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Split (1)

train · 219k rows

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	exact c1	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x phi) ⊢ 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.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	right	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ v = x ∨ fastAdmitsAux v u (binders ∪ {x}) phi	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v 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.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	apply phi_ih	case h v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ fastAdmitsAux v u (binders ∪ {x}) phi	case h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ v ∉ binders ∪ {x} case h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	simp	case h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ v ∉ binders ∪ {x}	case h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ 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.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	tauto	case h.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ 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.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	simp only [toIsBoundAux] at h2	case h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = toIsBoundAux binders (exists_ x (fastReplaceFree v u phi)) ⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)	case h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)) ⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	simp at h2	case h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) phi) = BoolFormula.forall_ (decide True) (toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)) ⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)	case h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) ⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.free_and_bound_unchanged_imp_fastAdmitsAux	[677, 1]	[740, 17]	exact h2	case h.h2 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), v ∉ binders → toIsBoundAux binders phi = toIsBoundAux binders (fastReplaceFree v u phi) → fastAdmitsAux v u binders phi binders : Finset VarName h1 : v ∉ binders c1 : ¬v = x h2 : toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u phi) ⊢ toIsBoundAux (binders ∪ {x}) phi = toIsBoundAux (binders ∪ {x}) (fastReplaceFree v u 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_self	[760, 1]	[769, 10]	induction F generalizing binders	F : Formula v : VarName binders : Finset VarName ⊢ admitsAux v v binders F	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ admitsAux v v binders (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ admitsAux v v binders (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName binders : Finset VarName ⊢ admitsAux v v binders (eq_ a✝¹ a✝) case true_ v : VarName binders : Finset VarName ⊢ admitsAux v v binders true_ case false_ v : VarName binders : Finset VarName ⊢ admitsAux v v binders false_ case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ admitsAux 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.admitsAux_self	[760, 1]	[769, 10]	all_goals simp only [admitsAux]	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ admitsAux v v binders (pred_const_ a✝¹ a✝) case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ admitsAux v v binders (pred_var_ a✝¹ a✝) case eq_ v a✝¹ a✝ : VarName binders : Finset VarName ⊢ admitsAux v v binders (eq_ a✝¹ a✝) case true_ v : VarName binders : Finset VarName ⊢ admitsAux v v binders true_ case false_ v : VarName binders : Finset VarName ⊢ admitsAux v v binders false_ case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝.not_ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.imp_ a✝) case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.and_ a✝) case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.or_ a✝) case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (a✝¹.iff_ a✝) case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (forall_ a✝¹ a✝) case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders (exists_ a✝¹ a✝) case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ admitsAux v v binders (def_ a✝¹ a✝)	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders case eq_ v a✝¹ a✝ : VarName binders : Finset VarName ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → v ∉ binders case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v (binders ∪ {a✝¹}) a✝ case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v (binders ∪ {a✝¹}) a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ 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.admitsAux_self	[760, 1]	[769, 10]	all_goals tauto	case pred_const_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders case pred_var_ v : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ v ∉ binders → v ∉ binders case eq_ v a✝¹ a✝ : VarName binders : Finset VarName ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → v ∉ binders case not_ v : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝ case imp_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case and_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case or_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case iff_ v : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), admitsAux v v binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v binders a✝¹ ∧ admitsAux v v binders a✝ case forall_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v (binders ∪ {a✝¹}) a✝ case exists_ v a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), admitsAux v v binders a✝ binders : Finset VarName ⊢ admitsAux v v (binders ∪ {a✝¹}) a✝ case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ 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.admitsAux_self	[760, 1]	[769, 10]	simp only [admitsAux]	case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ admitsAux v v binders (def_ a✝¹ a✝)	case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ 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.admitsAux_self	[760, 1]	[769, 10]	tauto	case def_ v : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName ⊢ v ∈ a✝ ∧ 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.admits_self	[772, 1]	[778, 23]	simp only [admits]	F : Formula v : VarName ⊢ admits v v F	F : Formula v : VarName ⊢ admitsAux v v ∅ F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admits_self	[772, 1]	[778, 23]	apply admitsAux_self	F : Formula v : VarName ⊢ admitsAux v v ∅ 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_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : ¬isFreeIn v F ⊢ admitsAux 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✝) ⊢ 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 : ¬isFreeIn v (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 : ¬isFreeIn v (eq_ a✝¹ a✝) ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v true_ ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v false_ ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝.not_ ⊢ admitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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 : ¬isFreeIn v (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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	all_goals simp only [isFreeIn] at h1 simp only [admitsAux]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬isFreeIn v (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 : ¬isFreeIn v (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 : ¬isFreeIn v (eq_ a✝¹ a✝) ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v true_ ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isFreeIn v false_ ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝.not_ ⊢ admitsAux v u binders a✝.not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v (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 : ¬isFreeIn v (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✝ ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(v = a✝¹ ∨ v = a✝) ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝ ⊢ admitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(¬v = a✝¹ ∧ isFreeIn v a✝) ⊢ admitsAux v u (binders ∪ {a✝¹}) a✝ case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(¬v = a✝¹ ∧ isFreeIn v 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✝ ⊢ 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	all_goals tauto	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(v = a✝¹ ∨ v = a✝) ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isFreeIn v a✝ ⊢ admitsAux v u binders a✝ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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), ¬isFreeIn v a✝¹ → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isFreeIn v a✝ → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isFreeIn v a✝¹ ∨ isFreeIn v 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✝ ⊢ 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 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✝) ⊢ 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✝ ⊢ 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	simp only [admitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ 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✝ ⊢ 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	by_cases c1 : v = x	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) ⊢ admitsAux v u (binders ∪ {x}) phi	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : v = x ⊢ admitsAux v u (binders ∪ {x}) phi case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	apply mem_binders_imp_admitsAux	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : v = x ⊢ admitsAux v u (binders ∪ {x}) phi	case pos.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : v = x ⊢ v ∈ binders ∪ {x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	simp	case pos.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : v = x ⊢ v ∈ binders ∪ {x}	case pos.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	tauto	case pos.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v 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.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	apply phi_ih	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : ¬v = x ⊢ admitsAux v u (binders ∪ {x}) phi	case neg.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : ¬v = x ⊢ ¬isFreeIn v phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isFreeIn_imp_admitsAux	[782, 1]	[802, 10]	tauto	case neg.h1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isFreeIn v phi → admitsAux v u binders phi binders : Finset VarName h1 : ¬(¬v = x ∧ isFreeIn v phi) c1 : ¬v = x ⊢ ¬isFreeIn v phi	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_admitsAux	[782, 1]	[802, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∉ a✝ ⊢ 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.not_isFreeIn_imp_admits	[805, 1]	[812, 46]	simp only [admits]	F : Formula v u : VarName h1 : ¬isFreeIn v F ⊢ admits v u F	F : Formula v u : VarName h1 : ¬isFreeIn v 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.not_isFreeIn_imp_admits	[805, 1]	[812, 46]	exact not_isFreeIn_imp_admitsAux F v u ∅ h1	F : Formula v u : VarName h1 : ¬isFreeIn v 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : ¬isBoundIn u F h2 : u ∉ binders ⊢ admitsAux 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 ⊢ 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 : ¬isBoundIn u (pred_var_ a✝¹ a✝) h2 : u ∉ binders ⊢ admitsAux 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 ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u true_ h2 : u ∉ binders ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u false_ h2 : u ∉ binders ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝.not_ h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.imp_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.and_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.or_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.iff_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (forall_ a✝¹ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (exists_ a✝¹ a✝) h2 : u ∉ binders ⊢ admitsAux 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 ⊢ 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	all_goals simp only [isBoundIn] at h1 simp only [admitsAux]	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 ⊢ 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 : ¬isBoundIn u (pred_var_ a✝¹ a✝) h2 : u ∉ binders ⊢ admitsAux 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 ⊢ admitsAux v u binders (eq_ a✝¹ a✝) case true_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u true_ h2 : u ∉ binders ⊢ admitsAux v u binders true_ case false_ v u : VarName binders : Finset VarName h1 : ¬isBoundIn u false_ h2 : u ∉ binders ⊢ admitsAux v u binders false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝.not_ h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.imp_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.and_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.or_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (a✝¹.iff_ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (forall_ a✝¹ a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u (exists_ a✝¹ a✝) h2 : u ∉ binders ⊢ admitsAux 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 ⊢ admitsAux 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✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝ h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(u = a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ binders ⊢ admitsAux 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✝ ∧ 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => push_neg at h1 cases h1 case intro h1_left h1_right => apply phi_ih (binders ∪ {x}) h1_right simp tauto	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h1 : ¬(u = x ∨ isBoundIn u phi) h2 : u ∉ 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 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✝ ∧ v ∉ binders → u ∉ binders case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ v ∈ a✝ ∧ v ∉ binders → u ∉ binders case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬False h2 : u ∉ binders ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ binders → u ∉ binders case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬isBoundIn u a✝ h2 : u ∉ binders ⊢ admitsAux 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 → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹ a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝ binders : Finset VarName h1 : ¬(isBoundIn u a✝¹ ∨ isBoundIn u a✝) h2 : u ∉ 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 : ¬False h2 : 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 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 ⊢ admitsAux 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 ⊢ 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	simp only [admitsAux]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : 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 : ¬False h2 : 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	push_neg at h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h1 : ¬(u = x ∨ isBoundIn u phi) h2 : u ∉ binders ⊢ admitsAux v u (binders ∪ {x}) phi	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1 : u ≠ x ∧ ¬isBoundIn u 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	cases h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1 : u ≠ x ∧ ¬isBoundIn u phi ⊢ admitsAux v u (binders ∪ {x}) phi	case intro v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders left✝ : u ≠ x right✝ : ¬isBoundIn u 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	case intro h1_left h1_right => apply phi_ih (binders ∪ {x}) h1_right simp tauto	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u 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.not_isBoundIn_imp_admitsAux	[816, 1]	[838, 10]	apply phi_ih (binders ∪ {x}) h1_right	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ admitsAux v u (binders ∪ {x}) phi	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux 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_admitsAux	[816, 1]	[838, 10]	simp	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux v u binders phi binders : Finset VarName h2 : u ∉ binders h1_left : u ≠ x h1_right : ¬isBoundIn u phi ⊢ u ∉ binders ∪ {x}	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux 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_admitsAux	[816, 1]	[838, 10]	tauto	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), ¬isBoundIn u phi → u ∉ binders → admitsAux 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_admitsAux	[816, 1]	[838, 10]	tauto	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬False h2 : 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.not_isBoundIn_imp_admits	[841, 1]	[849, 7]	simp only [admits]	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ admits v u F	F : Formula v u : VarName h1 : ¬isBoundIn 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.not_isBoundIn_imp_admits	[841, 1]	[849, 7]	apply not_isBoundIn_imp_admitsAux F v u ∅ h1	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ admitsAux v u ∅ F	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ u ∉ ∅
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admits	[841, 1]	[849, 7]	simp	F : Formula v u : VarName h1 : ¬isBoundIn u F ⊢ u ∉ ∅	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	induction F generalizing binders	F : Formula v t : VarName binders : Finset VarName h1 : ¬occursIn t F ⊢ admitsAux t v binders (replaceFreeAux v t binders F)	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_const_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝)) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_var_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝)) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬occursIn t (eq_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬occursIn t true_ ⊢ admitsAux t v binders (replaceFreeAux v t binders true_) case false_ v t : VarName binders : Finset VarName h1 : ¬occursIn t false_ ⊢ admitsAux t v binders (replaceFreeAux v t binders false_) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝.not_ ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝.not_) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.imp_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝)) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.and_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝)) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.or_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝)) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.iff_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (forall_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝)) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (exists_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝)) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	all_goals simp only [occursIn] at h1 simp only [replaceFreeAux] simp only [admitsAux]	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_const_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝)) case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (pred_var_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝)) case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬occursIn t (eq_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝)) case true_ v t : VarName binders : Finset VarName h1 : ¬occursIn t true_ ⊢ admitsAux t v binders (replaceFreeAux v t binders true_) case false_ v t : VarName binders : Finset VarName h1 : ¬occursIn t false_ ⊢ admitsAux t v binders (replaceFreeAux v t binders false_) case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝.not_ ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝.not_) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.imp_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝)) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.and_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝)) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.or_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝)) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (a✝¹.iff_ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝)) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (forall_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝)) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t (exists_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝)) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))	case pred_const_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders case pred_var_ v t : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders case eq_ v t a✝¹ a✝ : VarName binders : Finset VarName h1 : ¬(t = a✝¹ ∨ t = a✝) ⊢ ((t = if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) ∨ t = if v = a✝ ∧ a✝ ∉ binders then t else a✝) ∧ t ∉ binders → v ∉ binders case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝ ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case pred_const_ X xs \| pred_var_ X xs \| def_ X xs => simp intro x a1 a2 a3 by_cases c1 : v = x ∧ x ∉ binders case pos => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case neg => simp at c1 specialize a2 c1 subst a2 contradiction	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs ∧ t ∉ binders → v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case eq_ x y => push_neg at h1 cases h1 case intro h1_left h1_right => intro a1 split_ifs at a1 case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case _ c1 c2 => cases c2 case intro c2_left c2_right => subst c2_left exact c2_right case _ c1 c2 => tauto	v t x y : VarName binders : Finset VarName h1 : ¬(t = x ∨ t = y) ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	all_goals tauto	case not_ v t : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬occursIn t a✝ ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝) case imp_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case and_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case or_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case iff_ v t : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹) a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝) case forall_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝) case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	simp only [occursIn] at h1	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : ¬occursIn t (def_ a✝¹ a✝) ⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	simp only [replaceFreeAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ admitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	simp only [admitsAux]	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ admitsAux t v binders (def_ a✝¹ (List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝))	case def_ v t : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : t ∉ a✝ ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	simp	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) xs ∧ t ∉ binders → v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → t ∉ binders → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	intro x a1 a2 a3	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs ⊢ ∀ x ∈ xs, ((v = x → x ∈ binders) → x = t) → t ∉ binders → v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	by_cases c1 : v = x ∧ x ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders ⊢ v ∉ binders	case pos v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : v = x ∧ x ∉ binders ⊢ v ∉ binders case neg v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : ¬(v = x ∧ x ∉ binders) ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case pos => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : v = x ∧ x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case neg => simp at c1 specialize a2 c1 subst a2 contradiction	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : ¬(v = x ∧ x ∉ binders) ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	cases c1	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : v = x ∧ x ∉ binders ⊢ v ∉ binders	case intro v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders left✝ : v = x right✝ : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case intro c1_left c1_right => subst c1_left exact c1_right	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	subst c1_left	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs a3 : t ∉ binders a1 : v ∈ xs a2 : (v = v → v ∈ binders) → v = t c1_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	exact c1_right	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs a3 : t ∉ binders a1 : v ∈ xs a2 : (v = v → v ∈ binders) → v = t c1_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	simp at c1	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : ¬(v = x ∧ x ∉ binders) ⊢ v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : v = x → x ∈ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	specialize a2 c1	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a2 : (v = x → x ∈ binders) → x = t a3 : t ∉ binders c1 : v = x → x ∈ binders ⊢ v ∉ binders	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a3 : t ∉ binders c1 : v = x → x ∈ binders a2 : x = t ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	subst a2	v t : VarName X : DefName xs : List VarName binders : Finset VarName h1 : t ∉ xs x : VarName a1 : x ∈ xs a3 : t ∉ binders c1 : v = x → x ∈ binders a2 : x = t ⊢ v ∉ binders	v : VarName X : DefName xs : List VarName binders : Finset VarName x : VarName a1 : x ∈ xs c1 : v = x → x ∈ binders h1 : x ∉ xs a3 : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	contradiction	v : VarName X : DefName xs : List VarName binders : Finset VarName x : VarName a1 : x ∈ xs c1 : v = x → x ∈ binders h1 : x ∉ xs a3 : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	push_neg at h1	v t x y : VarName binders : Finset VarName h1 : ¬(t = x ∨ t = y) ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	cases h1	v t x y : VarName binders : Finset VarName h1 : t ≠ x ∧ t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders	case intro v t x y : VarName binders : Finset VarName left✝ : t ≠ x right✝ : t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case intro h1_left h1_right => intro a1 split_ifs at a1 case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right case _ c1 c2 => cases c2 case intro c2_left c2_right => subst c2_left exact c2_right case _ c1 c2 => tauto	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	intro a1	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y ⊢ ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders → v ∉ binders	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y a1 : ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	split_ifs at a1	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y a1 : ((t = if v = x ∧ x ∉ binders then t else x) ∨ t = if v = y ∧ y ∉ binders then t else y) ∧ t ∉ binders ⊢ v ∉ binders	case pos v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y h✝¹ : v = x ∧ x ∉ binders h✝ : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders ⊢ v ∉ binders case neg v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y h✝¹ : v = x ∧ x ∉ binders h✝ : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders ⊢ v ∉ binders case pos v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y h✝¹ : ¬(v = x ∧ x ∉ binders) h✝ : v = y ∧ y ∉ binders a1 : (t = x ∨ t = t) ∧ t ∉ binders ⊢ v ∉ binders case neg v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y h✝¹ : ¬(v = x ∧ x ∉ binders) h✝ : ¬(v = y ∧ y ∉ binders) a1 : (t = x ∨ t = y) ∧ t ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : v = x ∧ x ∉ binders c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case _ c1 c2 => cases c1 case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : v = x ∧ x ∉ binders c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case _ c1 c2 => cases c2 case intro c2_left c2_right => subst c2_left exact c2_right	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : v = y ∧ y ∉ binders a1 : (t = x ∨ t = t) ∧ t ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case _ c1 c2 => tauto	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = x ∨ t = y) ∧ t ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	cases c1	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : v = x ∧ x ∉ binders c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders ⊢ v ∉ binders	case intro v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders left✝ : v = x right✝ : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	subst c1_left	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	v t y : VarName binders : Finset VarName h1_right : t ≠ y c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders h1_left : t ≠ v c1_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	exact c1_right	v t y : VarName binders : Finset VarName h1_right : t ≠ y c2 : v = y ∧ y ∉ binders a1 : (t = t ∨ t = t) ∧ t ∉ binders h1_left : t ≠ v c1_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	cases c1	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : v = x ∧ x ∉ binders c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders ⊢ v ∉ binders	case intro v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders left✝ : v = x right✝ : x ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case intro c1_left c1_right => subst c1_left exact c1_right	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	subst c1_left	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders c1_left : v = x c1_right : x ∉ binders ⊢ v ∉ binders	v t y : VarName binders : Finset VarName h1_right : t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders h1_left : t ≠ v c1_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	exact c1_right	v t y : VarName binders : Finset VarName h1_right : t ≠ y c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = t ∨ t = y) ∧ t ∉ binders h1_left : t ≠ v c1_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	cases c2	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : v = y ∧ y ∉ binders a1 : (t = x ∨ t = t) ∧ t ∉ binders ⊢ v ∉ binders	case intro v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) a1 : (t = x ∨ t = t) ∧ t ∉ binders left✝ : v = y right✝ : y ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	case intro c2_left c2_right => subst c2_left exact c2_right	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) a1 : (t = x ∨ t = t) ∧ t ∉ binders c2_left : v = y c2_right : y ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	subst c2_left	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) a1 : (t = x ∨ t = t) ∧ t ∉ binders c2_left : v = y c2_right : y ∉ binders ⊢ v ∉ binders	v t x : VarName binders : Finset VarName h1_left : t ≠ x c1 : ¬(v = x ∧ x ∉ binders) a1 : (t = x ∨ t = t) ∧ t ∉ binders h1_right : t ≠ v c2_right : v ∉ binders ⊢ v ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	exact c2_right	v t x : VarName binders : Finset VarName h1_left : t ≠ x c1 : ¬(v = x ∧ x ∉ binders) a1 : (t = x ∨ t = t) ∧ t ∉ binders h1_right : t ≠ v c2_right : v ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	tauto	v t x y : VarName binders : Finset VarName h1_left : t ≠ x h1_right : t ≠ y c1 : ¬(v = x ∧ x ∉ binders) c2 : ¬(v = y ∧ y ∉ binders) a1 : (t = x ∨ t = y) ∧ t ∉ binders ⊢ v ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux	[853, 1]	[905, 10]	tauto	case exists_ v t a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝) binders : Finset VarName h1 : ¬(t = a✝¹ ∨ occursIn t a✝) ⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_admits	[908, 1]	[916, 44]	simp only [replaceFree]	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ admits t v (replaceFree v t F)	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ admits t v (replaceFreeAux v t ∅ F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_admits	[908, 1]	[916, 44]	simp only [admits]	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ admits t v (replaceFreeAux v t ∅ F)	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ admitsAux t v ∅ (replaceFreeAux v t ∅ F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.replaceFree_admits	[908, 1]	[916, 44]	exact replaceFreeAux_admitsAux F v t ∅ h1	F : Formula v t : VarName h1 : ¬occursIn t F ⊢ admitsAux t v ∅ (replaceFreeAux v t ∅ F)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	induction F generalizing S	F : Formula v u : VarName S T : Finset VarName h1 : admitsAux v u S F h2 : u ∉ T ⊢ admitsAux v u (S ∪ T) F	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (pred_const_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (pred_var_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝) case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : admitsAux v u S (eq_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (eq_ a✝¹ a✝) case true_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : admitsAux v u S true_ ⊢ admitsAux v u (S ∪ T) true_ case false_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : admitsAux v u S false_ ⊢ admitsAux v u (S ∪ T) false_ case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝.not_ ⊢ admitsAux v u (S ∪ T) a✝.not_ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.imp_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.imp_ a✝) case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.and_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.and_ a✝) case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.or_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.or_ a✝) case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.iff_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.iff_ a✝) case forall_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (forall_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (forall_ a✝¹ a✝) case exists_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (exists_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (exists_ a✝¹ a✝) case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (def_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	all_goals simp only [admitsAux] at h1 simp only [admitsAux]	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (pred_const_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (pred_const_ a✝¹ a✝) case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (pred_var_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (pred_var_ a✝¹ a✝) case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : admitsAux v u S (eq_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (eq_ a✝¹ a✝) case true_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : admitsAux v u S true_ ⊢ admitsAux v u (S ∪ T) true_ case false_ v u : VarName T : Finset VarName h2 : u ∉ T S : Finset VarName h1 : admitsAux v u S false_ ⊢ admitsAux v u (S ∪ T) false_ case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝.not_ ⊢ admitsAux v u (S ∪ T) a✝.not_ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.imp_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.imp_ a✝) case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.and_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.and_ a✝) case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.or_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.or_ a✝) case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (a✝¹.iff_ a✝) ⊢ admitsAux v u (S ∪ T) (a✝¹.iff_ a✝) case forall_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (forall_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (forall_ a✝¹ a✝) case exists_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S (exists_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (exists_ a✝¹ a✝) case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (def_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)	case pred_const_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S ⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T case pred_var_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : PredName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S ⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T case eq_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : VarName S : Finset VarName h1 : (v = a✝¹ ∨ v = a✝) ∧ v ∉ S → u ∉ S ⊢ (v = a✝¹ ∨ v = a✝) ∧ v ∉ S ∪ T → u ∉ S ∪ T case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case forall_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u (S ∪ {a✝¹}) a✝ ⊢ admitsAux v u (S ∪ T ∪ {a✝¹}) a✝ case exists_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u (S ∪ {a✝¹}) a✝ ⊢ admitsAux v u (S ∪ T ∪ {a✝¹}) a✝ case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S ⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	case pred_const_ X xs \| pred_var_ X xs \| eq_ x y \|def_ X xs => simp tauto	v u : VarName T : Finset VarName h2 : u ∉ T X : DefName xs : List VarName S : Finset VarName h1 : v ∈ xs ∧ v ∉ S → u ∉ S ⊢ v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	case forall_ x phi phi_ih \| exists_ x phi phi_ih => simp only [Finset.union_right_comm S T {x}] tauto	v u : VarName T : Finset VarName h2 : u ∉ T x : VarName phi : Formula phi_ih : ∀ (S : Finset VarName), admitsAux v u S phi → admitsAux v u (S ∪ T) phi S : Finset VarName h1 : admitsAux v u (S ∪ {x}) phi ⊢ admitsAux v u (S ∪ T ∪ {x}) phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	all_goals tauto	case not_ v u : VarName T : Finset VarName h2 : u ∉ T a✝ : Formula a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝ case imp_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case and_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case or_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝ case iff_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ a✝ : Formula a_ih✝¹ : ∀ (S : Finset VarName), admitsAux v u S a✝¹ → admitsAux v u (S ∪ T) a✝¹ a_ih✝ : ∀ (S : Finset VarName), admitsAux v u S a✝ → admitsAux v u (S ∪ T) a✝ S : Finset VarName h1 : admitsAux v u S a✝¹ ∧ admitsAux v u S a✝ ⊢ admitsAux v u (S ∪ T) a✝¹ ∧ admitsAux v u (S ∪ T) a✝	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	simp only [admitsAux] at h1	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : admitsAux v u S (def_ a✝¹ a✝) ⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S ⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	simp only [admitsAux]	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S ⊢ admitsAux v u (S ∪ T) (def_ a✝¹ a✝)	case def_ v u : VarName T : Finset VarName h2 : u ∉ T a✝¹ : DefName a✝ : List VarName S : Finset VarName h1 : v ∈ a✝ ∧ v ∉ S → u ∉ S ⊢ v ∈ a✝ ∧ v ∉ S ∪ T → u ∉ S ∪ T
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Rec/Admits.lean	FOL.NV.Sub.Var.One.Rec.admitsAux_add_binders	[920, 1]	[940, 10]	simp	v u : VarName T : Finset VarName h2 : u ∉ T X : DefName xs : List VarName S : Finset VarName h1 : v ∈ xs ∧ v ∉ S → u ∉ S ⊢ v ∈ xs ∧ v ∉ S ∪ T → u ∉ S ∪ T	v u : VarName T : Finset VarName h2 : u ∉ T X : DefName xs : List VarName S : Finset VarName h1 : v ∈ xs ∧ v ∉ S → u ∉ S ⊢ v ∈ xs → v ∉ S → v ∉ T → u ∉ S ∧ u ∉ T

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