.bagInter l\u2081 [] ** simp only [bagInter_nil] ** Qed", + "informal": "" + }, + { + "formal": "RelEmbedding.not_wellFounded_of_decreasing_seq ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : IsStrictOrder \u03b1 r f : (fun x x_1 => x > x_1) \u21aar r \u22a2 \u00acWellFounded r ** rw [wellFounded_iff_no_descending_seq, not_isEmpty_iff] ** \u03b1 : Type u_1 r : \u03b1 \u2192 \u03b1 \u2192 Prop inst\u271d : IsStrictOrder \u03b1 r f : (fun x x_1 => x > x_1) \u21aar r \u22a2 Nonempty ((fun x x_1 => x > x_1) \u21aar r) ** exact \u27e8f\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "mem_sphere_one_iff_norm ** \ud835\udcd5 : Type u_1 \ud835\udd5c : Type u_2 \u03b1 : Type u_3 \u03b9 : Type u_4 \u03ba : Type u_5 E : Type u_6 F : Type u_7 G : Type u_8 inst\u271d\u00b2 : SeminormedGroup E inst\u271d\u00b9 : SeminormedGroup F inst\u271d : SeminormedGroup G s : Set E a a\u2081 a\u2082 b b\u2081 b\u2082 : E r r\u2081 r\u2082 : \u211d \u22a2 a \u2208 sphere 1 r \u2194 \u2016a\u2016 = r ** simp [dist_eq_norm_div] ** Qed", + "informal": "" + }, + { + "formal": "WittVector.exists_frobenius_solution_fractionRing ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) ha : a \u2260 0 \u22a2 \u2203 b hb m, \u2191\u03c6 b * a = \u2191p ^ m * b ** revert ha ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) \u22a2 a \u2260 0 \u2192 \u2203 b hb m, \u2191\u03c6 b * a = \u2191p ^ m * b ** refine' Localization.induction_on a _ ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) \u22a2 \u2200 (y : \ud835\udd4e k \u00d7 { x // x \u2208 nonZeroDivisors (\ud835\udd4e k) }), Localization.mk y.1 y.2 \u2260 0 \u2192 \u2203 b hb m, \u2191\u03c6 b * Localization.mk y.1 y.2 = \u2191p ^ m * b ** rintro \u27e8r, q, hq\u27e9 hrq ** case mk.mk p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) r q : \ud835\udd4e k hq : q \u2208 nonZeroDivisors (\ud835\udd4e k) hrq : Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 \u2260 0 \u22a2 \u2203 b hb m, \u2191\u03c6 b * Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 = \u2191p ^ m * b ** have hq0 : q \u2260 0 := mem_nonZeroDivisors_iff_ne_zero.1 hq ** case mk.mk p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) r q : \ud835\udd4e k hq : q \u2208 nonZeroDivisors (\ud835\udd4e k) hrq : Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 \u2260 0 hq0 : q \u2260 0 \u22a2 \u2203 b hb m, \u2191\u03c6 b * Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 = \u2191p ^ m * b ** have hr0 : r \u2260 0 := fun h => hrq (by simp [h]) ** case mk.mk p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) r q : \ud835\udd4e k hq : q \u2208 nonZeroDivisors (\ud835\udd4e k) hrq : Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 \u2260 0 hq0 : q \u2260 0 hr0 : r \u2260 0 \u22a2 \u2203 b hb m, \u2191\u03c6 b * Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 = \u2191p ^ m * b ** obtain \u27e8m, r', hr', rfl\u27e9 := exists_eq_pow_p_mul r hr0 ** case mk.mk.intro.intro.intro p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) q : \ud835\udd4e k hq : q \u2208 nonZeroDivisors (\ud835\udd4e k) hq0 : q \u2260 0 m : \u2115 r' : \ud835\udd4e k hr' : coeff r' 0 \u2260 0 hrq : Localization.mk (\u2191p ^ m * r', { val := q, property := hq }).1 (\u2191p ^ m * r', { val := q, property := hq }).2 \u2260 0 hr0 : \u2191p ^ m * r' \u2260 0 \u22a2 \u2203 b hb m_1, \u2191\u03c6 b * Localization.mk (\u2191p ^ m * r', { val := q, property := hq }).1 (\u2191p ^ m * r', { val := q, property := hq }).2 = \u2191p ^ m_1 * b ** obtain \u27e8n, q', hq', rfl\u27e9 := exists_eq_pow_p_mul q hq0 ** case mk.mk.intro.intro.intro.intro.intro.intro p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) m : \u2115 r' : \ud835\udd4e k hr' : coeff r' 0 \u2260 0 hr0 : \u2191p ^ m * r' \u2260 0 n : \u2115 q' : \ud835\udd4e k hq' : coeff q' 0 \u2260 0 hq : \u2191p ^ n * q' \u2208 nonZeroDivisors (\ud835\udd4e k) hq0 : \u2191p ^ n * q' \u2260 0 hrq : Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 \u2260 0 \u22a2 \u2203 b hb m_1, \u2191\u03c6 b * Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 = \u2191p ^ m_1 * b ** let b := frobeniusRotation p hr' hq' ** case mk.mk.intro.intro.intro.intro.intro.intro p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) m : \u2115 r' : \ud835\udd4e k hr' : coeff r' 0 \u2260 0 hr0 : \u2191p ^ m * r' \u2260 0 n : \u2115 q' : \ud835\udd4e k hq' : coeff q' 0 \u2260 0 hq : \u2191p ^ n * q' \u2208 nonZeroDivisors (\ud835\udd4e k) hq0 : \u2191p ^ n * q' \u2260 0 hrq : Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 \u2260 0 b : \ud835\udd4e k := frobeniusRotation p hr' hq' \u22a2 \u2203 b hb m_1, \u2191\u03c6 b * Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 = \u2191p ^ m_1 * b ** refine' \u27e8algebraMap (\ud835\udd4e k) (FractionRing (\ud835\udd4e k)) b, _, m - n, _\u27e9 ** case mk.mk.intro.intro.intro.intro.intro.intro.refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) m : \u2115 r' : \ud835\udd4e k hr' : coeff r' 0 \u2260 0 hr0 : \u2191p ^ m * r' \u2260 0 n : \u2115 q' : \ud835\udd4e k hq' : coeff q' 0 \u2260 0 hq : \u2191p ^ n * q' \u2208 nonZeroDivisors (\ud835\udd4e k) hq0 : \u2191p ^ n * q' \u2260 0 hrq : Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 \u2260 0 b : \ud835\udd4e k := frobeniusRotation p hr' hq' \u22a2 \u2191\u03c6 (\u2191(algebraMap (\ud835\udd4e k) (FractionRing (\ud835\udd4e k))) b) * Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 = \u2191p ^ (\u2191m - \u2191n) * \u2191(algebraMap (\ud835\udd4e k) (FractionRing (\ud835\udd4e k))) b ** exact exists_frobenius_solution_fractionRing_aux p m n r' q' hr' hq' hq ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) r q : \ud835\udd4e k hq : q \u2208 nonZeroDivisors (\ud835\udd4e k) hrq : Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 \u2260 0 hq0 : q \u2260 0 h : r = 0 \u22a2 Localization.mk (r, { val := q, property := hq }).1 (r, { val := q, property := hq }).2 = 0 ** simp [h] ** case mk.mk.intro.intro.intro.intro.intro.intro.refine'_1 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d\u00b2 : Field k inst\u271d\u00b9 : CharP k p inst\u271d : IsAlgClosed k a : FractionRing (\ud835\udd4e k) m : \u2115 r' : \ud835\udd4e k hr' : coeff r' 0 \u2260 0 hr0 : \u2191p ^ m * r' \u2260 0 n : \u2115 q' : \ud835\udd4e k hq' : coeff q' 0 \u2260 0 hq : \u2191p ^ n * q' \u2208 nonZeroDivisors (\ud835\udd4e k) hq0 : \u2191p ^ n * q' \u2260 0 hrq : Localization.mk (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).1 (\u2191p ^ m * r', { val := \u2191p ^ n * q', property := hq }).2 \u2260 0 b : \ud835\udd4e k := frobeniusRotation p hr' hq' \u22a2 \u2191(algebraMap (\ud835\udd4e k) (FractionRing (\ud835\udd4e k))) b \u2260 0 ** simpa only [map_zero] using\n (IsFractionRing.injective (WittVector p k) (FractionRing (WittVector p k))).ne\n (frobeniusRotation_nonzero p hr' hq') ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.prodComparison_iso ** C : Type u\u2081 D : Type u\u2082 inst\u271d\u2076 : Category.{v\u2081, u\u2081} C inst\u271d\u2075 : Category.{v\u2081, u\u2082} D i : D \u2964 C inst\u271d\u2074 : HasFiniteProducts C inst\u271d\u00b3 : Reflective i inst\u271d\u00b2 : CartesianClosed C inst\u271d\u00b9 : HasFiniteProducts D inst\u271d : ExponentialIdeal i A B : C \u22a2 prodComparison (leftAdjoint i) A B \u226b \u2191(bijection i A B ((leftAdjoint i).obj (A \u2a2f B))) (\ud835\udfd9 ((leftAdjoint i).obj (A \u2a2f B))) = \ud835\udfd9 ((leftAdjoint i).obj (A \u2a2f B)) ** rw [\u2190 (bijection i _ _ _).injective.eq_iff, bijection_natural, \u2190 bijection_symm_apply_id,\n Equiv.apply_symm_apply, id_comp] ** C : Type u\u2081 D : Type u\u2082 inst\u271d\u2076 : Category.{v\u2081, u\u2081} C inst\u271d\u2075 : Category.{v\u2081, u\u2082} D i : D \u2964 C inst\u271d\u2074 : HasFiniteProducts C inst\u271d\u00b3 : Reflective i inst\u271d\u00b2 : CartesianClosed C inst\u271d\u00b9 : HasFiniteProducts D inst\u271d : ExponentialIdeal i A B : C \u22a2 \u2191(bijection i A B ((leftAdjoint i).obj (A \u2a2f B))) (\ud835\udfd9 ((leftAdjoint i).obj (A \u2a2f B))) \u226b prodComparison (leftAdjoint i) A B = \ud835\udfd9 ((leftAdjoint i).obj A \u2a2f (leftAdjoint i).obj B) ** rw [\u2190 bijection_natural, id_comp, \u2190 bijection_symm_apply_id, Equiv.apply_symm_apply] ** Qed", + "informal": "" + }, + { + "formal": "Cardinal.range_natCast ** \u03b1 \u03b2 : Type u x : Cardinal.{u_1} \u22a2 x \u2208 range Nat.cast \u2194 x \u2208 Iio \u2135\u2080 ** simp only [mem_Iio, mem_range, eq_comm, lt_aleph0] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.abs_areaForm_le ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E \u22a2 |\u2191(\u2191(areaForm o) x) y| \u2264 \u2016x\u2016 * \u2016y\u2016 ** simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.AEEqFun.compMeasurePreserving_mem_Lp ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b2 : Type u_5 inst\u271d : MeasurableSpace \u03b2 \u03bcb : Measure \u03b2 g : \u03b2 \u2192\u2098[\u03bcb] E hg : g \u2208 Lp E p f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f \u22a2 compMeasurePreserving g f hf \u2208 Lp E p ** rw [Lp.mem_Lp_iff_snorm_lt_top] at hg \u22a2 ** \u03b1 : Type u_1 E : Type u_2 F : Type u_3 G : Type u_4 m m0 : MeasurableSpace \u03b1 p : \u211d\u22650\u221e q : \u211d \u03bc \u03bd : Measure \u03b1 inst\u271d\u00b3 : NormedAddCommGroup E inst\u271d\u00b2 : NormedAddCommGroup F inst\u271d\u00b9 : NormedAddCommGroup G \u03b2 : Type u_5 inst\u271d : MeasurableSpace \u03b2 \u03bcb : Measure \u03b2 g : \u03b2 \u2192\u2098[\u03bcb] E hg : snorm (\u2191g) p \u03bcb < \u22a4 f : \u03b1 \u2192 \u03b2 hf : MeasurePreserving f \u22a2 snorm (\u2191(compMeasurePreserving g f hf)) p \u03bc < \u22a4 ** rwa [snorm_compMeasurePreserving] ** Qed", + "informal": "" + }, + { + "formal": "Stream'.WSeq.LiftRel.swap_lem ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w R : \u03b1 \u2192 \u03b2 \u2192 Prop s1 : WSeq \u03b1 s2 : WSeq \u03b2 h : LiftRel R s1 s2 \u22a2 LiftRel (swap R) s2 s1 ** refine' \u27e8swap (LiftRel R), h, fun {s t} (h : LiftRel R t s) => _\u27e9 ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w R : \u03b1 \u2192 \u03b2 \u2192 Prop s1 : WSeq \u03b1 s2 : WSeq \u03b2 h\u271d : LiftRel R s1 s2 s : WSeq \u03b2 t : WSeq \u03b1 h : LiftRel R t s \u22a2 Computation.LiftRel (LiftRelO (swap R) (swap (LiftRel R))) (destruct s) (destruct t) ** rw [\u2190 LiftRelO.swap, Computation.LiftRel.swap] ** \u03b1 : Type u \u03b2 : Type v \u03b3 : Type w R : \u03b1 \u2192 \u03b2 \u2192 Prop s1 : WSeq \u03b1 s2 : WSeq \u03b2 h\u271d : LiftRel R s1 s2 s : WSeq \u03b2 t : WSeq \u03b1 h : LiftRel R t s \u22a2 Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct t) (destruct s) ** apply liftRel_destruct h ** Qed", + "informal": "" + }, + { + "formal": "smul_ne_zero_iff ** \u03b1 : Type u_1 R : Type u_2 k : Type u_3 S : Type u_4 M : Type u_5 M\u2082 : Type u_6 M\u2083 : Type u_7 \u03b9 : Type u_8 inst\u271d\u00b3 : Zero R inst\u271d\u00b2 : Zero M inst\u271d\u00b9 : SMulWithZero R M inst\u271d : NoZeroSMulDivisors R M c : R x : M \u22a2 c \u2022 x \u2260 0 \u2194 c \u2260 0 \u2227 x \u2260 0 ** rw [Ne.def, smul_eq_zero, not_or] ** Qed", + "informal": "" + }, + { + "formal": "Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two ** A : Type u B : Type v C : Type z \u03b9 : Type w inst\u271d\u00b9\u2076 : DecidableEq \u03b9 inst\u271d\u00b9\u2075 : CommRing A inst\u271d\u00b9\u2074 : CommRing B inst\u271d\u00b9\u00b3 : Algebra A B inst\u271d\u00b9\u00b2 : CommRing C inst\u271d\u00b9\u00b9 : Algebra A C \u03b9' : Type u_1 inst\u271d\u00b9\u2070 : Fintype \u03b9' inst\u271d\u2079 : Fintype \u03b9 inst\u271d\u2078 : DecidableEq \u03b9' K : Type u L : Type v E : Type z inst\u271d\u2077 : Field K inst\u271d\u2076 : Field L inst\u271d\u2075 : Field E inst\u271d\u2074 : Algebra K L inst\u271d\u00b3 : Algebra K E inst\u271d\u00b2 : Module.Finite K L inst\u271d\u00b9 : IsAlgClosed E b : \u03b9 \u2192 L pb : PowerBasis K L inst\u271d : IsSeparable K L e : \u03b9 \u2243 (L \u2192\u2090[K] E) \u22a2 \u2191(algebraMap K E) (discr K b) = det (embeddingsMatrixReindex K E b e) ^ 2 ** rw [discr_def, RingHom.map_det, RingHom.mapMatrix_apply,\n traceMatrix_eq_embeddingsMatrixReindex_mul_trans, det_mul, det_transpose, pow_two] ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.VectorMeasure.map_add ** \u03b1 : Type u_1 \u03b2 : Type u_2 m inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M v\u271d : VectorMeasure \u03b1 M inst\u271d : ContinuousAdd M v w : VectorMeasure \u03b1 M f : \u03b1 \u2192 \u03b2 \u22a2 map (v + w) f = map v f + map w f ** by_cases hf : Measurable f ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 m inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M v\u271d : VectorMeasure \u03b1 M inst\u271d : ContinuousAdd M v w : VectorMeasure \u03b1 M f : \u03b1 \u2192 \u03b2 hf : Measurable f \u22a2 map (v + w) f = map v f + map w f ** ext i hi ** case pos.h \u03b1 : Type u_1 \u03b2 : Type u_2 m inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M v\u271d : VectorMeasure \u03b1 M inst\u271d : ContinuousAdd M v w : VectorMeasure \u03b1 M f : \u03b1 \u2192 \u03b2 hf : Measurable f i : Set \u03b2 hi : MeasurableSet i \u22a2 \u2191(map (v + w) f) i = \u2191(map v f + map w f) i ** simp [map_apply _ hf hi] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 m inst\u271d\u2074 : MeasurableSpace \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b2 M : Type u_3 inst\u271d\u00b2 : AddCommMonoid M inst\u271d\u00b9 : TopologicalSpace M v\u271d : VectorMeasure \u03b1 M inst\u271d : ContinuousAdd M v w : VectorMeasure \u03b1 M f : \u03b1 \u2192 \u03b2 hf : \u00acMeasurable f \u22a2 map (v + w) f = map v f + map w f ** simp [map, dif_neg hf] ** Qed", + "informal": "" + }, + { + "formal": "multiplicity.eq_coe_iff ** \u03b1 : Type u_1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : \u03b1 n : \u2115 \u22a2 multiplicity a b = \u2191n \u2194 a ^ n \u2223 b \u2227 \u00aca ^ (n + 1) \u2223 b ** exact\n \u27e8fun h =>\n let \u27e8h\u2081, h\u2082\u27e9 := eq_some_iff.1 h\n h\u2082 \u25b8 \u27e8pow_multiplicity_dvd _, is_greatest (by\n rw [PartENat.lt_coe_iff]\n exact \u27e8h\u2081, lt_succ_self _\u27e9)\u27e9,\n fun h => eq_some_iff.2 \u27e8\u27e8n, h.2\u27e9, Eq.symm <| unique' h.1 h.2\u27e9\u27e9 ** \u03b1 : Type u_1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : \u03b1 n : \u2115 h : multiplicity a b = \u2191n h\u2081 : (multiplicity a b).Dom h\u2082 : Part.get (multiplicity a b) h\u2081 = n \u22a2 multiplicity a b < \u2191(Part.get (multiplicity a b) h\u2081 + 1) ** rw [PartENat.lt_coe_iff] ** \u03b1 : Type u_1 inst\u271d\u00b9 : Monoid \u03b1 inst\u271d : DecidableRel fun x x_1 => x \u2223 x_1 a b : \u03b1 n : \u2115 h : multiplicity a b = \u2191n h\u2081 : (multiplicity a b).Dom h\u2082 : Part.get (multiplicity a b) h\u2081 = n \u22a2 \u2203 h, Part.get (multiplicity a b) h < Part.get (multiplicity a b) h\u2081 + 1 ** exact \u27e8h\u2081, lt_succ_self _\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Equiv.Perm.support_subtype_perm ** \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : Perm \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2194 \u2191f x \u2208 s \u22a2 support (subtypePerm f h) = filter (fun x => decide (\u2191f \u2191x \u2260 \u2191x) = true) (attach s) ** ext ** case a \u03b1 : Type u_1 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 f : Perm \u03b1 h : \u2200 (x : \u03b1), x \u2208 s \u2194 \u2191f x \u2208 s a\u271d : { x // x \u2208 s } \u22a2 a\u271d \u2208 support (subtypePerm f h) \u2194 a\u271d \u2208 filter (fun x => decide (\u2191f \u2191x \u2260 \u2191x) = true) (attach s) ** simp [Subtype.ext_iff] ** Qed", + "informal": "" + }, + { + "formal": "Ordnode.all_rotateL ** \u03b1 : Type u_1 P : \u03b1 \u2192 Prop l : Ordnode \u03b1 x : \u03b1 r : Ordnode \u03b1 \u22a2 All P (rotateL l x r) \u2194 All P l \u2227 P x \u2227 All P r ** cases r <;> simp [rotateL, all_node'] ** case node \u03b1 : Type u_1 P : \u03b1 \u2192 Prop l : Ordnode \u03b1 x : \u03b1 size\u271d : \u2115 l\u271d : Ordnode \u03b1 x\u271d : \u03b1 r\u271d : Ordnode \u03b1 \u22a2 All P (if size l\u271d < ratio * size r\u271d then node3L l x l\u271d x\u271d r\u271d else node4L l x l\u271d x\u271d r\u271d) \u2194 All P l \u2227 P x \u2227 All P (node size\u271d l\u271d x\u271d r\u271d) ** split_ifs <;>\nsimp [all_node3L, all_node4L, All, and_assoc] ** Qed", + "informal": "" + }, + { + "formal": "map_extChartAt_nhdsWithin' ** \ud835\udd5c : Type u_1 E : Type u_2 M : Type u_3 H : Type u_4 E' : Type u_5 M' : Type u_6 H' : Type u_7 inst\u271d\u00b9\u2070 : NontriviallyNormedField \ud835\udd5c inst\u271d\u2079 : NormedAddCommGroup E inst\u271d\u2078 : NormedSpace \ud835\udd5c E inst\u271d\u2077 : TopologicalSpace H inst\u271d\u2076 : TopologicalSpace M f f' : LocalHomeomorph M H I : ModelWithCorners \ud835\udd5c E H inst\u271d\u2075 : NormedAddCommGroup E' inst\u271d\u2074 : NormedSpace \ud835\udd5c E' inst\u271d\u00b3 : TopologicalSpace H' inst\u271d\u00b2 : TopologicalSpace M' I' : ModelWithCorners \ud835\udd5c E' H' x : M s t : Set M inst\u271d\u00b9 : ChartedSpace H M inst\u271d : ChartedSpace H' M' y : M hy : y \u2208 (extChartAt I x).source \u22a2 y \u2208 (chartAt H x).toLocalEquiv.source ** rwa [\u2190 extChartAt_source I] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.C_smul_derivation_apply ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Module R A inst\u271d : Module R[X] A D : Derivation R R[X] A a : R f : R[X] \u22a2 \u2191C a \u2022 \u2191D f = a \u2022 \u2191D f ** have : C a \u2022 D f = D (C a * f) := by simp ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Module R A inst\u271d : Module R[X] A D : Derivation R R[X] A a : R f : R[X] this : \u2191C a \u2022 \u2191D f = \u2191D (\u2191C a * f) \u22a2 \u2191C a \u2022 \u2191D f = a \u2022 \u2191D f ** rw [this, C_mul', D.map_smul] ** R : Type u_1 A : Type u_2 inst\u271d\u00b3 : CommSemiring R inst\u271d\u00b2 : AddCommMonoid A inst\u271d\u00b9 : Module R A inst\u271d : Module R[X] A D : Derivation R R[X] A a : R f : R[X] \u22a2 \u2191C a \u2022 \u2191D f = \u2191D (\u2191C a * f) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Set.Finite.infinite_compl ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x inst\u271d : Infinite \u03b1 s : Set \u03b1 hs : Set.Finite s h : Set.Finite s\u1d9c \u22a2 Set.Finite univ ** simpa using hs.union h ** Qed", + "informal": "" + }, + { + "formal": "UniqueFactorizationMonoid.exists_reduced_factors ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R \u22a2 \u2200 (a : R), a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b ** haveI := Classical.propDecidable ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a \u22a2 \u2200 (a : R), a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b ** intro a ** \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a : R \u22a2 a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b ** refine' induction_on_prime a _ _ _ ** case refine'_1 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a : R \u22a2 0 \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = 0 \u2227 c' * b' = b ** intros ** case refine'_1 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a : R x\u271d : 0 \u2260 0 b\u271d : R \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = 0 \u2227 c' * b' = b\u271d ** contradiction ** case refine'_2 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a : R \u22a2 \u2200 (x : R), IsUnit x \u2192 x \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = x \u2227 c' * b' = b ** intro a a_unit _ b ** case refine'_2 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a : R a_unit : IsUnit a x\u271d : a \u2260 0 b : R \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b ** use a, b, 1 ** case h \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a : R a_unit : IsUnit a x\u271d : a \u2260 0 b : R \u22a2 (\u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 IsUnit d) \u2227 1 * a = a \u2227 1 * b = b ** constructor ** case h.left \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a : R a_unit : IsUnit a x\u271d : a \u2260 0 b : R \u22a2 \u2200 {d : R}, d \u2223 a \u2192 d \u2223 b \u2192 IsUnit d ** intro p p_dvd_a _ ** case h.left \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d\u00b9 a : R a_unit : IsUnit a x\u271d : a \u2260 0 b p : R p_dvd_a : p \u2223 a a\u271d : p \u2223 b \u22a2 IsUnit p ** exact isUnit_of_dvd_unit p_dvd_a a_unit ** case h.right \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a : R a_unit : IsUnit a x\u271d : a \u2260 0 b : R \u22a2 1 * a = a \u2227 1 * b = b ** simp ** case refine'_3 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a : R \u22a2 \u2200 (a p : R), a \u2260 0 \u2192 Prime p \u2192 (a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b) \u2192 p * a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b ** intro a p a_ne_zero p_prime ih_a pa_ne_zero b ** case refine'_3 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b : R \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b ** by_cases p \u2223 b ** case pos \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b : R h : p \u2223 b \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b ** rcases h with \u27e8b, rfl\u27e9 ** case pos.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b : R \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = p * b ** obtain \u27e8a', b', c', no_factor, ha', hb'\u27e9 := ih_a a_ne_zero b ** case pos.intro.intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b a' b' c' : R no_factor : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d ha' : c' * a' = a hb' : c' * b' = b \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = p * b ** refine' \u27e8a', b', p * c', @no_factor, _, _\u27e9 ** case pos.intro.intro.intro.intro.intro.intro.refine'_1 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b a' b' c' : R no_factor : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d ha' : c' * a' = a hb' : c' * b' = b \u22a2 p * c' * a' = p * a ** rw [mul_assoc, ha'] ** case pos.intro.intro.intro.intro.intro.intro.refine'_2 \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b a' b' c' : R no_factor : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d ha' : c' * a' = a hb' : c' * b' = b \u22a2 p * c' * b' = p * b ** rw [mul_assoc, hb'] ** case neg \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a\u271d a p : R a_ne_zero : a \u2260 0 p_prime : Prime p ih_a : a \u2260 0 \u2192 \u2200 (b : R), \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = a \u2227 c' * b' = b pa_ne_zero : p * a \u2260 0 b : R h : \u00acp \u2223 b \u22a2 \u2203 a' b' c', (\u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c' * a' = p * a \u2227 c' * b' = b ** obtain \u27e8a', b', c', coprime, rfl, rfl\u27e9 := ih_a a_ne_zero b ** case neg.intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a p : R p_prime : Prime p a' b' c' : R coprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d a_ne_zero : c' * a' \u2260 0 ih_a : c' * a' \u2260 0 \u2192 \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b pa_ne_zero : p * (c' * a') \u2260 0 h : \u00acp \u2223 c' * b' \u22a2 \u2203 a'_1 b'_1 c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b'_1 \u2192 IsUnit d) \u2227 c'_1 * a'_1 = p * (c' * a') \u2227 c'_1 * b'_1 = c' * b' ** refine' \u27e8p * a', b', c', _, mul_left_comm _ _ _, rfl\u27e9 ** case neg.intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a p : R p_prime : Prime p a' b' c' : R coprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d a_ne_zero : c' * a' \u2260 0 ih_a : c' * a' \u2260 0 \u2192 \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b pa_ne_zero : p * (c' * a') \u2260 0 h : \u00acp \u2223 c' * b' \u22a2 \u2200 {d : R}, d \u2223 p * a' \u2192 d \u2223 b' \u2192 IsUnit d ** intro q q_dvd_pa' q_dvd_b' ** case neg.intro.intro.intro.intro.intro \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a p : R p_prime : Prime p a' b' c' : R coprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d a_ne_zero : c' * a' \u2260 0 ih_a : c' * a' \u2260 0 \u2192 \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b pa_ne_zero : p * (c' * a') \u2260 0 h : \u00acp \u2223 c' * b' q : R q_dvd_pa' : q \u2223 p * a' q_dvd_b' : q \u2223 b' \u22a2 IsUnit q ** cases' p_prime.left_dvd_or_dvd_right_of_dvd_mul q_dvd_pa' with p_dvd_q q_dvd_a' ** case neg.intro.intro.intro.intro.intro.inr \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a p : R p_prime : Prime p a' b' c' : R coprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d a_ne_zero : c' * a' \u2260 0 ih_a : c' * a' \u2260 0 \u2192 \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b pa_ne_zero : p * (c' * a') \u2260 0 h : \u00acp \u2223 c' * b' q : R q_dvd_pa' : q \u2223 p * a' q_dvd_b' : q \u2223 b' q_dvd_a' : q \u2223 a' \u22a2 IsUnit q ** exact coprime q_dvd_a' q_dvd_b' ** case neg.intro.intro.intro.intro.intro.inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this : (a : Prop) \u2192 Decidable a a p : R p_prime : Prime p a' b' c' : R coprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d a_ne_zero : c' * a' \u2260 0 ih_a : c' * a' \u2260 0 \u2192 \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b pa_ne_zero : p * (c' * a') \u2260 0 h : \u00acp \u2223 c' * b' q : R q_dvd_pa' : q \u2223 p * a' q_dvd_b' : q \u2223 b' p_dvd_q : p \u2223 q \u22a2 IsUnit q ** have : p \u2223 c' * b' := dvd_mul_of_dvd_right (p_dvd_q.trans q_dvd_b') _ ** case neg.intro.intro.intro.intro.intro.inl \u03b1 : Type u_1 R : Type u_2 inst\u271d\u00b9 : CancelCommMonoidWithZero R inst\u271d : UniqueFactorizationMonoid R this\u271d : (a : Prop) \u2192 Decidable a a p : R p_prime : Prime p a' b' c' : R coprime : \u2200 {d : R}, d \u2223 a' \u2192 d \u2223 b' \u2192 IsUnit d a_ne_zero : c' * a' \u2260 0 ih_a : c' * a' \u2260 0 \u2192 \u2200 (b : R), \u2203 a'_1 b' c'_1, (\u2200 {d : R}, d \u2223 a'_1 \u2192 d \u2223 b' \u2192 IsUnit d) \u2227 c'_1 * a'_1 = c' * a' \u2227 c'_1 * b' = b pa_ne_zero : p * (c' * a') \u2260 0 h : \u00acp \u2223 c' * b' q : R q_dvd_pa' : q \u2223 p * a' q_dvd_b' : q \u2223 b' p_dvd_q : p \u2223 q this : p \u2223 c' * b' \u22a2 IsUnit q ** contradiction ** Qed", + "informal": "" + }, + { + "formal": "Std.RBNode.setBlack_idem ** \u03b1 : Type u_1 t : RBNode \u03b1 \u22a2 setBlack (setBlack t) = setBlack t ** cases t <;> rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.condexpL1_smul ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' \u22a2 condexpL1 hm \u03bc (c \u2022 f) = c \u2022 condexpL1 hm \u03bc f ** refine' setToFun_smul _ _ c f ** \u03b1 : Type u_1 \u03b2 : Type u_2 F : Type u_3 F' : Type u_4 G : Type u_5 G' : Type u_6 \ud835\udd5c : Type u_7 p : \u211d\u22650\u221e inst\u271d\u00b9\u00b9 : IsROrC \ud835\udd5c inst\u271d\u00b9\u2070 : NormedAddCommGroup F inst\u271d\u2079 : NormedSpace \ud835\udd5c F inst\u271d\u2078 : NormedAddCommGroup F' inst\u271d\u2077 : NormedSpace \ud835\udd5c F' inst\u271d\u2076 : NormedSpace \u211d F' inst\u271d\u2075 : CompleteSpace F' inst\u271d\u2074 : NormedAddCommGroup G inst\u271d\u00b3 : NormedAddCommGroup G' inst\u271d\u00b2 : NormedSpace \u211d G' inst\u271d\u00b9 : CompleteSpace G' m m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 hm : m \u2264 m0 inst\u271d : SigmaFinite (Measure.trim \u03bc hm) f\u271d g : \u03b1 \u2192 F' s : Set \u03b1 c : \ud835\udd5c f : \u03b1 \u2192 F' \u22a2 \u2200 (c : \ud835\udd5c) (s : Set \u03b1) (x : F'), \u2191(condexpInd F' hm \u03bc s) (c \u2022 x) = c \u2022 \u2191(condexpInd F' hm \u03bc s) x ** exact fun c _ x => condexpInd_smul' c x ** Qed", + "informal": "" + }, + { + "formal": "MvPolynomial.totalDegree_neg ** R : Type u S : Type v \u03c3 : Type u_1 a\u271d a' a\u2081 a\u2082 : R e : \u2115 n m : \u03c3 s : \u03c3 \u2192\u2080 \u2115 inst\u271d : CommRing R p q a : MvPolynomial \u03c3 R \u22a2 totalDegree (-a) = totalDegree a ** simp only [totalDegree, support_neg] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.isAcyclic_iff_forall_edge_isBridge ** V : Type u G : SimpleGraph V \u22a2 IsAcyclic G \u2194 \u2200 \u2983e : Sym2 V\u2984, e \u2208 edgeSet G \u2192 IsBridge G e ** simp [isAcyclic_iff_forall_adj_isBridge, Sym2.forall] ** Qed", + "informal": "" + }, + { + "formal": "WittVector.remainder_vars ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 vars (remainder p n) \u2286 univ \u00d7\u02e2 range (n + 1) ** rw [remainder] ** p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 vars ((\u2211 x in range (n + 1), \u2191(rename (Prod.mk 0)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) * \u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) \u2286 univ \u00d7\u02e2 range (n + 1) ** apply Subset.trans (vars_mul _ _) ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 vars (\u2211 x in range (n + 1), \u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) \u2286 univ \u00d7\u02e2 range (n + 1) ** refine' Subset.trans (vars_sum_subset _ _) _ ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 (Finset.biUnion (range (n + 1)) fun i => vars (\u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | i => p ^ (n + 1 - i)) (\u2191p ^ i)))) \u2286 univ \u00d7\u02e2 range (n + 1) ** rw [biUnion_subset] ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n : \u2115 \u22a2 \u2200 (x : \u2115), x \u2208 range (n + 1) \u2192 vars (\u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) \u2286 univ \u00d7\u02e2 range (n + 1) ** intro x hx ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n x : \u2115 hx : x \u2208 range (n + 1) \u22a2 vars (\u2191(rename (Prod.mk 1)) (\u2191(monomial fun\u2080 | x => p ^ (n + 1 - x)) (\u2191p ^ x))) \u2286 univ \u00d7\u02e2 range (n + 1) ** rw [rename_monomial, vars_monomial, Finsupp.mapDomain_single] ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n x : \u2115 hx : x \u2208 range (n + 1) \u22a2 (fun\u2080 | (1, x) => p ^ (n + 1 - x)).support \u2286 univ \u00d7\u02e2 range (n + 1) ** apply Subset.trans Finsupp.support_single_subset ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n x : \u2115 hx : x \u2208 range (n + 1) \u22a2 {(1, x)} \u2286 univ \u00d7\u02e2 range (n + 1) ** simpa using mem_range.mp hx ** case refine'_2 p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n x : \u2115 hx : x \u2208 range (n + 1) \u22a2 \u2191p ^ x \u2260 0 ** apply pow_ne_zero ** case refine'_2.h p : \u2115 hp : Fact (Nat.Prime p) k : Type u_1 inst\u271d : CommRing k n x : \u2115 hx : x \u2208 range (n + 1) \u22a2 \u2191p \u2260 0 ** exact_mod_cast hp.out.ne_zero ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv ** B : Type u inst\u271d : Bicategory B a b c d e : B f : a \u27f6 b g : b \u27f6 c h : c \u27f6 d i : d \u27f6 e \u22a2 inv ((\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).hom) = inv ((\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv) ** simp ** Qed", + "informal": "" + }, + { + "formal": "Real.tendsto_abs_tan_of_cos_eq_zero ** x : \u211d hx : cos x = 0 \u22a2 Tendsto (fun x => |tan x|) (\ud835\udcdd[{x}\u1d9c] x) atTop ** have hx : Complex.cos x = 0 := by exact_mod_cast hx ** x : \u211d hx\u271d : cos x = 0 hx : Complex.cos \u2191x = 0 \u22a2 Tendsto (fun x => |tan x|) (\ud835\udcdd[{x}\u1d9c] x) atTop ** simp only [\u2190 Complex.abs_ofReal, Complex.ofReal_tan] ** x : \u211d hx\u271d : cos x = 0 hx : Complex.cos \u2191x = 0 \u22a2 Tendsto (fun x => \u2191Complex.abs (Complex.tan \u2191x)) (\ud835\udcdd[{x}\u1d9c] x) atTop ** refine' (Complex.tendsto_abs_tan_of_cos_eq_zero hx).comp _ ** x : \u211d hx\u271d : cos x = 0 hx : Complex.cos \u2191x = 0 \u22a2 Tendsto (fun x => \u2191x) (\ud835\udcdd[{x}\u1d9c] x) (\ud835\udcdd[{\u2191x}\u1d9c] \u2191x) ** refine' Tendsto.inf Complex.continuous_ofReal.continuousAt _ ** x : \u211d hx\u271d : cos x = 0 hx : Complex.cos \u2191x = 0 \u22a2 Tendsto (fun x => \u2191x) (\ud835\udcdf {x}\u1d9c) (\ud835\udcdf {\u2191x}\u1d9c) ** exact tendsto_principal_principal.2 fun y => mt Complex.ofReal_inj.1 ** x : \u211d hx : cos x = 0 \u22a2 Complex.cos \u2191x = 0 ** exact_mod_cast hx ** Qed", + "informal": "" + }, + { + "formal": "Finset.sup_biUnion ** F : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 \u03b3 : Type u_4 \u03b9 : Type u_5 \u03ba : Type u_6 inst\u271d\u00b2 : SemilatticeSup \u03b1 inst\u271d\u00b9 : OrderBot \u03b1 s\u271d s\u2081 s\u2082 : Finset \u03b2 f g : \u03b2 \u2192 \u03b1 a : \u03b1 inst\u271d : DecidableEq \u03b2 s : Finset \u03b3 t : \u03b3 \u2192 Finset \u03b2 c : \u03b1 \u22a2 sup (Finset.biUnion s t) f \u2264 c \u2194 (sup s fun x => sup (t x) f) \u2264 c ** simp [@forall_swap _ \u03b2] ** Qed", + "informal": "" + }, + { + "formal": "Set.Ici.isAtom_iff ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b\u271d : \u03b1 b : \u2191(Ici a) \u22a2 IsAtom b \u2194 a \u22d6 \u2191b ** rw [\u2190 bot_covby_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b\u271d : \u03b1 b : \u2191(Ici a) \u22a2 \u22a5 \u22d6 b \u2194 a \u22d6 \u2191b ** refine' (Set.OrdConnected.apply_covby_apply_iff (OrderEmbedding.subtype fun c => a \u2264 c) _).symm ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b\u271d : \u03b1 b : \u2191(Ici a) \u22a2 OrdConnected (range \u2191(OrderEmbedding.subtype fun c => a \u2264 c)) ** simpa only [OrderEmbedding.subtype_apply, Subtype.range_coe_subtype] using Set.ordConnected_Ici ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.Measure.count_singleton' ** \u03b1 : Type u_1 \u03b2 : Type ?u.20880 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 a : \u03b1 ha : MeasurableSet {a} \u22a2 \u2191\u2191count {a} = 1 ** rw [count_apply_finite' (Set.finite_singleton a) ha, Set.Finite.toFinset] ** \u03b1 : Type u_1 \u03b2 : Type ?u.20880 inst\u271d\u00b9 : MeasurableSpace \u03b1 inst\u271d : MeasurableSpace \u03b2 s : Set \u03b1 a : \u03b1 ha : MeasurableSet {a} \u22a2 \u2191(Finset.card (toFinset {a})) = 1 ** simp [@toFinset_card _ _ (Set.finite_singleton a).fintype,\n @Fintype.card_unique _ _ (Set.finite_singleton a).fintype] ** Qed", + "informal": "" + }, + { + "formal": "Orientation.eq_zero_or_eq_zero_of_kahler_eq_zero ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 \u22a2 x = 0 \u2228 y = 0 ** have : \u2016x\u2016 * \u2016y\u2016 = 0 := by simpa [hx] using (o.norm_kahler x y).symm ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 this : \u2016x\u2016 * \u2016y\u2016 = 0 \u22a2 x = 0 \u2228 y = 0 ** cases' eq_zero_or_eq_zero_of_mul_eq_zero this with h h ** E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 \u22a2 \u2016x\u2016 * \u2016y\u2016 = 0 ** simpa [hx] using (o.norm_kahler x y).symm ** case inl E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 this : \u2016x\u2016 * \u2016y\u2016 = 0 h : \u2016x\u2016 = 0 \u22a2 x = 0 \u2228 y = 0 ** left ** case inl.h E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 this : \u2016x\u2016 * \u2016y\u2016 = 0 h : \u2016x\u2016 = 0 \u22a2 x = 0 ** simpa using h ** case inr E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 this : \u2016x\u2016 * \u2016y\u2016 = 0 h : \u2016y\u2016 = 0 \u22a2 x = 0 \u2228 y = 0 ** right ** case inr.h E : Type u_1 inst\u271d\u00b2 : NormedAddCommGroup E inst\u271d\u00b9 : InnerProductSpace \u211d E inst\u271d : Fact (finrank \u211d E = 2) o : Orientation \u211d E (Fin 2) x y : E hx : \u2191(\u2191(kahler o) x) y = 0 this : \u2016x\u2016 * \u2016y\u2016 = 0 h : \u2016y\u2016 = 0 \u22a2 y = 0 ** simpa using h ** Qed", + "informal": "" + }, + { + "formal": "iUnion_Ici_eq_Ioi_iInf ** \u03b9 : Sort u \u03b1 : Type v \u03b2 : Type w inst\u271d\u00b9 : LinearOrder \u03b1 s : Set \u03b1 a : \u03b1 f\u271d : \u03b9 \u2192 \u03b1 R : Type u_1 inst\u271d : CompleteLinearOrder R f : \u03b9 \u2192 R no_least_elem : \u00ac\u2a05 i, f i \u2208 range f \u22a2 \u22c3 i, Ici (f i) = Ioi (\u2a05 i, f i) ** simp only [\u2190 IsGLB.biUnion_Ici_eq_Ioi (@isGLB_iInf _ _ _ f) no_least_elem, mem_range,\n iUnion_exists, iUnion_iUnion_eq'] ** Qed", + "informal": "" + }, + { + "formal": "Complex.tendsto_exp_nhds_zero_iff ** \u03b1 : Type u_1 l : Filter \u03b1 f : \u03b1 \u2192 \u2102 \u22a2 Tendsto (fun x => cexp (f x)) l (\ud835\udcdd 0) \u2194 Tendsto (fun x => (f x).re) l atBot ** simp_rw [\u2190comp_apply (f := exp), \u2190 tendsto_comap_iff, comap_exp_nhds_zero, tendsto_comap_iff] ** \u03b1 : Type u_1 l : Filter \u03b1 f : \u03b1 \u2192 \u2102 \u22a2 Tendsto (re \u2218 f) l atBot \u2194 Tendsto (fun x => (f x).re) l atBot ** rfl ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.lintegral_rw\u2082 ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f\u2081 f\u2081' : \u03b1 \u2192 \u03b2 f\u2082 f\u2082' : \u03b1 \u2192 \u03b3 h\u2081\u271d : f\u2081 =\u1d50[\u03bc] f\u2081' h\u2082\u271d : f\u2082 =\u1d50[\u03bc] f\u2082' g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e x\u271d : \u03b1 h\u2082 : f\u2082 x\u271d = f\u2082' x\u271d h\u2081 : f\u2081 x\u271d = f\u2081' x\u271d \u22a2 (fun a => g (f\u2081 a) (f\u2082 a)) x\u271d = (fun a => g (f\u2081' a) (f\u2082' a)) x\u271d ** dsimp only ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 m : MeasurableSpace \u03b1 \u03bc \u03bd : Measure \u03b1 f\u2081 f\u2081' : \u03b1 \u2192 \u03b2 f\u2082 f\u2082' : \u03b1 \u2192 \u03b3 h\u2081\u271d : f\u2081 =\u1d50[\u03bc] f\u2081' h\u2082\u271d : f\u2082 =\u1d50[\u03bc] f\u2082' g : \u03b2 \u2192 \u03b3 \u2192 \u211d\u22650\u221e x\u271d : \u03b1 h\u2082 : f\u2082 x\u271d = f\u2082' x\u271d h\u2081 : f\u2081 x\u271d = f\u2081' x\u271d \u22a2 g (f\u2081 x\u271d) (f\u2082 x\u271d) = g (f\u2081' x\u271d) (f\u2082' x\u271d) ** rw [h\u2081, h\u2082] ** Qed", + "informal": "" + }, + { + "formal": "or_rotate ** a b c : Prop \u22a2 a \u2228 b \u2228 c \u2194 b \u2228 c \u2228 a ** simp only [or_left_comm, Or.comm] ** Qed", + "informal": "" + }, + { + "formal": "Set.mem_Icc_Ico_Ioc_Ioo_of_subset_of_subset ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b \u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b} ** by_cases ha : a \u2208 s <;> by_cases hb : b \u2208 s ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : a \u2208 s hb : b \u2208 s \u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b} ** refine' Or.inl (Subset.antisymm hc _) ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : a \u2208 s hb : b \u2208 s \u22a2 Icc a b \u2286 s ** rwa [\u2190 Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha, \u2190 Icc_diff_right,\n diff_singleton_subset_iff, insert_eq_of_mem hb] at ho ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : a \u2208 s hb : \u00acb \u2208 s \u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b} ** refine' Or.inr <| Or.inl <| Subset.antisymm _ _ ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : a \u2208 s hb : \u00acb \u2208 s \u22a2 s \u2286 Ico a b ** rw [\u2190 Icc_diff_right] ** case neg.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : a \u2208 s hb : \u00acb \u2208 s \u22a2 s \u2286 Icc a b \\ {b} ** exact subset_diff_singleton hc hb ** case neg.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : a \u2208 s hb : \u00acb \u2208 s \u22a2 Ico a b \u2286 s ** rwa [\u2190 Ico_diff_left, diff_singleton_subset_iff, insert_eq_of_mem ha] at ho ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b} ** refine' Or.inr <| Or.inr <| Or.inl <| Subset.antisymm _ _ ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 s \u2286 Ioc a b ** rw [\u2190 Icc_diff_left] ** case pos.refine'_1 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 s \u2286 Icc a b \\ {a} ** exact subset_diff_singleton hc ha ** case pos.refine'_2 \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : b \u2208 s \u22a2 Ioc a b \u2286 s ** rwa [\u2190 Ioc_diff_right, diff_singleton_subset_iff, insert_eq_of_mem hb] at ho ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : \u00acb \u2208 s \u22a2 s \u2208 {Icc a b, Ico a b, Ioc a b, Ioo a b} ** refine' Or.inr <| Or.inr <| Or.inr <| Subset.antisymm _ ho ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : \u00acb \u2208 s \u22a2 s \u2286 Ioo a b ** rw [\u2190 Ico_diff_left, \u2190 Icc_diff_right] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : PartialOrder \u03b1 a b c : \u03b1 s : Set \u03b1 ho : Ioo a b \u2286 s hc : s \u2286 Icc a b ha : \u00aca \u2208 s hb : \u00acb \u2208 s \u22a2 s \u2286 (Icc a b \\ {b}) \\ {a} ** apply_rules [subset_diff_singleton] ** Qed", + "informal": "" + }, + { + "formal": "Nat.eleven_dvd_of_palindrome ** n : \u2115 p : List.Palindrome (digits 10 n) h : Even (List.length (digits 10 n)) \u22a2 11 \u2223 n ** let dig := (digits 10 n).map (Coe.coe : \u2115 \u2192 \u2124) ** n : \u2115 p : List.Palindrome (digits 10 n) h : Even (List.length (digits 10 n)) dig : List \u2124 := List.map Coe.coe (digits 10 n) \u22a2 11 \u2223 n ** replace h : Even dig.length := by rwa [List.length_map] ** n : \u2115 p : List.Palindrome (digits 10 n) dig : List \u2124 := List.map Coe.coe (digits 10 n) h : Even (List.length dig) \u22a2 11 \u2223 n ** refine' eleven_dvd_iff.2 \u27e80, (_ : dig.alternatingSum = 0)\u27e9 ** n : \u2115 p : List.Palindrome (digits 10 n) dig : List \u2124 := List.map Coe.coe (digits 10 n) h : Even (List.length dig) \u22a2 List.alternatingSum dig = 0 ** have := dig.alternatingSum_reverse ** n : \u2115 p : List.Palindrome (digits 10 n) dig : List \u2124 := List.map Coe.coe (digits 10 n) h : Even (List.length dig) this : List.alternatingSum (List.reverse dig) = (-1) ^ (List.length dig + 1) \u2022 List.alternatingSum dig \u22a2 List.alternatingSum dig = 0 ** rw [(p.map _).reverse_eq, _root_.pow_succ, h.neg_one_pow, mul_one, neg_one_zsmul] at this ** n : \u2115 p : List.Palindrome (digits 10 n) dig : List \u2124 := List.map Coe.coe (digits 10 n) h : Even (List.length dig) this : List.alternatingSum (List.map Coe.coe (digits 10 n)) = -List.alternatingSum dig \u22a2 List.alternatingSum dig = 0 ** exact eq_zero_of_neg_eq this.symm ** n : \u2115 p : List.Palindrome (digits 10 n) h : Even (List.length (digits 10 n)) dig : List \u2124 := List.map Coe.coe (digits 10 n) \u22a2 Even (List.length dig) ** rwa [List.length_map] ** Qed", + "informal": "" + }, + { + "formal": "WittVector.iterate_verschiebung_mul ** p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i] x * (\u2191verschiebung)^[j] y = (\u2191verschiebung)^[i + j] ((\u2191frobenius)^[j] x * (\u2191frobenius)^[i] y) ** calc\n _ = verschiebung^[i] (x * frobenius^[i] (verschiebung^[j] y)) := ?_\n _ = verschiebung^[i] (x * verschiebung^[j] (frobenius^[i] y)) := ?_\n _ = verschiebung^[i] (verschiebung^[j] (frobenius^[i] y) * x) := ?_\n _ = verschiebung^[i] (verschiebung^[j] (frobenius^[i] y * frobenius^[j] x)) := ?_\n _ = verschiebung^[i + j] (frobenius^[i] y * frobenius^[j] x) := ?_\n _ = _ := ?_ ** case calc_1 p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i] x * (\u2191verschiebung)^[j] y = (\u2191verschiebung)^[i] (x * (\u2191frobenius)^[i] ((\u2191verschiebung)^[j] y)) ** apply iterate_verschiebung_mul_left ** case calc_2 p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i] (x * (\u2191frobenius)^[i] ((\u2191verschiebung)^[j] y)) = (\u2191verschiebung)^[i] (x * (\u2191verschiebung)^[j] ((\u2191frobenius)^[i] y)) ** rw [verschiebung_frobenius_comm.iterate_iterate] ** case calc_3 p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i] (x * (\u2191verschiebung)^[j] ((\u2191frobenius)^[i] y)) = (\u2191verschiebung)^[i] ((\u2191verschiebung)^[j] ((\u2191frobenius)^[i] y) * x) ** rw [mul_comm] ** case calc_4 p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i] ((\u2191verschiebung)^[j] ((\u2191frobenius)^[i] y) * x) = (\u2191verschiebung)^[i] ((\u2191verschiebung)^[j] ((\u2191frobenius)^[i] y * (\u2191frobenius)^[j] x)) ** rw [iterate_verschiebung_mul_left] ** case calc_5 p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i] ((\u2191verschiebung)^[j] ((\u2191frobenius)^[i] y * (\u2191frobenius)^[j] x)) = (\u2191verschiebung)^[i + j] ((\u2191frobenius)^[i] y * (\u2191frobenius)^[j] x) ** rw [iterate_add_apply] ** case calc_6 p : \u2115 R : Type u_1 hp : Fact (Nat.Prime p) inst\u271d\u00b9 : CommRing R inst\u271d : CharP R p x y : \ud835\udd4e R i j : \u2115 \u22a2 (\u2191verschiebung)^[i + j] ((\u2191frobenius)^[i] y * (\u2191frobenius)^[j] x) = (\u2191verschiebung)^[i + j] ((\u2191frobenius)^[j] x * (\u2191frobenius)^[i] y) ** rw [mul_comm] ** Qed", + "informal": "" + }, + { + "formal": "Finset.affineCombination_affineCombinationSingleWeights ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082 : Finset \u03b9\u2082 inst\u271d : DecidableEq \u03b9 p : \u03b9 \u2192 P i : \u03b9 hi : i \u2208 s \u22a2 \u2191(affineCombination k s p) (affineCombinationSingleWeights k i) = p i ** refine' s.affineCombination_of_eq_one_of_eq_zero _ _ hi (by simp) _ ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082 : Finset \u03b9\u2082 inst\u271d : DecidableEq \u03b9 p : \u03b9 \u2192 P i : \u03b9 hi : i \u2208 s \u22a2 \u2200 (i2 : \u03b9), i2 \u2208 s \u2192 i2 \u2260 i \u2192 affineCombinationSingleWeights k i i2 = 0 ** rintro j - hj ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082 : Finset \u03b9\u2082 inst\u271d : DecidableEq \u03b9 p : \u03b9 \u2192 P i : \u03b9 hi : i \u2208 s j : \u03b9 hj : j \u2260 i \u22a2 affineCombinationSingleWeights k i j = 0 ** simp [hj] ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b3 : Ring k inst\u271d\u00b2 : AddCommGroup V inst\u271d\u00b9 : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082 : Finset \u03b9\u2082 inst\u271d : DecidableEq \u03b9 p : \u03b9 \u2192 P i : \u03b9 hi : i \u2208 s \u22a2 affineCombinationSingleWeights k i i = 1 ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasureTheory.isFiniteMeasure_withDensity ** \u03b1 : Type u_1 m0 : MeasurableSpace \u03b1 \u03bc : Measure \u03b1 f : \u03b1 \u2192 \u211d\u22650\u221e hf : \u222b\u207b (a : \u03b1), f a \u2202\u03bc \u2260 \u22a4 \u22a2 \u2191\u2191(withDensity \u03bc f) univ < \u22a4 ** rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ, lt_top_iff_ne_top] ** Qed", + "informal": "" + }, + { + "formal": "Finset.subset_map_iff ** \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x f : \u03b1 \u21aa \u03b2 s : Finset \u03b2 t : Finset \u03b1 \u22a2 s \u2286 map f t \u2194 \u2203 u x, s = map f u ** refine' \u27e8fun h => \u27e8_, preimage_subset h, _\u27e9, _\u27e9 ** case refine'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x f : \u03b1 \u21aa \u03b2 s : Finset \u03b2 t : Finset \u03b1 h : s \u2286 map f t \u22a2 s = map f (preimage s \u2191f (_ : InjOn (\u2191f) (\u2191f \u207b\u00b9' \u2191s))) ** rw [map_eq_image, image_preimage, filter_true_of_mem] ** case refine'_1 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x f : \u03b1 \u21aa \u03b2 s : Finset \u03b2 t : Finset \u03b1 h : s \u2286 map f t \u22a2 \u2200 (x : \u03b2), x \u2208 s \u2192 x \u2208 Set.range \u2191f ** exact fun x hx \u21a6 coe_map_subset_range _ _ (h hx) ** case refine'_2 \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x f : \u03b1 \u21aa \u03b2 s : Finset \u03b2 t : Finset \u03b1 \u22a2 (\u2203 u x, s = map f u) \u2192 s \u2286 map f t ** rintro \u27e8u, hut, rfl\u27e9 ** case refine'_2.intro.intro \u03b1 : Type u \u03b2 : Type v \u03b9 : Sort w \u03b3 : Type x f : \u03b1 \u21aa \u03b2 t u : Finset \u03b1 hut : u \u2286 t \u22a2 map f u \u2286 map f t ** exact map_subset_map.2 hut ** Qed", + "informal": "" + }, + { + "formal": "Finset.map_filter ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 f\u271d : \u03b1 \u21aa \u03b2 s : Finset \u03b1 f : \u03b1 \u2243 \u03b2 p : \u03b1 \u2192 Prop inst\u271d : DecidablePred p \u22a2 map (Equiv.toEmbedding f) (filter p s) = filter (p \u2218 \u2191f.symm) (map (Equiv.toEmbedding f) s) ** simp only [filter_map, Function.comp, Equiv.toEmbedding_apply, Equiv.symm_apply_apply] ** Qed", + "informal": "" + }, + { + "formal": "SimpleGraph.nonuniformWitness_subset ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : LinearOrderedField \ud835\udd5c G : SimpleGraph \u03b1 inst\u271d : DecidableRel G.Adj \u03b5 : \ud835\udd5c s t : Finset \u03b1 a b : \u03b1 h : \u00acIsUniform G \u03b5 s t \u22a2 nonuniformWitness G \u03b5 s t \u2286 s ** unfold nonuniformWitness ** \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : LinearOrderedField \ud835\udd5c G : SimpleGraph \u03b1 inst\u271d : DecidableRel G.Adj \u03b5 : \ud835\udd5c s t : Finset \u03b1 a b : \u03b1 h : \u00acIsUniform G \u03b5 s t \u22a2 (if WellOrderingRel s t then (nonuniformWitnesses G \u03b5 s t).1 else (nonuniformWitnesses G \u03b5 t s).2) \u2286 s ** split_ifs ** case pos \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : LinearOrderedField \ud835\udd5c G : SimpleGraph \u03b1 inst\u271d : DecidableRel G.Adj \u03b5 : \ud835\udd5c s t : Finset \u03b1 a b : \u03b1 h : \u00acIsUniform G \u03b5 s t h\u271d : WellOrderingRel s t \u22a2 (nonuniformWitnesses G \u03b5 s t).1 \u2286 s ** exact G.left_nonuniformWitnesses_subset h ** case neg \u03b1 : Type u_1 \ud835\udd5c : Type u_2 inst\u271d\u00b9 : LinearOrderedField \ud835\udd5c G : SimpleGraph \u03b1 inst\u271d : DecidableRel G.Adj \u03b5 : \ud835\udd5c s t : Finset \u03b1 a b : \u03b1 h : \u00acIsUniform G \u03b5 s t h\u271d : \u00acWellOrderingRel s t \u22a2 (nonuniformWitnesses G \u03b5 t s).2 \u2286 s ** exact G.right_nonuniformWitnesses_subset fun i => h i.symm ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Limits.biproduct.\u03b9_toSubtype ** J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J \u22a2 \u03b9 f j \u226b toSubtype f p = if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0 ** ext i ** case w J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p \u22a2 (\u03b9 f j \u226b toSubtype f p) \u226b \u03c0 (Subtype.restrict p f) i = (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i ** rw [biproduct.toSubtype, Category.assoc, biproduct.lift_\u03c0, biproduct.\u03b9_\u03c0] ** case w J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p \u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) = (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i ** by_cases h : p j ** case pos J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : p j \u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) = (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i ** rw [dif_pos h, biproduct.\u03b9_\u03c0] ** case pos J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : p j \u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) = if h_1 : { val := j, property := h } = i then eqToHom (_ : Subtype.restrict p f { val := j, property := h } = Subtype.restrict p f i) else 0 ** split_ifs with h\u2081 h\u2082 h\u2082 ** case pos J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : p j h\u2081 : j = \u2191i h\u2082 : { val := j, property := h } = i \u22a2 eqToHom (_ : f j = f \u2191i) = eqToHom (_ : Subtype.restrict p f { val := j, property := h } = Subtype.restrict p f i) case neg J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : p j h\u2081 : j = \u2191i h\u2082 : \u00ac{ val := j, property := h } = i \u22a2 eqToHom (_ : f j = f \u2191i) = 0 case pos J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : p j h\u2081 : \u00acj = \u2191i h\u2082 : { val := j, property := h } = i \u22a2 0 = eqToHom (_ : Subtype.restrict p f { val := j, property := h } = Subtype.restrict p f i) case neg J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : p j h\u2081 : \u00acj = \u2191i h\u2082 : \u00ac{ val := j, property := h } = i \u22a2 0 = 0 ** exacts [rfl, False.elim (h\u2082 (Subtype.ext h\u2081)), False.elim (h\u2081 (congr_arg Subtype.val h\u2082)), rfl] ** case neg J : Type w K : Type u_1 C : Type u inst\u271d\u2074 : Category.{v, u} C inst\u271d\u00b3 : HasZeroMorphisms C f : J \u2192 C inst\u271d\u00b2 : HasBiproduct f p : J \u2192 Prop inst\u271d\u00b9 : HasBiproduct (Subtype.restrict p f) inst\u271d : DecidablePred p j : J i : Subtype p h : \u00acp j \u22a2 (if h : j = \u2191i then eqToHom (_ : f j = f \u2191i) else 0) = (if h : p j then \u03b9 (Subtype.restrict p f) { val := j, property := h } else 0) \u226b \u03c0 (Subtype.restrict p f) i ** rw [dif_neg h, dif_neg (show j \u2260 i from fun h\u2082 => h (h\u2082.symm \u25b8 i.2)), zero_comp] ** Qed", + "informal": "" + }, + { + "formal": "Set.preimage_const_mul_Iio_of_neg ** \u03b1 : Type u_1 inst\u271d : LinearOrderedField \u03b1 a\u271d a c : \u03b1 h : c < 0 \u22a2 (fun x x_1 => x * x_1) c \u207b\u00b9' Iio a = Ioi (a / c) ** simpa only [mul_comm] using preimage_mul_const_Iio_of_neg a h ** Qed", + "informal": "" + }, + { + "formal": "Real.deriv_arcsin_aux ** x : \u211d h\u2081 : x \u2260 -1 h\u2082 : x \u2260 1 \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** cases' h\u2081.lt_or_lt with h\u2081 h\u2081 ** case inr x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082 : x \u2260 1 h\u2081 : -1 < x \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** cases' h\u2082.lt_or_lt with h\u2082 h\u2082 ** case inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082 : x \u2260 1 h\u2081 : x < -1 \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** have : 1 - x ^ 2 < 0 := by nlinarith [h\u2081] ** case inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082 : x \u2260 1 h\u2081 : x < -1 this : 1 - x ^ 2 < 0 \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** rw [sqrt_eq_zero'.2 this.le, div_zero] ** case inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082 : x \u2260 1 h\u2081 : x < -1 this : 1 - x ^ 2 < 0 \u22a2 HasStrictDerivAt arcsin 0 x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** have : arcsin =\u1da0[\ud835\udcdd x] fun _ => -(\u03c0 / 2) :=\n (gt_mem_nhds h\u2081).mono fun y hy => arcsin_of_le_neg_one hy.le ** case inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082 : x \u2260 1 h\u2081 : x < -1 this\u271d : 1 - x ^ 2 < 0 this : arcsin =\u1da0[\ud835\udcdd x] fun x => -(\u03c0 / 2) \u22a2 HasStrictDerivAt arcsin 0 x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** exact \u27e8(hasStrictDerivAt_const _ _).congr_of_eventuallyEq this.symm,\n contDiffAt_const.congr_of_eventuallyEq this\u27e9 ** x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082 : x \u2260 1 h\u2081 : x < -1 \u22a2 1 - x ^ 2 < 0 ** nlinarith [h\u2081] ** case inr.inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : x < 1 \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** have : 0 < sqrt (1 - x ^ 2) := sqrt_pos.2 (by nlinarith [h\u2081, h\u2082]) ** case inr.inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : x < 1 this : 0 < sqrt (1 - x ^ 2) \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** simp only [\u2190 cos_arcsin, one_div] at this \u22a2 ** case inr.inl x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : x < 1 this : 0 < cos (arcsin x) \u22a2 HasStrictDerivAt arcsin (cos (arcsin x))\u207b\u00b9 x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** exact \u27e8sinLocalHomeomorph.hasStrictDerivAt_symm \u27e8h\u2081, h\u2082\u27e9 this.ne' (hasStrictDerivAt_sin _),\n sinLocalHomeomorph.contDiffAt_symm_deriv this.ne' \u27e8h\u2081, h\u2082\u27e9 (hasDerivAt_sin _)\n contDiff_sin.contDiffAt\u27e9 ** x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : x < 1 \u22a2 0 < 1 - x ^ 2 ** nlinarith [h\u2081, h\u2082] ** case inr.inr x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : 1 < x \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** have : 1 - x ^ 2 < 0 := by nlinarith [h\u2082] ** case inr.inr x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : 1 < x this : 1 - x ^ 2 < 0 \u22a2 HasStrictDerivAt arcsin (1 / sqrt (1 - x ^ 2)) x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** rw [sqrt_eq_zero'.2 this.le, div_zero] ** case inr.inr x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : 1 < x this : 1 - x ^ 2 < 0 \u22a2 HasStrictDerivAt arcsin 0 x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** have : arcsin =\u1da0[\ud835\udcdd x] fun _ => \u03c0 / 2 := (lt_mem_nhds h\u2082).mono fun y hy => arcsin_of_one_le hy.le ** case inr.inr x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : 1 < x this\u271d : 1 - x ^ 2 < 0 this : arcsin =\u1da0[\ud835\udcdd x] fun x => \u03c0 / 2 \u22a2 HasStrictDerivAt arcsin 0 x \u2227 ContDiffAt \u211d \u22a4 arcsin x ** exact \u27e8(hasStrictDerivAt_const _ _).congr_of_eventuallyEq this.symm,\n contDiffAt_const.congr_of_eventuallyEq this\u27e9 ** x : \u211d h\u2081\u271d : x \u2260 -1 h\u2082\u271d : x \u2260 1 h\u2081 : -1 < x h\u2082 : 1 < x \u22a2 1 - x ^ 2 < 0 ** nlinarith [h\u2082] ** Qed", + "informal": "" + }, + { + "formal": "Function.mulSupport_comp_eq_of_range_subset ** \u03b1 : Type u_1 \u03b2 : Type u_2 A : Type u_3 B : Type u_4 M : Type u_5 N : Type u_6 P : Type u_7 R : Type u_8 S : Type u_9 G : Type u_10 M\u2080 : Type u_11 G\u2080 : Type u_12 \u03b9 : Sort u_13 inst\u271d\u00b2 : One M inst\u271d\u00b9 : One N inst\u271d : One P g : M \u2192 N f : \u03b1 \u2192 M hg : \u2200 {x : M}, x \u2208 range f \u2192 (g x = 1 \u2194 x = 1) x : \u03b1 \u22a2 (g \u2218 f) x = 1 \u2194 f x = 1 ** rw [Function.comp, hg (mem_range_self x)] ** Qed", + "informal": "" + }, + { + "formal": "Stream'.mem_of_mem_odd ** \u03b1 : Type u \u03b2 : Type v \u03b4 : Type w a : \u03b1 s : Stream' \u03b1 x\u271d : a \u2208 odd s n : \u2115 h : (fun b => a = b) (get (odd s) n) \u22a2 (fun b => a = b) (get s (2 * n + 1)) ** rw [h, get_odd] ** Qed", + "informal": "" + }, + { + "formal": "skewAdjointMatricesLieSubalgebraEquiv_apply ** R : Type u n : Type w inst\u271d\u00b2 : CommRing R inst\u271d\u00b9 : DecidableEq n inst\u271d : Fintype n J P : Matrix n n R h : Invertible P A : { x // x \u2208 skewAdjointMatricesLieSubalgebra J } \u22a2 \u2191(\u2191(skewAdjointMatricesLieSubalgebraEquiv J P h) A) = P\u207b\u00b9 * \u2191A * P ** simp [skewAdjointMatricesLieSubalgebraEquiv] ** Qed", + "informal": "" + }, + { + "formal": "ProbabilityTheory.kernel.compProd_null ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 hs : MeasurableSet s \u22a2 \u2191\u2191(\u2191(\u03ba \u2297\u2096 \u03b7) a) s = 0 \u2194 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0 ** rw [kernel.compProd_apply _ _ _ hs, lintegral_eq_zero_iff] ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 hs : MeasurableSet s \u22a2 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s}) =\u1d50[\u2191\u03ba a] 0 \u2194 (fun b => \u2191\u2191(\u2191\u03b7 (a, b)) (Prod.mk b \u207b\u00b9' s)) =\u1d50[\u2191\u03ba a] 0 ** rfl ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b9 : Type u_3 m\u03b1 : MeasurableSpace \u03b1 m\u03b2 : MeasurableSpace \u03b2 \u03b3 : Type u_4 m\u03b3 : MeasurableSpace \u03b3 s : Set (\u03b2 \u00d7 \u03b3) \u03ba : { x // x \u2208 kernel \u03b1 \u03b2 } inst\u271d\u00b9 : IsSFiniteKernel \u03ba \u03b7 : { x // x \u2208 kernel (\u03b1 \u00d7 \u03b2) \u03b3 } inst\u271d : IsSFiniteKernel \u03b7 a\u271d a : \u03b1 hs : MeasurableSet s \u22a2 Measurable fun b => \u2191\u2191(\u2191\u03b7 (a, b)) {c | (b, c) \u2208 s} ** exact kernel.measurable_kernel_prod_mk_left' hs a ** Qed", + "informal": "" + }, + { + "formal": "String.prev_of_valid' ** cs cs' : List Char \u22a2 prev { data := cs ++ cs' } { byteIdx := utf8Len cs } = { byteIdx := utf8Len (List.dropLast cs) } ** match cs, cs.eq_nil_or_concat with\n| _, .inl rfl => rfl\n| _, .inr \u27e8cs, c, rfl\u27e9 => simp [prev_of_valid] ** cs cs' : List Char \u22a2 prev { data := [] ++ cs' } { byteIdx := utf8Len [] } = { byteIdx := utf8Len (List.dropLast []) } ** rfl ** cs\u271d cs' cs : List Char c : Char \u22a2 prev { data := cs ++ [c] ++ cs' } { byteIdx := utf8Len (cs ++ [c]) } = { byteIdx := utf8Len (List.dropLast (cs ++ [c])) } ** simp [prev_of_valid] ** Qed", + "informal": "" + }, + { + "formal": "Finset.fold_image ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 op : \u03b2 \u2192 \u03b2 \u2192 \u03b2 hc : IsCommutative \u03b2 op ha : IsAssociative \u03b2 op f : \u03b1 \u2192 \u03b2 b : \u03b2 s\u271d : Finset \u03b1 a : \u03b1 inst\u271d : DecidableEq \u03b1 g : \u03b3 \u2192 \u03b1 s : Finset \u03b3 H : \u2200 (x : \u03b3), x \u2208 s \u2192 \u2200 (y : \u03b3), y \u2208 s \u2192 g x = g y \u2192 x = y \u22a2 fold op b f (image g s) = fold op b (f \u2218 g) s ** simp only [fold, image_val_of_injOn H, Multiset.map_map] ** Qed", + "informal": "" + }, + { + "formal": "CategoryTheory.Arrow.square_from_iso_invert ** T : Type u inst\u271d : Category.{v, u} T X Y : T i : X \u2245 Y p : Arrow T sq : mk i.hom \u27f6 p \u22a2 i.inv \u226b sq.left \u226b p.hom = sq.right ** simp only [Iso.inv_hom_id_assoc, Arrow.w, Arrow.mk_hom] ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.trailingDegree_monomial ** R : Type u S : Type v a b : R n m : \u2115 inst\u271d : Semiring R p q r : R[X] ha : a \u2260 0 \u22a2 trailingDegree (\u2191(monomial n) a) = \u2191n ** rw [trailingDegree, support_monomial n ha, min_singleton] ** R : Type u S : Type v a b : R n m : \u2115 inst\u271d : Semiring R p q r : R[X] ha : a \u2260 0 \u22a2 \u2191n = \u2191n ** rfl ** Qed", + "informal": "" + }, + { + "formal": "Polynomial.aroots_C ** R : Type u S : Type v T : Type w a\u271d b : R n : \u2115 inst\u271d\u2075 : CommRing R inst\u271d\u2074 : IsDomain R p q : R[X] inst\u271d\u00b3 : CommRing T inst\u271d\u00b2 : CommRing S inst\u271d\u00b9 : IsDomain S inst\u271d : Algebra T S a : T \u22a2 aroots (\u2191C a) S = 0 ** rw [aroots_def, map_C, roots_C] ** Qed", + "informal": "" + }, + { + "formal": "LinearMap.proj_comp_stdBasis ** R : Type u_1 \u03b9 : Type u_2 inst\u271d\u00b3 : Semiring R \u03c6 : \u03b9 \u2192 Type u_3 inst\u271d\u00b2 : (i : \u03b9) \u2192 AddCommMonoid (\u03c6 i) inst\u271d\u00b9 : (i : \u03b9) \u2192 Module R (\u03c6 i) inst\u271d : DecidableEq \u03b9 i j : \u03b9 \u22a2 comp (proj i) (stdBasis R \u03c6 j) = diag j i ** rw [stdBasis_eq_pi_diag, proj_pi] ** Qed", + "informal": "" + }, + { + "formal": "Int.gcd_least_linear ** a b : \u2124 ha : a \u2260 0 \u22a2 IsLeast {n | 0 < n \u2227 \u2203 x y, \u2191n = a * x + b * y} (gcd a b) ** simp_rw [\u2190 gcd_dvd_iff] ** a b : \u2124 ha : a \u2260 0 \u22a2 IsLeast {n | 0 < n \u2227 gcd a b \u2223 n} (gcd a b) ** constructor ** case left a b : \u2124 ha : a \u2260 0 \u22a2 gcd a b \u2208 {n | 0 < n \u2227 gcd a b \u2223 n} ** simpa [and_true_iff, dvd_refl, Set.mem_setOf_eq] using gcd_pos_of_ne_zero_left b ha ** case right a b : \u2124 ha : a \u2260 0 \u22a2 gcd a b \u2208 lowerBounds {n | 0 < n \u2227 gcd a b \u2223 n} ** simp only [lowerBounds, and_imp, Set.mem_setOf_eq] ** case right a b : \u2124 ha : a \u2260 0 \u22a2 \u2200 \u2983a_1 : \u2115\u2984, 0 < a_1 \u2192 gcd a b \u2223 a_1 \u2192 gcd a b \u2264 a_1 ** exact fun n hn_pos hn => Nat.le_of_dvd hn_pos hn ** Qed", + "informal": "" + }, + { + "formal": "List.formPerm_pow_length_eq_one_of_nodup ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x : \u03b1 hl : Nodup l \u22a2 formPerm l ^ length l = 1 ** ext x ** case H \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 \u22a2 \u2191(formPerm l ^ length l) x = \u21911 x ** by_cases hx : x \u2208 l ** case pos \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : x \u2208 l \u22a2 \u2191(formPerm l ^ length l) x = \u21911 x ** obtain \u27e8k, hk, rfl\u27e9 := nthLe_of_mem hx ** case pos.intro.intro \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x : \u03b1 hl : Nodup l k : \u2115 hk : k < length l hx : nthLe l k hk \u2208 l \u22a2 \u2191(formPerm l ^ length l) (nthLe l k hk) = \u21911 (nthLe l k hk) ** simp [formPerm_pow_apply_nthLe _ hl, Nat.mod_eq_of_lt hk] ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : \u00acx \u2208 l \u22a2 \u2191(formPerm l ^ length l) x = \u21911 x ** have : x \u2209 { x | (l.formPerm ^ l.length) x \u2260 x } := by\n intro H\n refine' hx _\n replace H := set_support_zpow_subset l.formPerm l.length H\n simpa using support_formPerm_le' _ H ** case neg \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : \u00acx \u2208 l this : \u00acx \u2208 {x | \u2191(formPerm l ^ length l) x \u2260 x} \u22a2 \u2191(formPerm l ^ length l) x = \u21911 x ** simpa using this ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : \u00acx \u2208 l \u22a2 \u00acx \u2208 {x | \u2191(formPerm l ^ length l) x \u2260 x} ** intro H ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : \u00acx \u2208 l H : x \u2208 {x | \u2191(formPerm l ^ length l) x \u2260 x} \u22a2 False ** refine' hx _ ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : \u00acx \u2208 l H : x \u2208 {x | \u2191(formPerm l ^ length l) x \u2260 x} \u22a2 x \u2208 l ** replace H := set_support_zpow_subset l.formPerm l.length H ** \u03b1 : Type u_1 \u03b2 : Type u_2 inst\u271d : DecidableEq \u03b1 l : List \u03b1 x\u271d : \u03b1 hl : Nodup l x : \u03b1 hx : \u00acx \u2208 l H : x \u2208 {x | \u2191(formPerm l) x \u2260 x} \u22a2 x \u2208 l ** simpa using support_formPerm_le' _ H ** Qed", + "informal": "" + }, + { + "formal": "Int.neg_pred ** a : \u2124 \u22a2 -pred a = succ (-a) ** rw [neg_eq_iff_eq_neg.mp (neg_succ (-a)), neg_neg] ** Qed", + "informal": "" + }, + { + "formal": "ContinuousLinearMap.comp_zero ** R\u2081 : Type u_1 R\u2082 : Type u_2 R\u2083 : Type u_3 inst\u271d\u00b9\u2077 : Semiring R\u2081 inst\u271d\u00b9\u2076 : Semiring R\u2082 inst\u271d\u00b9\u2075 : Semiring R\u2083 \u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082 \u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083 \u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083 M\u2081 : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M\u2081 inst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081 M'\u2081 : Type u_5 inst\u271d\u00b9\u00b2 : TopologicalSpace M'\u2081 inst\u271d\u00b9\u00b9 : AddCommMonoid M'\u2081 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : TopologicalSpace M\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 M\u2083 : Type u_7 inst\u271d\u2078 : TopologicalSpace M\u2083 inst\u271d\u2077 : AddCommMonoid M\u2083 M\u2084 : Type u_8 inst\u271d\u2076 : TopologicalSpace M\u2084 inst\u271d\u2075 : AddCommMonoid M\u2084 inst\u271d\u2074 : Module R\u2081 M\u2081 inst\u271d\u00b3 : Module R\u2081 M'\u2081 inst\u271d\u00b2 : Module R\u2082 M\u2082 inst\u271d\u00b9 : Module R\u2083 M\u2083 inst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083 \u22a2 comp g 0 = 0 ** ext ** case h R\u2081 : Type u_1 R\u2082 : Type u_2 R\u2083 : Type u_3 inst\u271d\u00b9\u2077 : Semiring R\u2081 inst\u271d\u00b9\u2076 : Semiring R\u2082 inst\u271d\u00b9\u2075 : Semiring R\u2083 \u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082 \u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083 \u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083 M\u2081 : Type u_4 inst\u271d\u00b9\u2074 : TopologicalSpace M\u2081 inst\u271d\u00b9\u00b3 : AddCommMonoid M\u2081 M'\u2081 : Type u_5 inst\u271d\u00b9\u00b2 : TopologicalSpace M'\u2081 inst\u271d\u00b9\u00b9 : AddCommMonoid M'\u2081 M\u2082 : Type u_6 inst\u271d\u00b9\u2070 : TopologicalSpace M\u2082 inst\u271d\u2079 : AddCommMonoid M\u2082 M\u2083 : Type u_7 inst\u271d\u2078 : TopologicalSpace M\u2083 inst\u271d\u2077 : AddCommMonoid M\u2083 M\u2084 : Type u_8 inst\u271d\u2076 : TopologicalSpace M\u2084 inst\u271d\u2075 : AddCommMonoid M\u2084 inst\u271d\u2074 : Module R\u2081 M\u2081 inst\u271d\u00b3 : Module R\u2081 M'\u2081 inst\u271d\u00b2 : Module R\u2082 M\u2082 inst\u271d\u00b9 : Module R\u2083 M\u2083 inst\u271d : RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083 x\u271d : M\u2081 \u22a2 \u2191(comp g 0) x\u271d = \u21910 x\u271d ** simp ** Qed", + "informal": "" + }, + { + "formal": "MeasurableEquiv.measurableSet_preimage ** \u03b1 : Type u_1 \u03b2 : Type u_2 \u03b3 : Type u_3 \u03b4 : Type u_4 \u03b4' : Type u_5 \u03b9 : Sort u\u03b9 s\u271d t u : Set \u03b1 inst\u271d\u00b3 : MeasurableSpace \u03b1 inst\u271d\u00b2 : MeasurableSpace \u03b2 inst\u271d\u00b9 : MeasurableSpace \u03b3 inst\u271d : MeasurableSpace \u03b4 e : \u03b1 \u2243\u1d50 \u03b2 s : Set \u03b2 h : MeasurableSet (\u2191e \u207b\u00b9' s) \u22a2 MeasurableSet s ** simpa only [symm_preimage_preimage] using e.symm.measurable h ** Qed", + "informal": "" + }, + { + "formal": "rank_mul_rank ** F\u271d : Type u K\u271d : Type v A\u271d : Type w inst\u271d\u00b2\u00b9 : CommRing F\u271d inst\u271d\u00b2\u2070 : Ring K\u271d inst\u271d\u00b9\u2079 : AddCommGroup A\u271d inst\u271d\u00b9\u2078 : Algebra F\u271d K\u271d inst\u271d\u00b9\u2077 : Module K\u271d A\u271d inst\u271d\u00b9\u2076 : Module F\u271d A\u271d inst\u271d\u00b9\u2075 : IsScalarTower F\u271d K\u271d A\u271d inst\u271d\u00b9\u2074 : StrongRankCondition F\u271d inst\u271d\u00b9\u00b3 : StrongRankCondition K\u271d inst\u271d\u00b9\u00b2 : Module.Free F\u271d K\u271d inst\u271d\u00b9\u00b9 : Module.Free K\u271d A\u271d F : Type u K A : Type v inst\u271d\u00b9\u2070 : CommRing F inst\u271d\u2079 : Ring K inst\u271d\u2078 : AddCommGroup A inst\u271d\u2077 : Algebra F K inst\u271d\u2076 : Module K A inst\u271d\u2075 : Module F A inst\u271d\u2074 : IsScalarTower F K A inst\u271d\u00b3 : StrongRankCondition F inst\u271d\u00b2 : StrongRankCondition K inst\u271d\u00b9 : Module.Free F K inst\u271d : Module.Free K A \u22a2 Module.rank F K * Module.rank K A = Module.rank F A ** convert lift_rank_mul_lift_rank F K A <;> rw [lift_id] ** Qed", + "informal": "" + }, + { + "formal": "Finset.weightedVSubOfPoint_const_smul ** k : Type u_1 V : Type u_2 P : Type u_3 inst\u271d\u00b2 : Ring k inst\u271d\u00b9 : AddCommGroup V inst\u271d : Module k V S : AffineSpace V P \u03b9 : Type u_4 s : Finset \u03b9 \u03b9\u2082 : Type u_5 s\u2082 : Finset \u03b9\u2082 w : \u03b9 \u2192 k p : \u03b9 \u2192 P b : P c : k \u22a2 \u2191(weightedVSubOfPoint s p b) (c \u2022 w) = c \u2022 \u2191(weightedVSubOfPoint s p b) w ** simp_rw [weightedVSubOfPoint_apply, smul_sum, Pi.smul_apply, smul_smul, smul_eq_mul] ** Qed", + "informal": "" + }, + { + "formal": "Std.RBNode.Balanced.insert ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 h : Balanced t c n \u22a2 \u2203 c' n', Balanced (RBNode.insert cmp t v) c' n' ** unfold insert ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 h : Balanced t c n \u22a2 \u2203 c' n', Balanced (match isRed t with | red => setBlack (ins cmp v t) | black => ins cmp v t) c' n' ** match ins cmp v t, h.ins cmp v with\n| _, .balanced h => split <;> [exact \u27e8_, h.setBlack\u27e9; exact \u27e8_, _, h\u27e9]\n| _, .redred _ ha hb => have .node red .. := t; exact \u27e8_, _, .black ha hb\u27e9 ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 h\u271d : Balanced t c n x\u271d : RBNode \u03b1 c\u271d : RBColor h : Balanced x\u271d c\u271d n \u22a2 \u2203 c' n', Balanced (match isRed t with | red => setBlack x\u271d | black => x\u271d) c' n' ** split <;> [exact \u27e8_, h.setBlack\u27e9; exact \u27e8_, _, h\u27e9] ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t : RBNode \u03b1 h : Balanced t c n a\u271d\u00b9 : RBNode \u03b1 c\u2081\u271d : RBColor b\u271d : RBNode \u03b1 c\u2082\u271d : RBColor x\u271d : \u03b1 a\u271d : isRed t = red ha : Balanced a\u271d\u00b9 c\u2081\u271d n hb : Balanced b\u271d c\u2082\u271d n \u22a2 \u2203 c' n', Balanced (match isRed t with | red => setBlack (node red a\u271d\u00b9 x\u271d b\u271d) | black => node red a\u271d\u00b9 x\u271d b\u271d) c' n' ** have .node red .. := t ** \u03b1 : Type u_1 c : RBColor n : Nat cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering v : \u03b1 t a\u271d\u00b9 : RBNode \u03b1 c\u2081\u271d : RBColor b\u271d : RBNode \u03b1 c\u2082\u271d : RBColor x\u271d : \u03b1 ha : Balanced a\u271d\u00b9 c\u2081\u271d n hb : Balanced b\u271d c\u2082\u271d n l\u271d : RBNode \u03b1 v\u271d : \u03b1 r\u271d : RBNode \u03b1 h : Balanced (node red l\u271d v\u271d r\u271d) c n a\u271d : isRed (node red l\u271d v\u271d r\u271d) = red \u22a2 \u2203 c' n', Balanced (match isRed (node red l\u271d v\u271d r\u271d) with | red => setBlack (node red a\u271d\u00b9 x\u271d b\u271d) | black => node red a\u271d\u00b9 x\u271d b\u271d) c' n' ** exact \u27e8_, _, .black ha hb\u27e9 ** Qed", + "informal": "" + }, + { + "formal": "Finset.sum_card_inter_le ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 B : Finset (Finset \u03b1) n : \u2115 h : \u2200 (a : \u03b1), a \u2208 s \u2192 card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n \u22a2 \u2211 t in B, card (s \u2229 t) \u2264 card s * n ** refine' le_trans _ (s.sum_le_card_nsmul _ _ h) ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 B : Finset (Finset \u03b1) n : \u2115 h : \u2200 (a : \u03b1), a \u2208 s \u2192 card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n \u22a2 \u2211 t in B, card (s \u2229 t) \u2264 \u2211 x in s, card (filter ((fun x x_1 => x \u2208 x_1) x) B) ** simp_rw [\u2190 filter_mem_eq_inter, card_eq_sum_ones, sum_filter] ** \u03b9 : Type u_1 \u03b1 : Type u_2 \u03b2 : Type u_3 M : Type u_4 N : Type u_5 G : Type u_6 k : Type u_7 R : Type u_8 inst\u271d : DecidableEq \u03b1 s : Finset \u03b1 B : Finset (Finset \u03b1) n : \u2115 h : \u2200 (a : \u03b1), a \u2208 s \u2192 card (filter ((fun x x_1 => x \u2208 x_1) a) B) \u2264 n \u22a2 (\u2211 x in B, \u2211 a in s, if a \u2208 x then 1 else 0) \u2264 \u2211 x in s, \u2211 a in B, if x \u2208 a then 1 else 0 ** exact sum_comm.le ** Qed", + "informal": "" + }, + { + "formal": "Real.deriv_arccos ** x : \u211d \u22a2 -deriv (fun y => arcsin y) x = -(1 / sqrt (1 - x ^ 2)) ** simp only [deriv_arcsin] ** Qed", + "informal": "" + }, + { + "formal": "Ideal.rank_pow_quot ** R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e \u22a2 Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) ** let Q : \u2115 \u2192 Prop :=\n fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) }\n = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) ** R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) \u22a2 Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) ** refine @Nat.decreasingInduction' Q i e (fun j lt_e _le_j ih => ?_) hi ?_ ** case refine_1 R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) j : \u2115 lt_e : j < e _le_j : i \u2264 j ih : Q (j + 1) \u22a2 Q j ** dsimp only ** case refine_1 R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) j : \u2115 lt_e : j < e _le_j : i \u2264 j ih : Q (j + 1) \u22a2 Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ j) } = (e - j) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) ** rw [rank_pow_quot_aux f p P _ lt_e, ih, \u2190 succ_nsmul, Nat.sub_succ, \u2190 Nat.succ_eq_add_one,\n Nat.succ_pred_eq_of_pos (Nat.sub_pos_of_lt lt_e)] ** R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) j : \u2115 lt_e : j < e _le_j : i \u2264 j ih : Q (j + 1) \u22a2 P \u2260 \u22a5 ** assumption ** case refine_2 R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) \u22a2 Q e ** dsimp only ** case refine_2 R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) \u22a2 Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ e) } = (e - e) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) ** rw [Nat.sub_self, zero_nsmul, map_quotient_self] ** case refine_2 R : Type u inst\u271d\u2075 : CommRing R S : Type v inst\u271d\u2074 : CommRing S f : R \u2192+* S p : Ideal R P : Ideal S hfp : NeZero e inst\u271d\u00b3 : IsDomain S inst\u271d\u00b2 : IsDedekindDomain S inst\u271d\u00b9 : IsMaximal p inst\u271d : IsPrime P hP0 : P \u2260 \u22a5 i : \u2115 hi : i \u2264 e Q : \u2115 \u2192 Prop := fun i => Module.rank (R \u29f8 p) { x // x \u2208 map (Quotient.mk (P ^ e)) (P ^ i) } = (e - i) \u2022 Module.rank (R \u29f8 p) (S \u29f8 P) \u22a2 Module.rank (R \u29f8 p) { x // x \u2208 \u22a5 } = 0 ** exact rank_bot (R \u29f8 p) (S \u29f8 P ^ e) ** Qed", + "informal": "" + }, + { + "formal": "IsPrimitiveRoot.power_basis_int'_dim ** p : \u2115+ k : \u2115 K : Type u inst\u271d\u00b9 : Field K inst\u271d : CharZero K \u03b6 : K hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p} \u211a K h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p \u22a2 (integralPowerBasis' h\u03b6).dim = \u03c6 \u2191p ** erw [@integralPowerBasis_dim p 1 K _ _ _ _ (by convert hcycl; rw [pow_one]) (by rwa [pow_one]),\n pow_one] ** p : \u2115+ k : \u2115 K : Type u inst\u271d\u00b9 : Field K inst\u271d : CharZero K \u03b6 : K hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p} \u211a K h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p \u22a2 IsCyclotomicExtension {p ^ 1} \u211a K ** convert hcycl ** case h.e'_1.h.e'_4 p : \u2115+ k : \u2115 K : Type u inst\u271d\u00b9 : Field K inst\u271d : CharZero K \u03b6 : K hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p} \u211a K h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p \u22a2 p ^ 1 = p ** rw [pow_one] ** p : \u2115+ k : \u2115 K : Type u inst\u271d\u00b9 : Field K inst\u271d : CharZero K \u03b6 : K hp : Fact (Nat.Prime \u2191p) hcycl : IsCyclotomicExtension {p} \u211a K h\u03b6 : IsPrimitiveRoot \u03b6 \u2191p \u22a2 IsPrimitiveRoot \u03b6 \u2191(p ^ 1) ** rwa [pow_one] ** Qed", + "informal": "" + } +] \ No newline at end of file