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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/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits	[388, 1]	[396, 7]	apply not_isBoundIn_imp_fastAdmitsAux F v u ∅ h1	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ fastAdmitsAux v u ∅ F	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ u ∉ ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmits	[388, 1]	[396, 7]	simp	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ u ∉ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	induction F generalizing binders	F : Formula v t : VarName binders : Finset VarName h1 : ¬occursIn t F h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t F)	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_const_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝)) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_var_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_var_ a✝¹ a✝)) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬occursIn t (eq_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (eq_ a✝¹ a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬occursIn t true_ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t true_) case false_ v t : VarName binders : Finset VarName h1 : ¬occursIn t false_ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t false_) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t a✝.not_ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝.not_) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.imp_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.imp_ a✝)) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.and_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.and_ a✝)) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.or_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.or_ a✝)) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.iff_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.iff_ a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (forall_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (forall_ a✝¹ a✝)) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (exists_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (exists_ a✝¹ a✝)) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	all_goals simp only [occursIn] at h1 simp only [fastReplaceFree]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_const_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_const_ a✝¹ a✝)) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_var_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (pred_var_ a✝¹ a✝)) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬occursIn t (eq_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (eq_ a✝¹ a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬occursIn t true_ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t true_) case false_ v t : VarName binders : Finset VarName h1 : ¬occursIn t false_ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t false_) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t a✝.not_ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝.not_) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.imp_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.imp_ a✝)) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.and_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.and_ a✝)) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.or_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.or_ a✝)) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.iff_ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (a✝¹.iff_ a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (forall_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (forall_ a✝¹ a✝)) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t (exists_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (exists_ a✝¹ a✝)) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : ¬(t = a✝¹ ∨ t = a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬False h2 : v ∉ binders ⊢ fastAdmitsAux t v binders true_ case false_ v t : VarName binders : Finset VarName h1 : ¬False h2 : v ∉ binders ⊢ fastAdmitsAux t v binders false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝).not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((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), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((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), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((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), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	any_goals simp only [fastAdmitsAux]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : ¬(t = a✝¹ ∨ t = a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (eq_ (if v = a✝¹ then t else a✝¹) (if v = a✝ then t else a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬False h2 : v ∉ binders ⊢ fastAdmitsAux t v binders true_ case false_ v t : VarName binders : Finset VarName h1 : ¬False h2 : v ∉ binders ⊢ fastAdmitsAux t v binders false_ case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝).not_ case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((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), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((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), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((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), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders ((fastReplaceFree v t a✝¹).iff_ (fastReplaceFree v t a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(t = a✝¹ ∨ t = a✝) h2 : v ∉ binders ⊢ ((t = if v = a✝¹ then t else a✝¹) ∨ t = if v = a✝ then t else a✝) → v ∉ binders case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => push_neg at h1 cases h1 case intro h1_left h1_right => split_ifs case pos c1 => simp only [fastAdmitsAux] subst c1 right apply not_isFreeIn_imp_fastAdmitsAux intro contra apply h1_right apply isFreeIn_imp_occursIn exact contra case neg c1 => simp only [fastAdmitsAux] right apply phi_ih (binders ∪ {x}) h1_right simp tauto	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h1 : ¬(t = x ∨ occursIn t phi) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	all_goals tauto	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(t = a✝¹ ∨ t = a✝) h2 : v ∉ binders ⊢ ((t = if v = a✝¹ then t else a✝¹) ∨ t = if v = a✝ then t else a✝) → v ∉ binders case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬occursIn t a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t a✝¹) ∧ fastAdmitsAux t v binders (fastReplaceFree v t a✝) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	simp only [occursIn] at h1	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	simp only [fastReplaceFree]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (fastReplaceFree v t (def_ a✝¹ a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (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/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	simp only [fastAdmitsAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x then t else x) a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	push_neg at h1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h1 : ¬(t = x ∨ occursIn t phi) h2 : v ∉ binders ⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1 : t ≠ x ∧ ¬occursIn t phi ⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	cases h1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1 : t ≠ x ∧ ¬occursIn t phi ⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))	case intro v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders left✝ : t ≠ x right✝ : ¬occursIn t phi ⊢ fastAdmitsAux t v binders (if v = x then exists_ x phi else exists_ x (fastReplaceFree v t phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	case intro h1_left h1_right => split_ifs case pos c1 => simp only [fastAdmitsAux] subst c1 right apply not_isFreeIn_imp_fastAdmitsAux intro contra apply h1_right apply isFreeIn_imp_occursIn exact contra case neg c1 => simp only [fastAdmitsAux] right apply phi_ih (binders ∪ {x}) h1_right simp tauto	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi ⊢ fastAdmitsAux t v binders (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/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	split_ifs	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi ⊢ fastAdmitsAux t v binders (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), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi h✝ : v = x ⊢ fastAdmitsAux t v binders (exists_ x phi) case neg v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi h✝ : ¬v = x ⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	case pos c1 => simp only [fastAdmitsAux] subst c1 right apply not_isFreeIn_imp_fastAdmitsAux intro contra apply h1_right apply isFreeIn_imp_occursIn exact contra	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : v = x ⊢ fastAdmitsAux t v binders (exists_ x phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	case neg c1 => simp only [fastAdmitsAux] right apply phi_ih (binders ∪ {x}) h1_right simp tauto	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	simp only [fastAdmitsAux]	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : v = x ⊢ fastAdmitsAux t v binders (exists_ x phi)	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : v = x ⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	subst c1	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : v = x ⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) phi	v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v ⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	right	v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v ⊢ t = v ∨ fastAdmitsAux t v (binders ∪ {v}) phi	case h v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v ⊢ fastAdmitsAux t v (binders ∪ {v}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	apply not_isFreeIn_imp_fastAdmitsAux	case h v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v ⊢ fastAdmitsAux t v (binders ∪ {v}) phi	case h.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v ⊢ ¬isFreeIn t phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	intro contra	case h.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v ⊢ ¬isFreeIn t phi	case h.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v contra : isFreeIn t phi ⊢ False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	apply h1_right	case h.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v contra : isFreeIn t phi ⊢ False	case h.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v contra : isFreeIn t phi ⊢ occursIn t phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	apply isFreeIn_imp_occursIn	case h.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v contra : isFreeIn t phi ⊢ occursIn t phi	case h.h1.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v contra : isFreeIn t phi ⊢ isFreeIn t phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	exact contra	case h.h1.h1 v t : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_right : ¬occursIn t phi h1_left : t ≠ v contra : isFreeIn t phi ⊢ isFreeIn t phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	simp only [fastAdmitsAux]	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ fastAdmitsAux t v binders (exists_ x (fastReplaceFree v t phi))	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	right	v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ t = x ∨ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)	case h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	apply phi_ih (binders ∪ {x}) h1_right	case h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ fastAdmitsAux t v (binders ∪ {x}) (fastReplaceFree v t phi)	case h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ v ∉ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	simp	case h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ v ∉ binders ∪ {x}	case h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ v ∉ binders ∧ ¬v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	tauto	case h v t x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬occursIn t phi → v ∉ binders → fastAdmitsAux t v binders (fastReplaceFree v t phi) binders : Finset VarName h2 : v ∉ binders h1_left : t ≠ x h1_right : ¬occursIn t phi c1 : ¬v = x ⊢ v ∉ binders ∧ ¬v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_aux_fastAdmitsAux	[400, 1]	[437, 10]	tauto	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ h2 : v ∉ binders ⊢ t ∈ List.map (fun x => if v = x then t else x) a✝ → v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits	[440, 1]	[448, 7]	simp only [fastAdmits]	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmits t v (fastReplaceFree v t F)	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits	[440, 1]	[448, 7]	apply fastReplaceFree_aux_fastAdmitsAux F v t ∅ h1	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmitsAux t v ∅ (fastReplaceFree v t F)	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ v ∉ ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastReplaceFree_fastAdmits	[440, 1]	[448, 7]	simp	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ v ∉ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	induction F generalizing binders	F : Formula v t : VarName binders : Finset VarName h1 : ¬occursIn t F ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders F)	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝)) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝)) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬occursIn t (eq_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬occursIn t true_ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders true_) case false_ v t : VarName binders : Finset VarName h1 : ¬occursIn t false_ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders false_) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝.not_ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝.not_) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.imp_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝)) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.and_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝)) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.or_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝)) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.iff_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (forall_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝)) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (exists_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝)) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	all_goals simp only [occursIn] at h1 simp only [replaceFreeAux] simp only [fastAdmitsAux]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝)) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝)) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬occursIn t (eq_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬occursIn t true_ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders true_) case false_ v t : VarName binders : Finset VarName h1 : ¬occursIn t false_ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders false_) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝.not_ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝.not_) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.imp_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝)) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.and_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝)) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.or_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝)) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.iff_ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (forall_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝)) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (exists_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝)) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(t = a✝¹ ∨ t = a✝) ⊢ ((t = if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) ∨ t = if v = a✝ ∧ a✝ ∉ binders then t else a✝) → v ∉ binders case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case eq_ x y => push_neg at h1 intros a1 split_ifs at a1 case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case _ c1 c2 => cases c2 case intro c2_left c2_right => subst c2_left exact c2_right case _ c1 c2 => tauto	v t x y : VarName binders : Finset VarName h1 : ¬(t = x ∨ t = y) ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	all_goals tauto	case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	simp only [occursIn] at h1	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	simp only [replaceFreeAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ fastAdmitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	simp only [fastAdmitsAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ fastAdmitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	simp	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs → v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	intro x a1 a2	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	by_cases c1 : v = x ∧ x ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t ⊢ v ∉ binders	case pos v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1 : v = x ∧ x ∉ binders ⊢ v ∉ binders case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1 : ¬(v = x ∧ x ∉ binders) ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	cases c1	case pos v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1 : v = x ∧ x ∉ binders ⊢ v ∉ binders	case pos.intro v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t left✝ : v = x right✝ : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case _ c1_left c1_right => subst c1_left exact c1_right	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	subst c1_left	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs a1 : v ∈ xs a2 : (v = v → v ∈ binders) → v = t c1_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	exact c1_right	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs a1 : v ∈ xs a2 : (v = v → v ∈ binders) → v = t c1_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	push_neg at c1	case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1 : ¬(v = x ∧ x ∉ binders) ⊢ v ∉ binders	case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1 : v = x → x ∈ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	specialize a2 c1	case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t c1 : v = x → x ∈ binders ⊢ v ∉ binders	case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs c1 : v = x → x ∈ binders a2 : x = t ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	subst a2	case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs c1 : v = x → x ∈ binders a2 : x = t ⊢ v ∉ binders	case neg v : VarName X : DefName xs : List VarName binders : Finset VarName x : VarName a1 : x ∈ xs c1 : v = x → x ∈ binders h1 : x ∉ xs ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	contradiction	case neg v : VarName X : DefName xs : List VarName binders : Finset VarName x : VarName a1 : x ∈ xs c1 : v = x → x ∈ binders h1 : x ∉ xs ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	push_neg at h1	v t x y : VarName binders : Finset VarName h1 : ¬(t = x ∨ t = y) ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	intros a1	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) → v ∉ binders	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	split_ifs at a1	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y a1 : (t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y ⊢ v ∉ binders	case pos v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y h✝¹ : v = x ∧ x ∉ binders h✝ : v = y ∧ y ∉ binders a1 : t = t ∨ t = t ⊢ v ∉ binders case neg v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y h✝¹ : v = x ∧ x ∉ binders h✝ : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y ⊢ v ∉ binders case pos v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y h✝¹ : ¬(v = x ∧ x ∉ binders) h✝ : v = y ∧ y ∉ binders a1 : t = x ∨ t = t ⊢ v ∉ binders case neg v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y h✝¹ : ¬(v = x ∧ x ∉ binders) h✝ : ¬(v = y ∧ y ∉ binders) a1 : t = x ∨ t = y ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : v = x ∧ x ∉ binders c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : v = x ∧ x ∉ binders c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case _ c1 c2 => cases c2 case intro c2_left c2_right => subst c2_left exact c2_right	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : v = y ∧ y ∉ binders a1 : t = x ∨ t = t ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case _ c1 c2 => tauto	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : ¬(v = y ∧ y ∉ binders) a1 : t = x ∨ t = y ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	cases c1	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : v = x ∧ x ∉ binders c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t ⊢ v ∉ binders	case intro v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t left✝ : v = x right✝ : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	subst c1_left	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	v t y : VarName binders : Finset VarName c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t h1 : t ≠ v ∧ t ≠ y c1_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	exact c1_right	v t y : VarName binders : Finset VarName c2 : v = y ∧ y ∉ binders a1 : t = t ∨ t = t h1 : t ≠ v ∧ t ≠ y c1_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	cases c1	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : v = x ∧ x ∉ binders c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y ⊢ v ∉ binders	case intro v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y left✝ : v = x right✝ : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	subst c1_left	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	v t y : VarName binders : Finset VarName c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y h1 : t ≠ v ∧ t ≠ y c1_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	exact c1_right	v t y : VarName binders : Finset VarName c2 : ¬(v = y ∧ y ∉ binders) a1 : t = t ∨ t = y h1 : t ≠ v ∧ t ≠ y c1_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	cases c2	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : v = y ∧ y ∉ binders a1 : t = x ∨ t = t ⊢ v ∉ binders	case intro v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) a1 : t = x ∨ t = t left✝ : v = y right✝ : y ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	case intro c2_left c2_right => subst c2_left exact c2_right	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) a1 : t = x ∨ t = t c2_left : v = y c2_right : y ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	subst c2_left	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) a1 : t = x ∨ t = t c2_left : v = y c2_right : y ∉ binders ⊢ v ∉ binders	v t x : VarName binders : Finset VarName c1 : ¬(v = x ∧ x ∉ binders) a1 : t = x ∨ t = t h1 : t ≠ x ∧ t ≠ v c2_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	exact c2_right	v t x : VarName binders : Finset VarName c1 : ¬(v = x ∧ x ∉ binders) a1 : t = x ∨ t = t h1 : t ≠ x ∧ t ≠ v c2_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	tauto	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : ¬(v = y ∧ y ∉ binders) a1 : t = x ∨ t = y ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_fastAdmitsAux	[452, 1]	[500, 10]	tauto	case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → fastAdmitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ t = a✝¹ ∨ fastAdmitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits	[503, 1]	[511, 48]	simp only [replaceFree]	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmits t v (replaceFree v t F)	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmits t v (replaceFreeAux v t ∅ F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits	[503, 1]	[511, 48]	simp only [fastAdmits]	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmits t v (replaceFreeAux v t ∅ F)	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_fastAdmits	[503, 1]	[511, 48]	exact replaceFreeAux_fastAdmitsAux F v t ∅ h1	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ fastAdmitsAux t v ∅ (replaceFreeAux v t ∅ F)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	induction F generalizing S	F : Formula v u : VarName S T : Finset VarName h1 : fastAdmitsAux v u S F h2 : u ∉ T ⊢ fastAdmitsAux v u (S ∪ T) F	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝) case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : fastAdmitsAux v u S (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝) case true_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : fastAdmitsAux v u S true_ ⊢ fastAdmitsAux v u (S ∪ T) true_ case false_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : fastAdmitsAux v u S false_ ⊢ fastAdmitsAux v u (S ∪ T) false_ case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝.not_ ⊢ fastAdmitsAux v u (S ∪ T) a✝.not_ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝) case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝) case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝) case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝) case forall_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝) case exists_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝) case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	all_goals simp only [fastAdmitsAux] at h1 simp only [fastAdmitsAux]	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝) case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : fastAdmitsAux v u S (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝) case true_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : fastAdmitsAux v u S true_ ⊢ fastAdmitsAux v u (S ∪ T) true_ case false_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : fastAdmitsAux v u S false_ ⊢ fastAdmitsAux v u (S ∪ T) false_ case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝.not_ ⊢ fastAdmitsAux v u (S ∪ T) a✝.not_ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝) case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝) case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝) case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝) case forall_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝) case exists_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝) case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ S ⊢ v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case forall_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝ ⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝ case exists_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝ ⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝ case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp cases h1 case inl h1 => tauto case inr h1 => specialize phi_ih (S ∪ {x}) h1 right simp only [Finset.union_right_comm_assoc] exact phi_ih	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	any_goals simp only [Finset.mem_union]	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ S ⊢ v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T) case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T) case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ S ⊢ v = a✝¹ ∨ v = a✝ → ¬(u ∈ S ∨ u ∈ T) case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	all_goals tauto	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T) case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T) case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ S ⊢ v = a✝¹ ∨ v = a✝ → ¬(u ∈ S ∨ u ∈ T) case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝¹ → fastAdmitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u S a✝ → fastAdmitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ ⊢ fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	simp only [fastAdmitsAux] at h1	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u S (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	simp only [fastAdmitsAux]	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝)	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	simp	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	cases h1	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	case inl v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h✝ : v = x ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi case inr v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h✝ : fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	case inl h1 => tauto	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : v = x ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	case inr h1 => specialize phi_ih (S ∪ {x}) h1 right simp only [Finset.union_right_comm_assoc] exact phi_ih	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	tauto	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : v = x ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	specialize phi_ih (S ∪ {x}) h1	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u S phi → fastAdmitsAux v u (S ∪ T) phi S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	right	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	case h v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi ⊢ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	simp only [Finset.union_right_comm_assoc]	case h v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi ⊢ fastAdmitsAux v u (S ∪ (T ∪ {x})) phi	case h v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi ⊢ fastAdmitsAux v u (S ∪ {x} ∪ T) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	exact phi_ih	case h v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula S : Finset VarName h1 : fastAdmitsAux v u (S ∪ {x}) phi phi_ih : fastAdmitsAux v u (S ∪ {x} ∪ T) phi ⊢ fastAdmitsAux v u (S ∪ {x} ∪ T) phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	simp only [Finset.mem_union]	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → u ∉ S ∪ T	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_add_binders	[515, 1]	[541, 10]	tauto	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ⊢ v ∈ a✝ → ¬(u ∈ S ∨ u ∈ T)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 10]	induction F generalizing S	F : Formula v u : VarName S T : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) F ⊢ fastAdmitsAux v u S F	case pred_const_ v u : VarName T : Finset VarName a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName T : Finset VarName a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (pred_var_ a✝¹ a✝) case eq_ v u : VarName T : Finset VarName a✝¹ a✝ : VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (eq_ a✝¹ a✝) case true_ v u : VarName T S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) true_ ⊢ fastAdmitsAux v u S true_ case false_ v u : VarName T S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) false_ ⊢ fastAdmitsAux v u S false_ case not_ v u : VarName T : Finset VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝.not_ ⊢ fastAdmitsAux v u S a✝.not_ case imp_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.imp_ a✝) case and_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.and_ a✝) case or_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.or_ a✝) case iff_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.iff_ a✝) case forall_ v u : VarName T : Finset VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (forall_ a✝¹ a✝) case exists_ v u : VarName T : Finset VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (exists_ a✝¹ a✝) case def_ v u : VarName T : Finset VarName a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 10]	all_goals simp only [fastAdmitsAux] at h1 simp only [fastAdmitsAux]	case pred_const_ v u : VarName T : Finset VarName a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName T : Finset VarName a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (pred_var_ a✝¹ a✝) case eq_ v u : VarName T : Finset VarName a✝¹ a✝ : VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (eq_ a✝¹ a✝) case true_ v u : VarName T S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) true_ ⊢ fastAdmitsAux v u S true_ case false_ v u : VarName T S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) false_ ⊢ fastAdmitsAux v u S false_ case not_ v u : VarName T : Finset VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝.not_ ⊢ fastAdmitsAux v u S a✝.not_ case imp_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.imp_ a✝) case and_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.and_ a✝) case or_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.or_ a✝) case iff_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u S (a✝¹.iff_ a✝) case forall_ v u : VarName T : Finset VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (forall_ a✝¹ a✝) case exists_ v u : VarName T : Finset VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (exists_ a✝¹ a✝) case def_ v u : VarName T : Finset VarName a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u S (def_ a✝¹ a✝)	case pred_const_ v u : VarName T : Finset VarName a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ∪ T ⊢ v ∈ a✝ → u ∉ S case pred_var_ v u : VarName T : Finset VarName a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ∪ T ⊢ v ∈ a✝ → u ∉ S case eq_ v u : VarName T : Finset VarName a✝¹ a✝ : VarName S : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ S ∪ T ⊢ v = a✝¹ ∨ v = a✝ → u ∉ S case not_ v u : VarName T : Finset VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝ ⊢ fastAdmitsAux v u S a✝ case imp_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ ⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ case and_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ ⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ case or_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ ⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ case iff_ v u : VarName T : Finset VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝¹ → fastAdmitsAux v u S a✝¹ a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) a✝¹ ∧ fastAdmitsAux v u (S ∪ T) a✝ ⊢ fastAdmitsAux v u S a✝¹ ∧ fastAdmitsAux v u S a✝ case forall_ v u : VarName T : Finset VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝ ⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝ case exists_ v u : VarName T : Finset VarName a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) a✝ → fastAdmitsAux v u S a✝ S : Finset VarName h1 : v = a✝¹ ∨ fastAdmitsAux v u (S ∪ T ∪ {a✝¹}) a✝ ⊢ v = a✝¹ ∨ fastAdmitsAux v u (S ∪ {a✝¹}) a✝ case def_ v u : VarName T : Finset VarName a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ → u ∉ S ∪ T ⊢ v ∈ a✝ → u ∉ S
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 10]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => simp at h1 tauto	v u : VarName T : Finset VarName X : DefName xs : List VarName S : Finset VarName h1 : v ∈ xs → u ∉ S ∪ T ⊢ v ∈ xs → u ∉ S	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 10]	case eq_ x y => simp at h1 tauto	v u : VarName T : Finset VarName x y : VarName S : Finset VarName h1 : v = x ∨ v = y → u ∉ S ∪ T ⊢ v = x ∨ v = y → u ∉ S	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 10]	case not_ phi phi_ih => tauto	v u : VarName T : Finset VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) phi ⊢ fastAdmitsAux v u S phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 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 => tauto	v u : VarName T : Finset VarName phi psi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi psi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) psi → fastAdmitsAux v u S psi S : Finset VarName h1 : fastAdmitsAux v u (S ∪ T) phi ∧ fastAdmitsAux v u (S ∪ T) psi ⊢ fastAdmitsAux v u S phi ∧ fastAdmitsAux v u S psi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_del_binders	[544, 1]	[575, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp only [Finset.union_right_comm S T {x}] at h1 tauto	v u : VarName T : Finset VarName x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), fastAdmitsAux v u (S ∪ T) phi → fastAdmitsAux v u S phi S : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (S ∪ T ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (S ∪ {x}) phi	no goals

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