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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/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	tauto	v : Var phi psi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet psi_ih : occursIn v psi ∧ v.isBound ↔ v ∈ psi.boundVarSet ⊢ (occursIn v phi ∨ occursIn v psi) ∧ v.isBound ↔ v ∈ phi.boundVarSet ∨ v ∈ psi.boundVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [Formula.boundVarSet]	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ (forall_ a✝ phi).boundVarSet	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ phi.boundVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	simp only [occursIn]	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v (forall_ a✝ phi) ∧ v.isBound ↔ v ∈ phi.boundVarSet	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet	[267, 1]	[320, 17]	exact phi_ih	v : Var a✝ : String phi : Formula phi_ih : occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet ⊢ occursIn v phi ∧ v.isBound ↔ v ∈ phi.boundVarSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet'	[323, 1]	[331, 36]	simp only [freeVarSet']	v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ F.freeVarSet'	v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ Finset.filter isFree F.varSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet'	[323, 1]	[331, 36]	simp	v : Var F : Formula ⊢ occursIn v F ∧ v.isFree ↔ v ∈ Finset.filter isFree F.varSet	v : Var F : Formula ⊢ v.isFree → (occursIn v F ↔ v ∈ F.varSet)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet'	[323, 1]	[331, 36]	intro _	v : Var F : Formula ⊢ v.isFree → (occursIn v F ↔ v ∈ F.varSet)	v : Var F : Formula a✝ : v.isFree ⊢ occursIn v F ↔ v ∈ F.varSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isFreeIn_iff_mem_freeVarSet'	[323, 1]	[331, 36]	exact occursIn_iff_mem_varSet v F	v : Var F : Formula a✝ : v.isFree ⊢ occursIn v F ↔ v ∈ F.varSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet'	[334, 1]	[342, 36]	simp only [boundVarSet']	v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ F.boundVarSet'	v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ Finset.filter isBound F.varSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet'	[334, 1]	[342, 36]	simp	v : Var F : Formula ⊢ occursIn v F ∧ v.isBound ↔ v ∈ Finset.filter isBound F.varSet	v : Var F : Formula ⊢ v.isBound → (occursIn v F ↔ v ∈ F.varSet)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet'	[334, 1]	[342, 36]	intro _	v : Var F : Formula ⊢ v.isBound → (occursIn v F ↔ v ∈ F.varSet)	v : Var F : Formula a✝ : v.isBound ⊢ occursIn v F ↔ v ∈ F.varSet
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.isBoundIn_iff_mem_boundVarSet'	[334, 1]	[342, 36]	exact occursIn_iff_mem_varSet v F	v : Var F : Formula a✝ : v.isBound ⊢ occursIn v F ↔ v ∈ F.varSet	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	cases v	v : Var ⊢ v.isFree ↔ ∃ x, v = free_ x	case free_ a✝ : String ⊢ (free_ a✝).isFree ↔ ∃ x, free_ a✝ = free_ x case bound_ a✝ : ℕ ⊢ (bound_ a✝).isFree ↔ ∃ x, bound_ a✝ = free_ x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	case free_ x => simp only [isFree] simp	x : String ⊢ (free_ x).isFree ↔ ∃ x_1, free_ x = free_ x_1	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	case bound_ i => simp only [isFree] simp	i : ℕ ⊢ (bound_ i).isFree ↔ ∃ x, bound_ i = free_ x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	simp only [isFree]	x : String ⊢ (free_ x).isFree ↔ ∃ x_1, free_ x = free_ x_1	x : String ⊢ True ↔ ∃ x_1, free_ x = free_ x_1
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	simp	x : String ⊢ True ↔ ∃ x_1, free_ x = free_ x_1	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	simp only [isFree]	i : ℕ ⊢ (bound_ i).isFree ↔ ∃ x, bound_ i = free_ x	i : ℕ ⊢ False ↔ ∃ x, False
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/LN/Binders.lean	LN.IsFreeIffExistsString	[346, 1]	[356, 9]	simp	i : ℕ ⊢ False ↔ ∃ x, False	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders F ⊢ fastAdmitsAux v u binders F	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders true_ ⊢ fastAdmitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders false_ ⊢ fastAdmitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝.not_ ⊢ fastAdmitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u 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.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	all_goals simp only [admitsAux] at h2 simp only [fastAdmitsAux]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders true_ ⊢ fastAdmitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders false_ ⊢ fastAdmitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝.not_ ⊢ fastAdmitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders h2 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders ⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {a✝¹}) a✝ ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {a✝¹}) a✝ ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	all_goals tauto	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders h2 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders ⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	simp only [admitsAux] at h2	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u binders (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ fastAdmitsAux v u 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.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	simp only [fastAdmitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	by_cases c1 : v = x	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	left	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	case pos.h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	exact c1	case pos.h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ 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.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	right	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	case neg.h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ fastAdmitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	apply phi_ih	case neg.h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ fastAdmitsAux v u (binders ∪ {x}) phi	case neg.h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ v ∉ binders ∪ {x} case neg.h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ admitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	simp	case neg.h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ v ∉ binders ∪ {x}	case neg.h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) 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.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	tauto	case neg.h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) 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.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	exact h2	case neg.h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → admitsAux v u binders phi → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders h2 : admitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ admitsAux v u (binders ∪ {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.admitsAux_imp_fastAdmitsAux	[217, 1]	[240, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders h2 : v ∈ a✝ ∧ v ∉ binders → u ∉ binders ⊢ v ∈ a✝ → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders F	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u 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.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	all_goals simp only [admitsAux]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (def_ a✝¹ a✝)	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u (binders ∪ {a✝¹}) a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => apply phi_ih simp left exact h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u (binders ∪ {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.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	all_goals tauto	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∈ binders ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	simp only [admitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	apply phi_ih	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : v ∈ binders ⊢ admitsAux v u (binders ∪ {x}) phi	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	simp	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∪ {x}	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : v ∈ binders ⊢ 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.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	left	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ binders ∨ v = x	case h1.h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : 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.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	exact h1	case h1.h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∈ binders → admitsAux v u binders phi binders : Finset VarName h1 : 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.mem_binders_imp_admitsAux	[243, 1]	[259, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders F ⊢ admitsAux v u binders F	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) ⊢ admitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) ⊢ admitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝) ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders true_ ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders false_ ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝.not_ ⊢ admitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝) ⊢ admitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝) ⊢ admitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝) ⊢ admitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝) ⊢ admitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝) ⊢ admitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝) ⊢ admitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝) ⊢ admitsAux v u 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.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	all_goals simp only [fastAdmitsAux] at h1 simp only [admitsAux]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) ⊢ admitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) ⊢ admitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (eq_ a✝¹ a✝) ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders true_ ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : fastAdmitsAux v u binders false_ ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝.not_ ⊢ admitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.imp_ a✝) ⊢ admitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.and_ a✝) ⊢ admitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.or_ a✝) ⊢ admitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (a✝¹.iff_ a✝) ⊢ admitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (forall_ a✝¹ a✝) ⊢ admitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders (exists_ a✝¹ a✝) ⊢ admitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝) ⊢ admitsAux v u binders (def_ a✝¹ a✝)	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ ⊢ admitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ ⊢ admitsAux v u (binders ∪ {a✝¹}) a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => cases h1 case inl h1 => apply mem_binders_imp_admitsAux subst h1 simp case inr h1 => apply phi_ih exact h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi ⊢ admitsAux v u (binders ∪ {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_imp_admitsAux	[262, 1]	[284, 10]	all_goals tauto	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), fastAdmitsAux v u binders a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ ⊢ admitsAux v u binders a✝¹ ∧ admitsAux v u binders a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	simp only [fastAdmitsAux] at h1	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : fastAdmitsAux v u binders (def_ a✝¹ a✝) ⊢ admitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ admitsAux v u 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.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	simp only [admitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ admitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	cases h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi ⊢ admitsAux v u (binders ∪ {x}) phi	case inl v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h✝ : v = x ⊢ admitsAux v u (binders ∪ {x}) phi case inr v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h✝ : fastAdmitsAux v u (binders ∪ {x}) phi ⊢ admitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	case inl h1 => apply mem_binders_imp_admitsAux subst h1 simp	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : v = x ⊢ admitsAux v u (binders ∪ {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_imp_admitsAux	[262, 1]	[284, 10]	case inr h1 => apply phi_ih exact h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : fastAdmitsAux v u (binders ∪ {x}) phi ⊢ admitsAux v u (binders ∪ {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_imp_admitsAux	[262, 1]	[284, 10]	apply mem_binders_imp_admitsAux	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : v = x ⊢ admitsAux v u (binders ∪ {x}) phi	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : 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.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	subst h1	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : v = x ⊢ v ∈ binders ∪ {x}	case h1 v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName ⊢ v ∈ binders ∪ {v}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	simp	case h1 v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName ⊢ v ∈ binders ∪ {v}	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	apply phi_ih	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : fastAdmitsAux v u (binders ∪ {x}) phi ⊢ admitsAux v u (binders ∪ {x}) phi	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : fastAdmitsAux v u (binders ∪ {x}) phi ⊢ fastAdmitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_imp_admitsAux	[262, 1]	[284, 10]	exact h1	case h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), fastAdmitsAux v u binders phi → admitsAux v u binders phi binders : Finset VarName h1 : fastAdmitsAux v u (binders ∪ {x}) phi ⊢ fastAdmitsAux v u (binders ∪ {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_imp_admitsAux	[262, 1]	[284, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits	[287, 1]	[297, 46]	simp only [admits]	F : Formula v u : VarName ⊢ admits v u F ↔ fastAdmits v u F	F : Formula v u : VarName ⊢ admitsAux v u ∅ F ↔ fastAdmits v u F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits	[287, 1]	[297, 46]	simp only [fastAdmits]	F : Formula v u : VarName ⊢ admitsAux v u ∅ F ↔ fastAdmits v u F	F : Formula v u : VarName ⊢ admitsAux v u ∅ F ↔ fastAdmitsAux v u ∅ F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits	[287, 1]	[297, 46]	constructor	F : Formula v u : VarName ⊢ admitsAux v u ∅ F ↔ fastAdmitsAux v u ∅ F	case mp F : Formula v u : VarName ⊢ admitsAux v u ∅ F → fastAdmitsAux v u ∅ F case mpr F : Formula v u : VarName ⊢ fastAdmitsAux v u ∅ F → admitsAux v u ∅ F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits	[287, 1]	[297, 46]	apply admitsAux_imp_fastAdmitsAux	case mp F : Formula v u : VarName ⊢ admitsAux v u ∅ F → fastAdmitsAux v u ∅ F	case mp.h1 F : Formula v u : VarName ⊢ v ∉ ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits	[287, 1]	[297, 46]	simp	case mp.h1 F : Formula v u : VarName ⊢ v ∉ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_iff_fastAdmits	[287, 1]	[297, 46]	exact fastAdmitsAux_imp_admitsAux F v u ∅	case mpr F : Formula v u : VarName ⊢ fastAdmitsAux v u ∅ F → admitsAux v u ∅ 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_self	[302, 1]	[321, 10]	induction F generalizing binders	F : Formula v : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders F	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (eq_ a✝¹ a✝) case true_ v : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders true_ case false_ v : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders false_ case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v 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.fastAdmitsAux_self	[302, 1]	[321, 10]	all_goals simp only [fastAdmitsAux]	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (eq_ a✝¹ a✝) case true_ v : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders true_ case false_ v : VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders false_ case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ a✝ → v ∉ binders case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ a✝ → v ∉ binders case eq_ v a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = a✝¹ ∨ v = a✝ → v ∉ binders case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ v = a✝¹ ∨ fastAdmitsAux v v (binders ∪ {a✝¹}) a✝ case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ v = a✝¹ ∨ fastAdmitsAux v v (binders ∪ {a✝¹}) a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ a✝ → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	all_goals tauto	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ a✝ → v ∉ binders case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ a✝ → v ∉ binders case eq_ v a✝¹ a✝ : VarName binders : Finset VarName h1 : v ∉ binders ⊢ v = a✝¹ ∨ v = a✝ → v ∉ binders case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders a✝ binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders a✝¹ ∧ fastAdmitsAux v v binders a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ 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.fastAdmitsAux_self	[302, 1]	[321, 10]	simp only [fastAdmitsAux]	case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ fastAdmitsAux v v binders (def_ a✝¹ a✝)	case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ a✝ → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	by_cases c1 : v = x	v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders ⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi	case pos v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi case neg v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v = x ∨ fastAdmitsAux v 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.fastAdmitsAux_self	[302, 1]	[321, 10]	left	case pos v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi	case pos.h v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	exact c1	case pos.h v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : v = x ⊢ v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	right	case neg v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v = x ∨ fastAdmitsAux v v (binders ∪ {x}) phi	case neg.h v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ fastAdmitsAux v 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.fastAdmitsAux_self	[302, 1]	[321, 10]	apply phi_ih	case neg.h v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ fastAdmitsAux v v (binders ∪ {x}) phi	case neg.h.h1 v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	simp	case neg.h.h1 v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∪ {x}	case neg.h.h1 v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∧ ¬v = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	tauto	case neg.h.h1 v x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → fastAdmitsAux v v binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x ⊢ v ∉ binders ∧ ¬v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmitsAux_self	[302, 1]	[321, 10]	tauto	case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ binders ⊢ v ∈ 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.fastAdmits_self	[324, 1]	[331, 7]	simp only [fastAdmits]	F : Formula v : VarName ⊢ fastAdmits v v F	F : Formula v : VarName ⊢ fastAdmitsAux v v ∅ F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmits_self	[324, 1]	[331, 7]	apply fastAdmitsAux_self	F : Formula v : VarName ⊢ fastAdmitsAux v v ∅ F	case h1 F : Formula v : VarName ⊢ v ∉ ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.fastAdmits_self	[324, 1]	[331, 7]	simp	case h1 F : Formula v : VarName ⊢ v ∉ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux	[335, 1]	[348, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : ¬isFreeIn v F ⊢ fastAdmitsAux v u binders F	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬isFreeIn v (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v true_ ⊢ fastAdmitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v false_ ⊢ fastAdmitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝.not_ ⊢ fastAdmitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u 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.not_isFreeIn_imp_fastAdmitsAux	[335, 1]	[348, 10]	all_goals simp only [isFreeIn] at h1 simp only [fastAdmitsAux]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (pred_const_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (pred_var_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬isFreeIn v (eq_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v true_ ⊢ fastAdmitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v false_ ⊢ fastAdmitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝.not_ ⊢ fastAdmitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.imp_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.and_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.or_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (a✝¹.iff_ a✝) ⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (forall_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (exists_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(v = a✝¹ ∨ v = a✝) ⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝ ⊢ fastAdmitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux	[335, 1]	[348, 10]	all_goals tauto	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(v = a✝¹ ∨ v = a✝) ⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝ ⊢ fastAdmitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v a✝) ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux	[335, 1]	[348, 10]	simp only [isFreeIn] at h1	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (def_ a✝¹ a✝) ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ fastAdmitsAux v u 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.not_isFreeIn_imp_fastAdmitsAux	[335, 1]	[348, 10]	simp only [fastAdmitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmitsAux	[335, 1]	[348, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmits	[351, 1]	[358, 50]	simp only [fastAdmits]	F : Formula v u : VarName h1 : ¬isFreeIn v F ⊢ fastAdmits v u F	F : Formula v u : VarName h1 : ¬isFreeIn v F ⊢ fastAdmitsAux v u ∅ F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_fastAdmits	[351, 1]	[358, 50]	exact not_isFreeIn_imp_fastAdmitsAux F v u ∅ h1	F : Formula v u : VarName h1 : ¬isFreeIn v F ⊢ fastAdmitsAux v u ∅ F	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : ¬isBoundIn u F h2 : u ∉ binders ⊢ fastAdmitsAux v u binders F	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬isBoundIn u (eq_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u true_ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u false_ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝.not_ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.imp_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.and_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.or_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.iff_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (forall_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (exists_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (def_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u 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.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	all_goals simp only [isBoundIn] at h1 simp only [fastAdmitsAux]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬isBoundIn u (eq_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u true_ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u false_ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝.not_ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.imp_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.imp_ a✝) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.and_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.and_ a✝) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.or_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.or_ a✝) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.iff_ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (a✝¹.iff_ a✝) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (forall_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (forall_ a✝¹ a✝) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (exists_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (exists_ a✝¹ a✝) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (def_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ v = a✝¹ ∨ fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	all_goals tauto	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v = a✝¹ ∨ v = a✝ → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝ h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → fastAdmitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → fastAdmitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders a✝¹ ∧ fastAdmitsAux v u binders a✝ case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	simp only [isBoundIn] at h1	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬isBoundIn u (def_ a✝¹ a✝) h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ fastAdmitsAux v u 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.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	simp only [fastAdmitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ fastAdmitsAux v u binders (def_ a✝¹ a✝)	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	push_neg at h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h1 : ¬(u = x ∨ isBoundIn u phi) h2 : u ∉ binders ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1 : u ≠ x ∧ ¬isBoundIn u phi ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	cases h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1 : u ≠ x ∧ ¬isBoundIn u phi ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	case intro v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders left✝ : u ≠ x right✝ : ¬isBoundIn u phi ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	right	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ fastAdmitsAux v u (binders ∪ {x}) phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	apply phi_ih (binders ∪ {x}) h1_right	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ fastAdmitsAux v u (binders ∪ {x}) phi	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ u ∉ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	simp	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ u ∉ binders ∪ {x}	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ u ∉ binders ∧ ¬u = x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	tauto	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → fastAdmitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ u ∉ binders ∧ ¬u = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_fastAdmitsAux	[362, 1]	[385, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ → u ∉ binders	no goals
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 only [fastAdmits]	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ fastAdmits v u F	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ fastAdmitsAux v u ∅ F

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