internlm/Lean-Github · Datasets at Hugging Face

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	case nil => apply Exists.intro [] simp	h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ ∃ xs, [] = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	case cons hd tl ih => simp at h1 cases h1 case intro h1_left h1_right => specialize ih h1_right apply Exists.elim ih intro xs a1 cases hd case free_ x => apply Exists.intro (x :: xs) simp exact a1 case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left	hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v ⊢ ∃ xs, hd :: tl = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	apply Exists.intro []	h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ ∃ xs, [] = List.map free_ xs	h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ [] = List.map free_ []
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	simp	h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ [] = List.map free_ []	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	simp at h1	hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v ⊢ ∃ xs, hd :: tl = List.map free_ xs	hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	cases h1	hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs	case intro hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs left✝ : Var.lc_at 0 hd right✝ : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	case intro h1_left h1_right => specialize ih h1_right apply Exists.elim ih intro xs a1 cases hd case free_ x => apply Exists.intro (x :: xs) simp exact a1 case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left	hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	specialize ih h1_right	hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs	hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	apply Exists.elim ih	hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs	hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	intro xs a1	hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs	hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	cases hd	hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs	case free_ tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs a✝ : String h1_left : Var.lc_at 0 (free_ a✝) ⊢ ∃ xs, free_ a✝ :: tl = List.map free_ xs case bound_ tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs a✝ : ℕ h1_left : Var.lc_at 0 (bound_ a✝) ⊢ ∃ xs, bound_ a✝ :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	case free_ x => apply Exists.intro (x :: xs) simp exact a1	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ ∃ xs, free_ x :: tl = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : Var.lc_at 0 (bound_ i) ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	apply Exists.intro (x :: xs)	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ ∃ xs, free_ x :: tl = List.map free_ xs	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ free_ x :: tl = List.map free_ (x :: xs)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	simp	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ free_ x :: tl = List.map free_ (x :: xs)	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	exact a1	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ tl = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	simp only [Var.lc_at] at h1_left	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : Var.lc_at 0 (bound_ i) ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : i < 0 ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.free_var_list_to_string_list	[186, 1]	[210, 24]	simp at h1_left	tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : i < 0 ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	cases v	j : ℕ z : String v : Var ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}	case free_ j : ℕ z a✝ : String ⊢ (Var.open j (free_ z) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ {free_ z} case bound_ j : ℕ z : String a✝ : ℕ ⊢ (Var.open j (free_ z) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	case free_ x => simp only [Var.open] simp only [Var.freeVarSet] simp	j : ℕ z x : String ⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	case bound_ i => simp only [Var.open] split_ifs case _ c1 => simp only [Var.freeVarSet] simp case _ c1 c2 => simp only [Var.freeVarSet] simp case _ c1 c2 => simp only [Var.freeVarSet] simp	j : ℕ z : String i : ℕ ⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp only [Var.open]	j : ℕ z x : String ⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}	j : ℕ z x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp only [Var.freeVarSet]	j : ℕ z x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}	j : ℕ z x : String ⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp	j : ℕ z x : String ⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp only [Var.open]	j : ℕ z : String i : ℕ ⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	j : ℕ z : String i : ℕ ⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	split_ifs	j : ℕ z : String i : ℕ ⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	case pos j : ℕ z : String i : ℕ h✝ : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} case pos j : ℕ z : String i : ℕ h✝¹ : ¬i < j h✝ : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} case neg j : ℕ z : String i : ℕ h✝¹ : ¬i < j h✝ : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	case _ c1 => simp only [Var.freeVarSet] simp	j : ℕ z : String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	case _ c1 c2 => simp only [Var.freeVarSet] simp	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	case _ c1 c2 => simp only [Var.freeVarSet] simp	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp only [Var.freeVarSet]	j : ℕ z : String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	j : ℕ z : String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp	j : ℕ z : String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp only [Var.freeVarSet]	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ {free_ z} ⊆ ∅ ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ {free_ z} ⊆ ∅ ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp only [Var.freeVarSet]	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ ∅ ⊆ ∅ ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenFreeVarSet	[216, 1]	[238, 11]	simp	j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ ∅ ⊆ ∅ ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	induction F generalizing j	j : ℕ z : String F : Formula ⊢ (Formula.open j (free_ z) F).freeVarSet ⊆ F.freeVarSet ∪ {free_ z}	case pred_ z a✝¹ : String a✝ : List Var j : ℕ ⊢ (Formula.open j (free_ z) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ {free_ z} case not_ z : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ {free_ z} case imp_ z : String a✝¹ a✝ : Formula a_ih✝¹ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ {free_ z} a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ {free_ z} case forall_ z a✝¹ : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	case not_ phi phi_ih => simp only [Formula.open] simp only [Formula.freeVarSet] apply phi_ih	z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	case forall_ x phi phi_ih => simp only [Formula.open] simp only [Formula.freeVarSet] apply phi_ih	z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.open]	z X : String vs : List Var j : ℕ ⊢ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}	z X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.freeVarSet]	z X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}	z X : String vs : List Var j : ℕ ⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp	z X : String vs : List Var j : ℕ ⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}	z X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	intro v a1	z X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}	z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	trans Var.freeVarSet v ∪ {free_ z}	z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}	z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z} z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	exact VarOpenFreeVarSet j z v	z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	apply Finset.union_subset_union_left	z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}	case h z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	apply Finset.subset_biUnion_of_mem	case h z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet	case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp	case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset	case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	exact a1	case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.open]	z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}	z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.freeVarSet]	z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}	z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	apply phi_ih	z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.open]	z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}	z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.freeVarSet]	z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}	z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	apply Finset.union_subset_union_left_right	z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}	case h1 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} case h2 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	exact phi_ih j	case h1 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	exact psi_ih j	case h2 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.open]	z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}	z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	simp only [Formula.freeVarSet]	z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}	z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenFreeVarSet	[241, 1]	[273, 17]	apply phi_ih	z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	cases v	j : ℕ zs : List String v : Var ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset	case free_ j : ℕ zs : List String a✝ : String ⊢ (Var.openList j (List.map free_ zs) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case bound_ j : ℕ zs : List String a✝ : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	case free_ x => simp only [Var.openList] simp only [Var.freeVarSet] simp	j : ℕ zs : List String x : String ⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	case bound_ i => simp only [Var.openList] split_ifs case pos c1 => simp only [Var.freeVarSet] simp case pos c1 c2 => simp simp only [Var.freeVarSet] simp apply List.get_mem case neg c1 c2 => simp only [Var.freeVarSet] simp	j : ℕ zs : List String i : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp only [Var.openList]	j : ℕ zs : List String x : String ⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp only [Var.freeVarSet]	j : ℕ zs : List String x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String x : String ⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp	j : ℕ zs : List String x : String ⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp only [Var.openList]	j : ℕ zs : List String i : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String i : ℕ ⊢ (if i < j then bound_ i else if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j] else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	split_ifs	j : ℕ zs : List String i : ℕ ⊢ (if i < j then bound_ i else if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j] else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	case pos j : ℕ zs : List String i : ℕ h✝ : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset case pos j : ℕ zs : List String i : ℕ h✝¹ : ¬i < j h✝ : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset case neg j : ℕ zs : List String i : ℕ h✝¹ : ¬i < j h✝ : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	case pos c1 => simp only [Var.freeVarSet] simp	j : ℕ zs : List String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	case pos c1 c2 => simp simp only [Var.freeVarSet] simp apply List.get_mem	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	case neg c1 c2 => simp only [Var.freeVarSet] simp	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp only [Var.freeVarSet]	j : ℕ zs : List String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp	j : ℕ zs : List String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp only [Var.freeVarSet]	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	apply List.get_mem	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp only [Var.freeVarSet]	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.VarOpenListFreeVarSet	[279, 1]	[303, 11]	simp	j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	induction F generalizing j	j : ℕ zs : List String F : Formula ⊢ (Formula.openList j (List.map free_ zs) F).freeVarSet ⊆ F.freeVarSet ∪ (List.map free_ zs).toFinset	case pred_ zs : List String a✝¹ : String a✝ : List Var j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case not_ zs : List String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ (List.map free_ zs).toFinset case imp_ zs : List String a✝¹ a✝ : Formula a_ih✝¹ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ (List.map free_ zs).toFinset a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case forall_ zs : List String a✝¹ : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	case not_ phi phi_ih => simp only [Formula.openList] simp only [Formula.freeVarSet] apply phi_ih	zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	case forall_ x phi phi_ih => simp only [Formula.openList] simp only [Formula.freeVarSet] apply phi_ih	zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.openList]	zs : List String X : String vs : List Var j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.freeVarSet]	zs : List String X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String X : String vs : List Var j : ℕ ⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp	zs : List String X : String vs : List Var j : ℕ ⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.openList j (List.map free_ zs) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	intro v a1	zs : List String X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.openList j (List.map free_ zs) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	trans v.freeVarSet ∪ (zs.map free_).toFinset	zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	exact VarOpenListFreeVarSet j zs v	zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	apply Finset.union_subset_union_left	zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset	case h zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	apply Finset.subset_biUnion_of_mem	case h zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet	case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp	case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset	case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	exact a1	case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.openList]	zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.freeVarSet]	zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	apply phi_ih	zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.openList]	zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.freeVarSet]	zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	apply Finset.union_subset_union_left_right	zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset	case h1 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset case h2 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	exact phi_ih j	case h1 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	exact psi_ih j	case h2 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Paper.lean	LN.FormulaOpenListFreeVarSet	[306, 1]	[338, 17]	simp only [Formula.openList]	zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset	zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

case nil => apply Exists.intro [] simp

h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ ∃ xs, [] = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

case cons hd tl ih => simp at h1 cases h1 case intro h1_left h1_right => specialize ih h1_right apply Exists.elim ih intro xs a1 cases hd case free_ x => apply Exists.intro (x :: xs) simp exact a1 case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left

hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v ⊢ ∃ xs, hd :: tl = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

apply Exists.intro []

h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ ∃ xs, [] = List.map free_ xs

h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ [] = List.map free_ []

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

simp

h1 : ∀ v ∈ [], Var.lc_at 0 v ⊢ [] = List.map free_ []

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

simp at h1

hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : ∀ v ∈ hd :: tl, Var.lc_at 0 v ⊢ ∃ xs, hd :: tl = List.map free_ xs

hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

cases h1

hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1 : Var.lc_at 0 hd ∧ ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs

case intro hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs left✝ : Var.lc_at 0 hd right✝ : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

case intro h1_left h1_right => specialize ih h1_right apply Exists.elim ih intro xs a1 cases hd case free_ x => apply Exists.intro (x :: xs) simp exact a1 case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left

hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

specialize ih h1_right

hd : Var tl : List Var ih : (∀ v ∈ tl, Var.lc_at 0 v) → ∃ xs, tl = List.map free_ xs h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ⊢ ∃ xs, hd :: tl = List.map free_ xs

hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

apply Exists.elim ih

hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs

hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

intro xs a1

hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs ⊢ ∀ (a : List String), tl = List.map free_ a → ∃ xs, hd :: tl = List.map free_ xs

hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

cases hd

hd : Var tl : List Var h1_left : Var.lc_at 0 hd h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs ⊢ ∃ xs, hd :: tl = List.map free_ xs

case free_ tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs a✝ : String h1_left : Var.lc_at 0 (free_ a✝) ⊢ ∃ xs, free_ a✝ :: tl = List.map free_ xs case bound_ tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs a✝ : ℕ h1_left : Var.lc_at 0 (bound_ a✝) ⊢ ∃ xs, bound_ a✝ :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

case free_ x => apply Exists.intro (x :: xs) simp exact a1

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ ∃ xs, free_ x :: tl = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

case bound_ i => simp only [Var.lc_at] at h1_left simp at h1_left

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : Var.lc_at 0 (bound_ i) ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

apply Exists.intro (x :: xs)

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ ∃ xs, free_ x :: tl = List.map free_ xs

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ free_ x :: tl = List.map free_ (x :: xs)

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

simp

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ free_ x :: tl = List.map free_ (x :: xs)

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

exact a1

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs x : String h1_left : Var.lc_at 0 (free_ x) ⊢ tl = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

simp only [Var.lc_at] at h1_left

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : Var.lc_at 0 (bound_ i) ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : i < 0 ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.free_var_list_to_string_list

[186, 1]

[210, 24]

simp at h1_left

tl : List Var h1_right : ∀ a ∈ tl, Var.lc_at 0 a ih : ∃ xs, tl = List.map free_ xs xs : List String a1 : tl = List.map free_ xs i : ℕ h1_left : i < 0 ⊢ ∃ xs, bound_ i :: tl = List.map free_ xs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

cases v

j : ℕ z : String v : Var ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}

case free_ j : ℕ z a✝ : String ⊢ (Var.open j (free_ z) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ {free_ z} case bound_ j : ℕ z : String a✝ : ℕ ⊢ (Var.open j (free_ z) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

case free_ x => simp only [Var.open] simp only [Var.freeVarSet] simp

j : ℕ z x : String ⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

case bound_ i => simp only [Var.open] split_ifs case _ c1 => simp only [Var.freeVarSet] simp case _ c1 c2 => simp only [Var.freeVarSet] simp case _ c1 c2 => simp only [Var.freeVarSet] simp

j : ℕ z : String i : ℕ ⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp only [Var.open]

j : ℕ z x : String ⊢ (Var.open j (free_ z) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}

j : ℕ z x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp only [Var.freeVarSet]

j : ℕ z x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ {free_ z}

j : ℕ z x : String ⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp

j : ℕ z x : String ⊢ {free_ x} ⊆ {free_ x} ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp only [Var.open]

j : ℕ z : String i : ℕ ⊢ (Var.open j (free_ z) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

j : ℕ z : String i : ℕ ⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

split_ifs

j : ℕ z : String i : ℕ ⊢ (if i < j then bound_ i else if i = j then free_ z else bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

case pos j : ℕ z : String i : ℕ h✝ : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} case pos j : ℕ z : String i : ℕ h✝¹ : ¬i < j h✝ : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z} case neg j : ℕ z : String i : ℕ h✝¹ : ¬i < j h✝ : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

case _ c1 => simp only [Var.freeVarSet] simp

j : ℕ z : String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

case _ c1 c2 => simp only [Var.freeVarSet] simp

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

case _ c1 c2 => simp only [Var.freeVarSet] simp

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp only [Var.freeVarSet]

j : ℕ z : String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

j : ℕ z : String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp

j : ℕ z : String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp only [Var.freeVarSet]

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ (free_ z).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ {free_ z} ⊆ ∅ ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : i = j ⊢ {free_ z} ⊆ ∅ ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp only [Var.freeVarSet]

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ (bound_ (i - 1)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ {free_ z}

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ ∅ ⊆ ∅ ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenFreeVarSet

[216, 1]

[238, 11]

simp

j : ℕ z : String i : ℕ c1 : ¬i < j c2 : ¬i = j ⊢ ∅ ⊆ ∅ ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

induction F generalizing j

j : ℕ z : String F : Formula ⊢ (Formula.open j (free_ z) F).freeVarSet ⊆ F.freeVarSet ∪ {free_ z}

case pred_ z a✝¹ : String a✝ : List Var j : ℕ ⊢ (Formula.open j (free_ z) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ {free_ z} case not_ z : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ {free_ z} case imp_ z : String a✝¹ a✝ : Formula a_ih✝¹ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ {free_ z} a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ {free_ z} case forall_ z a✝¹ : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.open j (free_ z) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

case not_ phi phi_ih => simp only [Formula.open] simp only [Formula.freeVarSet] apply phi_ih

z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

case forall_ x phi phi_ih => simp only [Formula.open] simp only [Formula.freeVarSet] apply phi_ih

z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.open]

z X : String vs : List Var j : ℕ ⊢ (Formula.open j (free_ z) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}

z X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.freeVarSet]

z X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.open j (free_ z)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ {free_ z}

z X : String vs : List Var j : ℕ ⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp

z X : String vs : List Var j : ℕ ⊢ (List.map (Var.open j (free_ z)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

z X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

intro v a1

z X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.open j (free_ z) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

trans Var.freeVarSet v ∪ {free_ z}

z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z} z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

exact VarOpenFreeVarSet j z v

z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.open j (free_ z) v).freeVarSet ⊆ v.freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

apply Finset.union_subset_union_left

z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ {free_ z} ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ {free_ z}

case h z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

apply Finset.subset_biUnion_of_mem

case h z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet

case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp

case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset

case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

exact a1

case h.xs z X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.open]

z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}

z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.freeVarSet]

z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ {free_ z}

z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

apply phi_ih

z : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.open]

z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}

z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.freeVarSet]

z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ ((Formula.open j (free_ z) phi).imp_ (Formula.open j (free_ z) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ {free_ z}

z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

apply Finset.union_subset_union_left_right

z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ∪ (Formula.open j (free_ z) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ {free_ z}

case h1 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} case h2 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

exact phi_ih j

case h1 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

exact psi_ih j

case h2 z : String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} psi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) psi).freeVarSet ⊆ psi.freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.open]

z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open j (free_ z) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}

z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

simp only [Formula.freeVarSet]

z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (forall_ x (Formula.open (j + 1) (free_ z) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ {free_ z}

z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenFreeVarSet

[241, 1]

[273, 17]

apply phi_ih

z x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.open j (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z} j : ℕ ⊢ (Formula.open (j + 1) (free_ z) phi).freeVarSet ⊆ phi.freeVarSet ∪ {free_ z}

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

cases v

j : ℕ zs : List String v : Var ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset

case free_ j : ℕ zs : List String a✝ : String ⊢ (Var.openList j (List.map free_ zs) (free_ a✝)).freeVarSet ⊆ (free_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case bound_ j : ℕ zs : List String a✝ : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ a✝)).freeVarSet ⊆ (bound_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

case free_ x => simp only [Var.openList] simp only [Var.freeVarSet] simp

j : ℕ zs : List String x : String ⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

case bound_ i => simp only [Var.openList] split_ifs case pos c1 => simp only [Var.freeVarSet] simp case pos c1 c2 => simp simp only [Var.freeVarSet] simp apply List.get_mem case neg c1 c2 => simp only [Var.freeVarSet] simp

j : ℕ zs : List String i : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp only [Var.openList]

j : ℕ zs : List String x : String ⊢ (Var.openList j (List.map free_ zs) (free_ x)).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp only [Var.freeVarSet]

j : ℕ zs : List String x : String ⊢ (free_ x).freeVarSet ⊆ (free_ x).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String x : String ⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp

j : ℕ zs : List String x : String ⊢ {free_ x} ⊆ {free_ x} ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp only [Var.openList]

j : ℕ zs : List String i : ℕ ⊢ (Var.openList j (List.map free_ zs) (bound_ i)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String i : ℕ ⊢ (if i < j then bound_ i else if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j] else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

split_ifs

j : ℕ zs : List String i : ℕ ⊢ (if i < j then bound_ i else if x : i - j < (List.map free_ zs).length then (List.map free_ zs)[i - j] else bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

case pos j : ℕ zs : List String i : ℕ h✝ : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset case pos j : ℕ zs : List String i : ℕ h✝¹ : ¬i < j h✝ : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset case neg j : ℕ zs : List String i : ℕ h✝¹ : ¬i < j h✝ : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

case pos c1 => simp only [Var.freeVarSet] simp

j : ℕ zs : List String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

case pos c1 c2 => simp simp only [Var.freeVarSet] simp apply List.get_mem

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

case neg c1 c2 => simp only [Var.freeVarSet] simp

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp only [Var.freeVarSet]

j : ℕ zs : List String i : ℕ c1 : i < j ⊢ (bound_ i).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp

j : ℕ zs : List String i : ℕ c1 : i < j ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (List.map free_ zs)[i - j].freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp only [Var.freeVarSet]

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ (free_ (zs.get ⟨i - j, ⋯⟩)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ {free_ (zs.get ⟨i - j, ⋯⟩)} ⊆ ∅ ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

apply List.get_mem

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : i - j < (List.map free_ zs).length ⊢ zs.get ⟨i - j, ⋯⟩ ∈ zs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp only [Var.freeVarSet]

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ (bound_ (i - j - (List.map free_ zs).length + j)).freeVarSet ⊆ (bound_ i).freeVarSet ∪ (List.map free_ zs).toFinset

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.VarOpenListFreeVarSet

[279, 1]

[303, 11]

simp

j : ℕ zs : List String i : ℕ c1 : ¬i < j c2 : ¬i - j < (List.map free_ zs).length ⊢ ∅ ⊆ ∅ ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

induction F generalizing j

j : ℕ zs : List String F : Formula ⊢ (Formula.openList j (List.map free_ zs) F).freeVarSet ⊆ F.freeVarSet ∪ (List.map free_ zs).toFinset

case pred_ zs : List String a✝¹ : String a✝ : List Var j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (pred_ a✝¹ a✝)).freeVarSet ⊆ (pred_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case not_ zs : List String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) a✝.not_).freeVarSet ⊆ a✝.not_.freeVarSet ∪ (List.map free_ zs).toFinset case imp_ zs : List String a✝¹ a✝ : Formula a_ih✝¹ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝¹).freeVarSet ⊆ a✝¹.freeVarSet ∪ (List.map free_ zs).toFinset a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (a✝¹.imp_ a✝)).freeVarSet ⊆ (a✝¹.imp_ a✝).freeVarSet ∪ (List.map free_ zs).toFinset case forall_ zs : List String a✝¹ : String a✝ : Formula a_ih✝ : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) a✝).freeVarSet ⊆ a✝.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ a✝¹ a✝)).freeVarSet ⊆ (forall_ a✝¹ a✝).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

case not_ phi phi_ih => simp only [Formula.openList] simp only [Formula.freeVarSet] apply phi_ih

zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

case forall_ x phi phi_ih => simp only [Formula.openList] simp only [Formula.freeVarSet] apply phi_ih

zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.openList]

zs : List String X : String vs : List Var j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (pred_ X vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.freeVarSet]

zs : List String X : String vs : List Var j : ℕ ⊢ (pred_ X (List.map (Var.openList j (List.map free_ zs)) vs)).freeVarSet ⊆ (pred_ X vs).freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String X : String vs : List Var j : ℕ ⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp

zs : List String X : String vs : List Var j : ℕ ⊢ (List.map (Var.openList j (List.map free_ zs)) vs).toFinset.biUnion Var.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.openList j (List.map free_ zs) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

intro v a1

zs : List String X : String vs : List Var j : ℕ ⊢ ∀ a ∈ vs, (Var.openList j (List.map free_ zs) a).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

trans v.freeVarSet ∪ (zs.map free_).toFinset

zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

exact VarOpenListFreeVarSet j zs v

zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ (Var.openList j (List.map free_ zs) v).freeVarSet ⊆ v.freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

apply Finset.union_subset_union_left

zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ∪ (List.map free_ zs).toFinset ⊆ vs.toFinset.biUnion Var.freeVarSet ∪ (List.map free_ zs).toFinset

case h zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

apply Finset.subset_biUnion_of_mem

case h zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v.freeVarSet ⊆ vs.toFinset.biUnion Var.freeVarSet

case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp

case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs.toFinset

case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

exact a1

case h.xs zs : List String X : String vs : List Var j : ℕ v : Var a1 : v ∈ vs ⊢ v ∈ vs

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.openList]

zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi.not_).freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.freeVarSet]

zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).not_.freeVarSet ⊆ phi.not_.freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

apply phi_ih

zs : List String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.openList]

zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (phi.imp_ psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.freeVarSet]

zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ ((Formula.openList j (List.map free_ zs) phi).imp_ (Formula.openList j (List.map free_ zs) psi)).freeVarSet ⊆ (phi.imp_ psi).freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

apply Finset.union_subset_union_left_right

zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ∪ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ phi.freeVarSet ∪ psi.freeVarSet ∪ (List.map free_ zs).toFinset

case h1 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset case h2 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

exact phi_ih j

case h1 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

exact psi_ih j

case h2 zs : List String phi psi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset psi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) psi).freeVarSet ⊆ psi.freeVarSet ∪ (List.map free_ zs).toFinset

no goals

https://github.com/pthomas505/FOL.git

097a4abea51b641d144539b9a0f7516f3b9d818c

FOL/LN/Paper.lean

LN.FormulaOpenListFreeVarSet

[306, 1]

[338, 17]

simp only [Formula.openList]

zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (Formula.openList j (List.map free_ zs) (forall_ x phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset

zs : List String x : String phi : Formula phi_ih : ∀ (j : ℕ), (Formula.openList j (List.map free_ zs) phi).freeVarSet ⊆ phi.freeVarSet ∪ (List.map free_ zs).toFinset j : ℕ ⊢ (forall_ x (Formula.openList (j + 1) (List.map free_ zs) phi)).freeVarSet ⊆ (forall_ x phi).freeVarSet ∪ (List.map free_ zs).toFinset