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Split (1)

train · 219k rows

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/NV/Sub/Prop/All/Rec/Sub.lean	FOL.NV.Sub.Prop.All.Rec.substitution_is_valid	[185, 1]	[196, 11]	simp only [substitution_theorem D I V E τ F]	F : Formula τ : PredName → PredName h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D I V E (sub τ F)	F : Formula τ : PredName → PredName h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds } V E F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Prop/All/Rec/Sub.lean	FOL.NV.Sub.Prop.All.Rec.substitution_is_valid	[185, 1]	[196, 11]	apply h1	F : Formula τ : PredName → PredName h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F D : Type I : Interpretation D V : VarAssignment D E : Env ⊢ Holds D { nonempty := ⋯, pred_const_ := I.pred_const_, pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds } V E F	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	induction F generalizing binders	F : Formula v t : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders F = F	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝ case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝ case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = eq_ a✝¹ a✝ case true_ v t : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders true_ = true_ case false_ v t : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders false_ = false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders a✝.not_ = a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = a✝¹.imp_ a✝ case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = a✝¹.and_ a✝ case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = a✝¹.or_ a✝ case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = a✝¹.iff_ a✝ case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = exists_ a✝¹ a✝ case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	any_goals simp only [replaceFreeAux]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝ case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝ case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = eq_ a✝¹ a✝ case true_ v t : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders true_ = true_ case false_ v t : VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders false_ = false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders a✝.not_ = a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = a✝¹.imp_ a✝ case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = a✝¹.and_ a✝ case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = a✝¹.or_ a✝ case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = a✝¹.iff_ a✝ case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = exists_ a✝¹ a✝ case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = def_ a✝¹ a✝	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ pred_const_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = pred_const_ a✝¹ a✝ case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ pred_var_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = pred_var_ a✝¹ a✝ case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ eq_ (if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) (if v = a✝ ∧ a✝ ∉ binders then t else a✝) = eq_ a✝¹ a✝ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders a✝).not_ = a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders a✝¹).imp_ (replaceFreeAux v t binders a✝) = a✝¹.imp_ a✝ case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders a✝¹).and_ (replaceFreeAux v t binders a✝) = a✝¹.and_ a✝ case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders a✝¹).or_ (replaceFreeAux v t binders a✝) = a✝¹.or_ a✝ case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝¹ = a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders a✝¹).iff_ (replaceFreeAux v t binders a✝) = a✝¹.iff_ a✝ case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ forall_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) = forall_ a✝¹ a✝ case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders a✝ = a✝ binders : Finset VarName h1 : v ∈ binders ⊢ exists_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) = exists_ a✝¹ a✝ case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => simp simp only [List.map_eq_self_iff] simp intro x _ a2 a3 subst a2 contradiction	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) = def_ X xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	case eq_ x y => simp constructor case left \| right => intro a1 a2 subst a1 contradiction	v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) = eq_ x y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	case not_ phi phi_ih => tauto	v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders phi).not_ = phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	case imp_ phi psi phi_ih psi_ih \| and_ phi psi phi_ih psi_ih \| or_ phi psi phi_ih psi_ih \| iff_ phi psi phi_ih psi_ih => simp tauto	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) = phi.iff_ psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp apply phi_ih simp left exact h1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp only [replaceFreeAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = def_ a✝¹ a✝	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) = def_ X xs	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs = xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp only [List.map_eq_self_iff]	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs = xs	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ ∀ x ∈ xs, (if v = x ∧ x ∉ binders then t else x) = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ ∀ x ∈ xs, (if v = x ∧ x ∉ binders then t else x) = x	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ ∀ x ∈ xs, v = x → x ∉ binders → t = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	intro x _ a2 a3	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ ∀ x ∈ xs, v = x → x ∉ binders → t = x	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders x : VarName a✝ : x ∈ xs a2 : v = x a3 : x ∉ binders ⊢ t = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	subst a2	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders x : VarName a✝ : x ∈ xs a2 : v = x a3 : x ∉ binders ⊢ t = x	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders a✝ : v ∈ xs a3 : v ∉ binders ⊢ t = v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	contradiction	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ binders a✝ : v ∈ xs a3 : v ∉ binders ⊢ t = v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp	v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) = eq_ x y	v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ (v = x → x ∉ binders → t = x) ∧ (v = y → y ∉ binders → t = y)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	constructor	v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ (v = x → x ∉ binders → t = x) ∧ (v = y → y ∉ binders → t = y)	case left v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ v = x → x ∉ binders → t = x case right v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ v = y → y ∉ binders → t = y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	case left \| right => intro a1 a2 subst a1 contradiction	v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ v = y → y ∉ binders → t = y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	intro a1 a2	v t x y : VarName binders : Finset VarName h1 : v ∈ binders ⊢ v = y → y ∉ binders → t = y	v t x y : VarName binders : Finset VarName h1 : v ∈ binders a1 : v = y a2 : y ∉ binders ⊢ t = y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	subst a1	v t x y : VarName binders : Finset VarName h1 : v ∈ binders a1 : v = y a2 : y ∉ binders ⊢ t = y	v t x : VarName binders : Finset VarName h1 : v ∈ binders a2 : v ∉ binders ⊢ t = v
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	contradiction	v t x : VarName binders : Finset VarName h1 : v ∈ binders a2 : v ∉ binders ⊢ t = v	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	tauto	v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders phi).not_ = phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi binders : Finset VarName h1 : v ∈ binders ⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) = phi.iff_ psi	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders phi = phi ∧ replaceFreeAux v t binders psi = psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	tauto	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi psi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders psi = psi binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t binders phi = phi ∧ replaceFreeAux v t binders psi = psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	apply phi_ih	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi	case h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	simp	case h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∪ {x}	case h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∨ v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	left	case h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∨ v = x	case h1.h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_mem_binders	[122, 1]	[160, 13]	exact h1	case h1.h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → replaceFreeAux v t binders phi = phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	induction F generalizing binders	F : Formula v t : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders F = fastReplaceFree v t F	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = fastReplaceFree v t (pred_const_ a✝¹ a✝) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = fastReplaceFree v t (pred_var_ a✝¹ a✝) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = fastReplaceFree v t (eq_ a✝¹ a✝) case true_ v t : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders true_ = fastReplaceFree v t true_ case false_ v t : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders false_ = fastReplaceFree v t false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders a✝.not_ = fastReplaceFree v t a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = fastReplaceFree v t (a✝¹.imp_ a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = fastReplaceFree v t (a✝¹.and_ a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = fastReplaceFree v t (a✝¹.or_ a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = fastReplaceFree v t (a✝¹.iff_ a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = fastReplaceFree v t (forall_ a✝¹ a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = fastReplaceFree v t (exists_ a✝¹ a✝) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = fastReplaceFree v t (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	any_goals simp only [replaceFreeAux] simp only [fastReplaceFree]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (pred_const_ a✝¹ a✝) = fastReplaceFree v t (pred_const_ a✝¹ a✝) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (pred_var_ a✝¹ a✝) = fastReplaceFree v t (pred_var_ a✝¹ a✝) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (eq_ a✝¹ a✝) = fastReplaceFree v t (eq_ a✝¹ a✝) case true_ v t : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders true_ = fastReplaceFree v t true_ case false_ v t : VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders false_ = fastReplaceFree v t false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders a✝.not_ = fastReplaceFree v t a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.imp_ a✝) = fastReplaceFree v t (a✝¹.imp_ a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.and_ a✝) = fastReplaceFree v t (a✝¹.and_ a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.or_ a✝) = fastReplaceFree v t (a✝¹.or_ a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (a✝¹.iff_ a✝) = fastReplaceFree v t (a✝¹.iff_ a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (forall_ a✝¹ a✝) = fastReplaceFree v t (forall_ a✝¹ a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (exists_ a✝¹ a✝) = fastReplaceFree v t (exists_ a✝¹ a✝) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = fastReplaceFree v t (def_ a✝¹ a✝)	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ pred_const_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ pred_var_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ eq_ (if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) (if v = a✝ ∧ a✝ ∉ binders then t else a✝) = eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders a✝).not_ = (fastReplaceFree v t a✝).not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders a✝¹).imp_ (replaceFreeAux v t binders a✝) = (fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders a✝¹).and_ (replaceFreeAux v t binders a✝) = (fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders a✝¹).or_ (replaceFreeAux v t binders a✝) = (fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝¹ = fastReplaceFree v t a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders a✝¹).iff_ (replaceFreeAux v t binders a✝) = (fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ forall_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) = if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders a✝ = fastReplaceFree v t a✝ binders : Finset VarName h1 : v ∉ binders ⊢ exists_ a✝¹ (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) = if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => congr! case _ x => constructor case mp => tauto case mpr => intro a1 subst a1 tauto	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) = def_ X (List.map (fun x => if v = x then t else x) xs)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case eq_ x y => congr! case _ \| _ => constructor case mp => tauto case mpr => intro a1 subst a1 tauto	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) = eq_ (if v = x then t else x) (if v = y then t else y)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case not_ phi phi_ih => tauto	v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders phi).not_ = (fastReplaceFree v t phi).not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case imp_ phi psi phi_ih psi_ih \| and_ phi psi phi_ih psi_ih \| or_ phi psi phi_ih psi_ih \| iff_ phi psi phi_ih psi_ih => simp tauto	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) = (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => split_ifs case pos c1 => congr! 1 apply replaceFreeAux_mem_binders simp right exact c1 case neg c1 => congr! 1 apply phi_ih simp tauto	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	simp only [replaceFreeAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders (def_ a✝¹ a✝) = fastReplaceFree v t (def_ a✝¹ a✝)	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = fastReplaceFree v t (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	simp only [fastReplaceFree]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = fastReplaceFree v t (def_ a✝¹ a✝)	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝) = def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	congr!	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ def_ X (List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs) = def_ X (List.map (fun x => if v = x then t else x) xs)	case h.e'_2.h.e'_3.h.h₁.a v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x✝ : VarName ⊢ v = x✝ ∧ x✝ ∉ binders ↔ v = x✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case _ x => constructor case mp => tauto case mpr => intro a1 subst a1 tauto	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x ∧ x ∉ binders ↔ v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	constructor	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x ∧ x ∉ binders ↔ v = x	case mp v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x ∧ x ∉ binders → v = x case mpr v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x → v = x ∧ x ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case mp => tauto	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x ∧ x ∉ binders → v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case mpr => intro a1 subst a1 tauto	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x → v = x ∧ x ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x ∧ x ∉ binders → v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	intro a1	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName ⊢ v = x → v = x ∧ x ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName a1 : v = x ⊢ v = x ∧ x ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	subst a1	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders x : VarName a1 : v = x ⊢ v = x ∧ x ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = v ∧ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = v ∧ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	congr!	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ eq_ (if v = x ∧ x ∉ binders then t else x) (if v = y ∧ y ∉ binders then t else y) = eq_ (if v = x then t else x) (if v = y then t else y)	case h.e'_1.h₁.a v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = x ∧ x ∉ binders ↔ v = x case h.e'_2.h₁.a v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y ∧ y ∉ binders ↔ v = y
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case _ \| _ => constructor case mp => tauto case mpr => intro a1 subst a1 tauto	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y ∧ y ∉ binders ↔ v = y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	constructor	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y ∧ y ∉ binders ↔ v = y	case mp v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y ∧ y ∉ binders → v = y case mpr v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y → v = y ∧ y ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case mp => tauto	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y ∧ y ∉ binders → v = y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case mpr => intro a1 subst a1 tauto	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y → v = y ∧ y ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y ∧ y ∉ binders → v = y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	intro a1	v t x y : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = y → v = y ∧ y ∉ binders	v t x y : VarName binders : Finset VarName h1 : v ∉ binders a1 : v = y ⊢ v = y ∧ y ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	subst a1	v t x y : VarName binders : Finset VarName h1 : v ∉ binders a1 : v = y ⊢ v = y ∧ y ∉ binders	v t x : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = v ∧ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	v t x : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = v ∧ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders phi).not_ = (fastReplaceFree v t phi).not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	simp	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi binders : Finset VarName h1 : v ∉ binders ⊢ (replaceFreeAux v t binders phi).iff_ (replaceFreeAux v t binders psi) = (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi)	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders phi = fastReplaceFree v t phi ∧ replaceFreeAux v t binders psi = fastReplaceFree v t psi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	v t : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi psi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders psi = fastReplaceFree v t psi binders : Finset VarName h1 : v ∉ binders ⊢ replaceFreeAux v t binders phi = fastReplaceFree v t phi ∧ replaceFreeAux v t binders psi = fastReplaceFree v t psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	split_ifs	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)	case pos v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders h✝ : v = x ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi case neg v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders h✝ : ¬v = x ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case pos c1 => congr! 1 apply replaceFreeAux_mem_binders simp right exact c1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	case neg c1 => congr! 1 apply phi_ih simp tauto	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	congr! 1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x phi	case h.e'_2 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	apply replaceFreeAux_mem_binders	case h.e'_2 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ replaceFreeAux v t (binders ∪ {x}) phi = phi	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v ∈ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	simp	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v ∈ binders ∪ {x}	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v ∈ binders ∨ v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	right	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v ∈ binders ∨ v = x	case h.e'_2.h1.h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	exact c1	case h.e'_2.h1.h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	congr! 1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ exists_ x (replaceFreeAux v t (binders ∪ {x}) phi) = exists_ x (fastReplaceFree v t phi)	case h.e'_2 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ replaceFreeAux v t (binders ∪ {x}) phi = fastReplaceFree v t phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	apply phi_ih	case h.e'_2 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ replaceFreeAux v t (binders ∪ {x}) phi = fastReplaceFree v t phi	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	simp	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∪ {x}	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∧ ¬v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_eq_fastReplaceFree	[163, 1]	[216, 12]	tauto	case h.e'_2.h1 v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → replaceFreeAux v t binders phi = fastReplaceFree v t phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∧ ¬v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_eq_fastReplaceFree	[219, 1]	[226, 7]	simp only [replaceFree]	F : Formula v t : VarName ⊢ replaceFree v t F = fastReplaceFree v t F	F : Formula v t : VarName ⊢ replaceFreeAux v t ∅ F = fastReplaceFree v t F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_eq_fastReplaceFree	[219, 1]	[226, 7]	apply replaceFreeAux_eq_fastReplaceFree	F : Formula v t : VarName ⊢ replaceFreeAux v t ∅ F = fastReplaceFree v t F	case h1 F : Formula v t : VarName ⊢ v ∉ ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_eq_fastReplaceFree	[219, 1]	[226, 7]	simp	case h1 F : Formula v t : VarName ⊢ v ∉ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	induction F	F : Formula v : VarName ⊢ fastReplaceFree v v F = F	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ fastReplaceFree v v (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝ case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ fastReplaceFree v v (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝ case eq_ v a✝¹ a✝ : VarName ⊢ fastReplaceFree v v (eq_ a✝¹ a✝) = eq_ a✝¹ a✝ case true_ v : VarName ⊢ fastReplaceFree v v true_ = true_ case false_ v : VarName ⊢ fastReplaceFree v v false_ = false_ case not_ v : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v a✝.not_ = a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.imp_ a✝) = a✝¹.imp_ a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.and_ a✝) = a✝¹.and_ a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.or_ a✝) = a✝¹.or_ a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.iff_ a✝) = a✝¹.iff_ a✝ case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (exists_ a✝¹ a✝) = exists_ a✝¹ a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	any_goals simp only [fastReplaceFree]	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ fastReplaceFree v v (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝ case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ fastReplaceFree v v (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝ case eq_ v a✝¹ a✝ : VarName ⊢ fastReplaceFree v v (eq_ a✝¹ a✝) = eq_ a✝¹ a✝ case true_ v : VarName ⊢ fastReplaceFree v v true_ = true_ case false_ v : VarName ⊢ fastReplaceFree v v false_ = false_ case not_ v : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v a✝.not_ = a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.imp_ a✝) = a✝¹.imp_ a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.and_ a✝) = a✝¹.and_ a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.or_ a✝) = a✝¹.or_ a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (a✝¹.iff_ a✝) = a✝¹.iff_ a✝ case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ fastReplaceFree v v (exists_ a✝¹ a✝) = exists_ a✝¹ a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ pred_const_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = pred_const_ a✝¹ a✝ case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName ⊢ pred_var_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = pred_var_ a✝¹ a✝ case eq_ v a✝¹ a✝ : VarName ⊢ eq_ (if v = a✝¹ then v else a✝¹) (if v = a✝ then v else a✝) = eq_ a✝¹ a✝ case not_ v : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (fastReplaceFree v v a✝).not_ = a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (fastReplaceFree v v a✝¹).imp_ (fastReplaceFree v v a✝) = a✝¹.imp_ a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (fastReplaceFree v v a✝¹).and_ (fastReplaceFree v v a✝) = a✝¹.and_ a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (fastReplaceFree v v a✝¹).or_ (fastReplaceFree v v a✝) = a✝¹.or_ a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : fastReplaceFree v v a✝¹ = a✝¹ a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (fastReplaceFree v v a✝¹).iff_ (fastReplaceFree v v a✝) = a✝¹.iff_ a✝ case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v v a✝)) = forall_ a✝¹ a✝ case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : fastReplaceFree v v a✝ = a✝ ⊢ (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v v a✝)) = exists_ a✝¹ a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ def_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => simp simp only [List.map_eq_self_iff] simp	v : VarName X : DefName xs : List VarName ⊢ def_ X (List.map (fun x => if v = x then v else x) xs) = def_ X xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	case eq_ x y => simp	v x y : VarName ⊢ eq_ (if v = x then v else x) (if v = y then v else y) = eq_ x y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	case not_ phi phi_ih => congr!	v : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi ⊢ (fastReplaceFree v v phi).not_ = phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	case imp_ phi psi phi_ih psi_ih \| and_ phi psi phi_ih psi_ih \| or_ phi psi phi_ih psi_ih \| iff_ phi psi phi_ih psi_ih => congr!	v : VarName phi psi : Formula phi_ih : fastReplaceFree v v phi = phi psi_ih : fastReplaceFree v v psi = psi ⊢ (fastReplaceFree v v phi).iff_ (fastReplaceFree v v psi) = phi.iff_ psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp intro _ exact phi_ih	v x : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi ⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v v phi)) = exists_ x phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	simp only [fastReplaceFree]	case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ fastReplaceFree v v (def_ a✝¹ a✝) = def_ a✝¹ a✝	case def_ v : VarName a✝¹ : DefName a✝ : List VarName ⊢ def_ a✝¹ (List.map (fun x => if v = x then v else x) a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	simp	v : VarName X : DefName xs : List VarName ⊢ def_ X (List.map (fun x => if v = x then v else x) xs) = def_ X xs	v : VarName X : DefName xs : List VarName ⊢ List.map (fun x => if v = x then v else x) xs = xs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	simp only [List.map_eq_self_iff]	v : VarName X : DefName xs : List VarName ⊢ List.map (fun x => if v = x then v else x) xs = xs	v : VarName X : DefName xs : List VarName ⊢ ∀ x ∈ xs, (if v = x then v else x) = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	simp	v : VarName X : DefName xs : List VarName ⊢ ∀ x ∈ xs, (if v = x then v else x) = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	simp	v x y : VarName ⊢ eq_ (if v = x then v else x) (if v = y then v else y) = eq_ x y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	congr!	v : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi ⊢ (fastReplaceFree v v phi).not_ = phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	congr!	v : VarName phi psi : Formula phi_ih : fastReplaceFree v v phi = phi psi_ih : fastReplaceFree v v psi = psi ⊢ (fastReplaceFree v v phi).iff_ (fastReplaceFree v v psi) = phi.iff_ psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	simp	v x : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi ⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v v phi)) = exists_ x phi	v x : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi ⊢ ¬v = x → fastReplaceFree v v phi = phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	intro _	v x : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi ⊢ ¬v = x → fastReplaceFree v v phi = phi	v x : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi a✝ : ¬v = x ⊢ fastReplaceFree v v phi = phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_self	[230, 1]	[255, 17]	exact phi_ih	v x : VarName phi : Formula phi_ih : fastReplaceFree v v phi = phi a✝ : ¬v = x ⊢ fastReplaceFree v v phi = phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	induction F	F : Formula v t : VarName h1 : ¬isFreeIn v F ⊢ fastReplaceFree v t F = F	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝) ⊢ fastReplaceFree v t (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝ case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝) ⊢ fastReplaceFree v t (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝ case eq_ v t a✝¹ a✝ : VarName h1 : ¬isFreeIn v (eq_ a✝¹ a✝) ⊢ fastReplaceFree v t (eq_ a✝¹ a✝) = eq_ a✝¹ a✝ case true_ v t : VarName h1 : ¬isFreeIn v true_ ⊢ fastReplaceFree v t true_ = true_ case false_ v t : VarName h1 : ¬isFreeIn v false_ ⊢ fastReplaceFree v t false_ = false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v a✝.not_ ⊢ fastReplaceFree v t a✝.not_ = a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.imp_ a✝) ⊢ fastReplaceFree v t (a✝¹.imp_ a✝) = a✝¹.imp_ a✝ case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.and_ a✝) ⊢ fastReplaceFree v t (a✝¹.and_ a✝) = a✝¹.and_ a✝ case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.or_ a✝) ⊢ fastReplaceFree v t (a✝¹.or_ a✝) = a✝¹.or_ a✝ case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.iff_ a✝) ⊢ fastReplaceFree v t (a✝¹.iff_ a✝) = a✝¹.iff_ a✝ case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (forall_ a✝¹ a✝) ⊢ fastReplaceFree v t (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (exists_ a✝¹ a✝) ⊢ fastReplaceFree v t (exists_ a✝¹ a✝) = exists_ a✝¹ a✝ case def_ v t : VarName a✝¹ : DefName a✝ : List VarName h1 : ¬isFreeIn v (def_ a✝¹ a✝) ⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	any_goals simp only [isFreeIn] at h1 simp only [fastReplaceFree]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝) ⊢ fastReplaceFree v t (pred_const_ a✝¹ a✝) = pred_const_ a✝¹ a✝ case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝) ⊢ fastReplaceFree v t (pred_var_ a✝¹ a✝) = pred_var_ a✝¹ a✝ case eq_ v t a✝¹ a✝ : VarName h1 : ¬isFreeIn v (eq_ a✝¹ a✝) ⊢ fastReplaceFree v t (eq_ a✝¹ a✝) = eq_ a✝¹ a✝ case true_ v t : VarName h1 : ¬isFreeIn v true_ ⊢ fastReplaceFree v t true_ = true_ case false_ v t : VarName h1 : ¬isFreeIn v false_ ⊢ fastReplaceFree v t false_ = false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v a✝.not_ ⊢ fastReplaceFree v t a✝.not_ = a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.imp_ a✝) ⊢ fastReplaceFree v t (a✝¹.imp_ a✝) = a✝¹.imp_ a✝ case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.and_ a✝) ⊢ fastReplaceFree v t (a✝¹.and_ a✝) = a✝¹.and_ a✝ case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.or_ a✝) ⊢ fastReplaceFree v t (a✝¹.or_ a✝) = a✝¹.or_ a✝ case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (a✝¹.iff_ a✝) ⊢ fastReplaceFree v t (a✝¹.iff_ a✝) = a✝¹.iff_ a✝ case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (forall_ a✝¹ a✝) ⊢ fastReplaceFree v t (forall_ a✝¹ a✝) = forall_ a✝¹ a✝ case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v (exists_ a✝¹ a✝) ⊢ fastReplaceFree v t (exists_ a✝¹ a✝) = exists_ a✝¹ a✝ case def_ v t : VarName a✝¹ : DefName a✝ : List VarName h1 : ¬isFreeIn v (def_ a✝¹ a✝) ⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName h1 : v ∉ a✝ ⊢ pred_const_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = pred_const_ a✝¹ a✝ case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName h1 : v ∉ a✝ ⊢ pred_var_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = pred_var_ a✝¹ a✝ case eq_ v t a✝¹ a✝ : VarName h1 : ¬(v = a✝¹ ∨ v = a✝) ⊢ eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝) = eq_ a✝¹ a✝ case not_ v t : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬isFreeIn v a✝ ⊢ (fastReplaceFree v t a✝).not_ = a✝.not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ (fastReplaceFree v t a✝¹).imp_ (fastReplaceFree v t a✝) = a✝¹.imp_ a✝ case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ (fastReplaceFree v t a✝¹).and_ (fastReplaceFree v t a✝) = a✝¹.and_ a✝ case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ (fastReplaceFree v t a✝¹).or_ (fastReplaceFree v t a✝) = a✝¹.or_ a✝ case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ¬isFreeIn v a✝¹ → fastReplaceFree v t a✝¹ = a✝¹ a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ (fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝) = a✝¹.iff_ a✝ case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (fastReplaceFree v t a✝)) = forall_ a✝¹ a✝ case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ¬isFreeIn v a✝ → fastReplaceFree v t a✝ = a✝ h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (fastReplaceFree v t a✝)) = exists_ a✝¹ a✝ case def_ v t : VarName a✝¹ : DefName a✝ : List VarName h1 : v ∉ a✝ ⊢ def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝) = def_ a✝¹ a✝
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => simp simp only [List.map_eq_self_iff] simp intro x a1 a2 subst a2 contradiction	v t : VarName X : DefName xs : List VarName h1 : v ∉ xs ⊢ def_ X (List.map (fun x => if v = x then t else x) xs) = def_ X xs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	case eq_ x y => simp tauto	v t x y : VarName h1 : ¬(v = x ∨ v = y) ⊢ eq_ (if v = x then t else x) (if v = y then t else y) = eq_ x y	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	case not_ phi phi_ih => tauto	v t : VarName phi : Formula phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi h1 : ¬isFreeIn v phi ⊢ (fastReplaceFree v t phi).not_ = phi.not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	case imp_ phi psi phi_ih psi_ih \| and_ phi psi phi_ih psi_ih \| or_ phi psi phi_ih psi_ih \| iff_ phi psi phi_ih psi_ih => simp tauto	v t : VarName phi psi : Formula phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi psi_ih : ¬isFreeIn v psi → fastReplaceFree v t psi = psi h1 : ¬(isFreeIn v phi ∨ isFreeIn v psi) ⊢ (fastReplaceFree v t phi).iff_ (fastReplaceFree v t psi) = phi.iff_ psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp tauto	v t x : VarName phi : Formula phi_ih : ¬isFreeIn v phi → fastReplaceFree v t phi = phi h1 : ¬(¬v = x ∧ isFreeIn v phi) ⊢ (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi)) = exists_ x phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/ReplaceFree.lean	FOL.NV.Sub.Var.One.Rec.not_free_in_fastReplaceFree_self	[258, 1]	[290, 10]	simp only [isFreeIn] at h1	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName h1 : ¬isFreeIn v (def_ a✝¹ a✝) ⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName h1 : v ∉ a✝ ⊢ fastReplaceFree v t (def_ a✝¹ a✝) = def_ a✝¹ a✝

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