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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/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	apply Exists.elim right	α : Type R : RegExp α cs : List α right : ∃ r ∈ RegExp.languageOf α R, ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ r ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs : List α right : ∃ r ∈ RegExp.languageOf α R, ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ r ++ a.join = cs ⊢ ∀ (a : List α), (a ∈ RegExp.languageOf α R ∧ ∃ a_1, (∀ r ∈ a_1, r ∈ RegExp.languageOf α R) ∧ a ++ a_1.join = cs) → ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	intro xs a2	α : Type R : RegExp α cs : List α right : ∃ r ∈ RegExp.languageOf α R, ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ r ++ a.join = cs ⊢ ∀ (a : List α), (a ∈ RegExp.languageOf α R ∧ ∃ a_1, (∀ r ∈ a_1, r ∈ RegExp.languageOf α R) ∧ a ++ a_1.join = cs) → ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs : List α right : ∃ r ∈ RegExp.languageOf α R, ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ r ++ a.join = cs xs : List α a2 : xs ∈ RegExp.languageOf α R ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	clear right	α : Type R : RegExp α cs : List α right : ∃ r ∈ RegExp.languageOf α R, ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ r ++ a.join = cs xs : List α a2 : xs ∈ RegExp.languageOf α R ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs xs : List α a2 : xs ∈ RegExp.languageOf α R ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	cases a2	α : Type R : RegExp α cs xs : List α a2 : xs ∈ RegExp.languageOf α R ∧ ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	case intro α : Type R : RegExp α cs xs : List α left✝ : xs ∈ RegExp.languageOf α R right✝ : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	case _ a2_left a2_right => apply Exists.elim a2_right intro ys a3 clear a2_right cases a3 case _ a3_left a3_right => apply Exists.intro ([xs] ++ ys) simp tauto	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	apply Exists.elim a2_right	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∀ (a : List (List α)), (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs → ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	intro ys a3	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ⊢ ∀ (a : List (List α)), (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs → ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ys : List (List α) a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α R) ∧ xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	clear a2_right	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R a2_right : ∃ a, (∀ r ∈ a, r ∈ RegExp.languageOf α R) ∧ xs ++ a.join = cs ys : List (List α) a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α R) ∧ xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α R) ∧ xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	cases a3	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3 : (∀ r ∈ ys, r ∈ RegExp.languageOf α R) ∧ xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	case intro α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) left✝ : ∀ r ∈ ys, r ∈ RegExp.languageOf α R right✝ : xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	case _ a3_left a3_right => apply Exists.intro ([xs] ++ ys) simp tauto	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α R a3_right : xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	apply Exists.intro ([xs] ++ ys)	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α R a3_right : xs ++ ys.join = cs ⊢ ∃ rs, (∀ r ∈ rs, r ∈ RegExp.languageOf α R) ∧ rs.join = cs	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α R a3_right : xs ++ ys.join = cs ⊢ (∀ r ∈ [xs] ++ ys, r ∈ RegExp.languageOf α R) ∧ ([xs] ++ ys).join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	simp	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α R a3_right : xs ++ ys.join = cs ⊢ (∀ r ∈ [xs] ++ ys, r ∈ RegExp.languageOf α R) ∧ ([xs] ++ ys).join = cs	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α R a3_right : xs ++ ys.join = cs ⊢ (xs ∈ RegExp.languageOf α R ∧ ∀ a ∈ ys, a ∈ RegExp.languageOf α R) ∧ xs ++ ys.join = cs
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/Parsing/RegExp.lean	equiv_language_of_closure	[36, 1]	[83, 18]	tauto	α : Type R : RegExp α cs xs : List α a2_left : xs ∈ RegExp.languageOf α R ys : List (List α) a3_left : ∀ r ∈ ys, r ∈ RegExp.languageOf α R a3_right : xs ++ ys.join = cs ⊢ (xs ∈ RegExp.languageOf α R ∧ ∀ a ∈ ys, a ∈ RegExp.languageOf α R) ∧ xs ++ ys.join = cs	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	subst h2	F F' : Formula v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders F h2 : Rec.fastReplaceFree v u F = F' ⊢ IsSub F v u F'	F : Formula v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders F ⊢ IsSub F v u (Rec.fastReplaceFree v u F)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	induction F generalizing binders	F : Formula v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders F ⊢ IsSub F v u (Rec.fastReplaceFree v u F)	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) ⊢ IsSub (pred_const_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (pred_const_ a✝¹ a✝)) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) ⊢ IsSub (pred_var_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (pred_var_ a✝¹ a✝)) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (eq_ a✝¹ a✝) ⊢ IsSub (eq_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (eq_ a✝¹ a✝)) case true_ v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders true_ ⊢ IsSub true_ v u (Rec.fastReplaceFree v u true_) case false_ v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders false_ ⊢ IsSub false_ v u (Rec.fastReplaceFree v u false_) case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝.not_ ⊢ IsSub a✝.not_ v u (Rec.fastReplaceFree v u a✝.not_) case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.imp_ a✝) ⊢ IsSub (a✝¹.imp_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.imp_ a✝)) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.and_ a✝) ⊢ IsSub (a✝¹.and_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.and_ a✝)) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.or_ a✝) ⊢ IsSub (a✝¹.or_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.or_ a✝)) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.iff_ a✝) ⊢ IsSub (a✝¹.iff_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.iff_ a✝)) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (forall_ a✝¹ a✝) ⊢ IsSub (forall_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (forall_ a✝¹ a✝)) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (exists_ a✝¹ a✝) ⊢ IsSub (exists_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (exists_ a✝¹ a✝)) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (def_ a✝¹ a✝) ⊢ IsSub (def_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	all_goals simp only [Rec.fastAdmitsAux] at h1 simp only [Rec.fastReplaceFree]	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (pred_const_ a✝¹ a✝) ⊢ IsSub (pred_const_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (pred_const_ a✝¹ a✝)) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (pred_var_ a✝¹ a✝) ⊢ IsSub (pred_var_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (pred_var_ a✝¹ a✝)) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (eq_ a✝¹ a✝) ⊢ IsSub (eq_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (eq_ a✝¹ a✝)) case true_ v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders true_ ⊢ IsSub true_ v u (Rec.fastReplaceFree v u true_) case false_ v u : VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders false_ ⊢ IsSub false_ v u (Rec.fastReplaceFree v u false_) case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝.not_ ⊢ IsSub a✝.not_ v u (Rec.fastReplaceFree v u a✝.not_) case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.imp_ a✝) ⊢ IsSub (a✝¹.imp_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.imp_ a✝)) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.and_ a✝) ⊢ IsSub (a✝¹.and_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.and_ a✝)) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.or_ a✝) ⊢ IsSub (a✝¹.or_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.or_ a✝)) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (a✝¹.iff_ a✝) ⊢ IsSub (a✝¹.iff_ a✝) v u (Rec.fastReplaceFree v u (a✝¹.iff_ a✝)) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (forall_ a✝¹ a✝) ⊢ IsSub (forall_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (forall_ a✝¹ a✝)) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (exists_ a✝¹ a✝) ⊢ IsSub (exists_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (exists_ a✝¹ a✝)) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (def_ a✝¹ a✝) ⊢ IsSub (def_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (def_ a✝¹ a✝))	case pred_const_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ IsSub (pred_const_ a✝¹ a✝) v u (pred_const_ a✝¹ (List.map (fun x => if v = x then u else x) a✝)) case pred_var_ v u : VarName a✝¹ : PredName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ IsSub (pred_var_ a✝¹ a✝) v u (pred_var_ a✝¹ (List.map (fun x => if v = x then u else x) a✝)) case eq_ v u a✝¹ a✝ : VarName binders : Finset VarName h1 : v = a✝¹ ∨ v = a✝ → u ∉ binders ⊢ IsSub (eq_ a✝¹ a✝) v u (eq_ (if v = a✝¹ then u else a✝¹) (if v = a✝ then u else a✝)) case true_ v u : VarName binders : Finset VarName h1 : True ⊢ IsSub true_ v u true_ case false_ v u : VarName binders : Finset VarName h1 : True ⊢ IsSub false_ v u false_ case not_ v u : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝ ⊢ IsSub a✝.not_ v u (Rec.fastReplaceFree v u a✝).not_ case imp_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝¹ ∧ Rec.fastAdmitsAux v u binders a✝ ⊢ IsSub (a✝¹.imp_ a✝) v u ((Rec.fastReplaceFree v u a✝¹).imp_ (Rec.fastReplaceFree v u a✝)) case and_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝¹ ∧ Rec.fastAdmitsAux v u binders a✝ ⊢ IsSub (a✝¹.and_ a✝) v u ((Rec.fastReplaceFree v u a✝¹).and_ (Rec.fastReplaceFree v u a✝)) case or_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝¹ ∧ Rec.fastAdmitsAux v u binders a✝ ⊢ IsSub (a✝¹.or_ a✝) v u ((Rec.fastReplaceFree v u a✝¹).or_ (Rec.fastReplaceFree v u a✝)) case iff_ v u : VarName a✝¹ a✝ : Formula a_ih✝¹ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝¹ → IsSub a✝¹ v u (Rec.fastReplaceFree v u a✝¹) a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders a✝¹ ∧ Rec.fastAdmitsAux v u binders a✝ ⊢ IsSub (a✝¹.iff_ a✝) v u ((Rec.fastReplaceFree v u a✝¹).iff_ (Rec.fastReplaceFree v u a✝)) case forall_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : v = a✝¹ ∨ Rec.fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ ⊢ IsSub (forall_ a✝¹ a✝) v u (if v = a✝¹ then forall_ a✝¹ a✝ else forall_ a✝¹ (Rec.fastReplaceFree v u a✝)) case exists_ v u a✝¹ : VarName a✝ : Formula a_ih✝ : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders a✝ → IsSub a✝ v u (Rec.fastReplaceFree v u a✝) binders : Finset VarName h1 : v = a✝¹ ∨ Rec.fastAdmitsAux v u (binders ∪ {a✝¹}) a✝ ⊢ IsSub (exists_ a✝¹ a✝) v u (if v = a✝¹ then exists_ a✝¹ a✝ else exists_ a✝¹ (Rec.fastReplaceFree v u a✝)) case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ IsSub (def_ a✝¹ a✝) v u (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case pred_const_ X xs \| pred_var_ X xs => first \| apply IsSub.pred_const_ \| apply IsSub.pred_var_	v u : VarName X : PredName xs : List VarName binders : Finset VarName h1 : v ∈ xs → u ∉ binders ⊢ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case eq_ x y => apply IsSub.eq_	v u x y : VarName binders : Finset VarName h1 : v = x ∨ v = y → u ∉ binders ⊢ IsSub (eq_ x y) v u (eq_ (if v = x then u else x) (if v = y then u else y))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case true_ \| false_ => first \| apply IsSub.true_ \| apply IsSub.false_	v u : VarName binders : Finset VarName h1 : True ⊢ IsSub false_ v u false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case not_ phi phi_ih => apply IsSub.not_ exact phi_ih binders h1	v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders phi ⊢ IsSub phi.not_ v u (Rec.fastReplaceFree v u phi).not_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case def_ X xs => apply IsSub.def_	v u : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ xs → u ∉ binders ⊢ IsSub (def_ X xs) v u (def_ X (List.map (fun x => if v = x then u else x) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [Rec.fastAdmitsAux] at h1	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders (def_ a✝¹ a✝) ⊢ IsSub (def_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (def_ a✝¹ a✝))	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ IsSub (def_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (def_ a✝¹ a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [Rec.fastReplaceFree]	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ IsSub (def_ a✝¹ a✝) v u (Rec.fastReplaceFree v u (def_ a✝¹ a✝))	case def_ v u : VarName a✝¹ : DefName a✝ : List VarName binders : Finset VarName h1 : v ∈ a✝ → u ∉ binders ⊢ IsSub (def_ a✝¹ a✝) v u (def_ a✝¹ (List.map (fun x => if v = x then u else x) a✝))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.pred_const_ \| apply IsSub.pred_var_	v u : VarName X : PredName xs : List VarName binders : Finset VarName h1 : v ∈ xs → u ∉ binders ⊢ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.pred_const_	v u : VarName X : PredName xs : List VarName binders : Finset VarName h1 : v ∈ xs → u ∉ binders ⊢ IsSub (pred_const_ X xs) v u (pred_const_ X (List.map (fun x => if v = x then u else x) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.pred_var_	v u : VarName X : PredName xs : List VarName binders : Finset VarName h1 : v ∈ xs → u ∉ binders ⊢ IsSub (pred_var_ X xs) v u (pred_var_ X (List.map (fun x => if v = x then u else x) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.eq_	v u x y : VarName binders : Finset VarName h1 : v = x ∨ v = y → u ∉ binders ⊢ IsSub (eq_ x y) v u (eq_ (if v = x then u else x) (if v = y then u else y))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.true_ \| apply IsSub.false_	v u : VarName binders : Finset VarName h1 : True ⊢ IsSub false_ v u false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.true_	v u : VarName binders : Finset VarName h1 : True ⊢ IsSub true_ v u true_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.false_	v u : VarName binders : Finset VarName h1 : True ⊢ IsSub false_ v u false_	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.not_	v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders phi ⊢ IsSub phi.not_ v u (Rec.fastReplaceFree v u phi).not_	case a v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact phi_ih binders h1	case a v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	cases h1	v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u binders phi ∧ Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi))	case intro v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName left✝ : Rec.fastAdmitsAux v u binders phi right✝ : Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.imp_ \| apply IsSub.and_ \| apply IsSub.or_ \| apply IsSub.iff_	v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi))	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub psi v u (Rec.fastReplaceFree v u psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.imp_	v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.imp_ psi) v u ((Rec.fastReplaceFree v u phi).imp_ (Rec.fastReplaceFree v u psi))	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub psi v u (Rec.fastReplaceFree v u psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.and_	v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.and_ psi) v u ((Rec.fastReplaceFree v u phi).and_ (Rec.fastReplaceFree v u psi))	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub psi v u (Rec.fastReplaceFree v u psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.or_	v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.or_ psi) v u ((Rec.fastReplaceFree v u phi).or_ (Rec.fastReplaceFree v u psi))	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub psi v u (Rec.fastReplaceFree v u psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.iff_	v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub (phi.iff_ psi) v u ((Rec.fastReplaceFree v u phi).iff_ (Rec.fastReplaceFree v u psi))	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi) case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub psi v u (Rec.fastReplaceFree v u psi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact phi_ih binders h1_left	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact psi_ih binders h1_right	case a v u : VarName phi psi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) psi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders psi → IsSub psi v u (Rec.fastReplaceFree v u psi) binders : Finset VarName h1_left : Rec.fastAdmitsAux v u binders phi h1_right : Rec.fastAdmitsAux v u binders psi ⊢ IsSub psi v u (Rec.fastReplaceFree v u psi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	cases h1	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x ∨ Rec.fastAdmitsAux v u (binders ∪ {x}) phi ⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi))	case inl v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h✝ : v = x ⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi)) case inr v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h✝ : Rec.fastAdmitsAux v u (binders ∪ {x}) phi ⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case inl h1 => split_ifs case pos c1 \| neg c1 => first \| apply IsSub.forall_not_free_in \| apply IsSub.exists_not_free_in subst h1 simp only [isFreeIn] simp	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x ⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	split_ifs	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x ⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi))	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 h✝ : v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi) case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x h✝ : ¬v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case pos c1 \| neg c1 => first \| apply IsSub.forall_not_free_in \| apply IsSub.exists_not_free_in subst h1 simp only [isFreeIn] simp	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.forall_not_free_in \| apply IsSub.exists_not_free_in	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ ¬isFreeIn v (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	subst h1	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ ¬isFreeIn v (exists_ x phi)	case a v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName c1 : ¬v = v ⊢ ¬isFreeIn v (exists_ v phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [isFreeIn]	case a v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName c1 : ¬v = v ⊢ ¬isFreeIn v (exists_ v phi)	case a v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName c1 : ¬v = v ⊢ ¬(¬True ∧ isFreeIn v phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp	case a v u : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName c1 : ¬v = v ⊢ ¬(¬True ∧ isFreeIn v phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.forall_not_free_in	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ IsSub (forall_ x phi) v u (forall_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ ¬isFreeIn v (forall_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.exists_not_free_in	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : v = x c1 : ¬v = x ⊢ ¬isFreeIn v (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	split_ifs	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi ⊢ IsSub (exists_ x phi) v u (if v = x then exists_ x phi else exists_ x (Rec.fastReplaceFree v u phi))	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi h✝ : v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi) case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi h✝ : ¬v = x ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	case pos c1 => first \| apply IsSub.forall_not_free_in \| apply IsSub.exists_not_free_in simp only [isFreeIn] tauto	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.forall_not_free_in \| apply IsSub.exists_not_free_in	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ ¬isFreeIn v (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [isFreeIn]	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ ¬isFreeIn v (exists_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ ¬(¬v = x ∧ isFreeIn v phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	tauto	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ ¬(¬v = x ∧ isFreeIn v phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.forall_not_free_in	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ IsSub (forall_ x phi) v u (forall_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ ¬isFreeIn v (forall_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.exists_not_free_in	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	case a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : v = x ⊢ ¬isFreeIn v (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	by_cases c2 : isFreeIn v phi	v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi)) case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.forall_free_in \| apply IsSub.exists_free_in	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v (exists_ x phi) case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [isFreeIn]	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v (exists_ x phi) case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬v = x ∧ isFreeIn v phi case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	constructor	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬v = x ∧ isFreeIn v phi case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	case pos.a.left v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬v = x case pos.a.right v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v phi case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	. exact c2	case pos.a.right v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v phi case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	. have s1 : ¬ u ∈ binders ∪ {x} exact Rec.fastAdmitsAux_isFreeIn phi v u (binders ∪ {x}) h1 c2 simp at s1 tauto	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.forall_free_in	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub (forall_ x phi) v u (forall_ x (Rec.fastReplaceFree v u phi))	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v (forall_ x phi) case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.exists_free_in	case pos v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v (exists_ x phi) case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact c1	case pos.a.left v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬v = x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact c2	case pos.a.right v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ isFreeIn v phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	have s1 : ¬ u ∈ binders ∪ {x}	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ ¬x = u	case s1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ u ∉ binders ∪ {x} case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi s1 : u ∉ binders ∪ {x} ⊢ ¬x = u
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact Rec.fastAdmitsAux_isFreeIn phi v u (binders ∪ {x}) h1 c2	case s1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ u ∉ binders ∪ {x} case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi s1 : u ∉ binders ∪ {x} ⊢ ¬x = u	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi s1 : u ∉ binders ∪ {x} ⊢ ¬x = u
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp at s1	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi s1 : u ∉ binders ∪ {x} ⊢ ¬x = u	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi s1 : u ∉ binders ∧ ¬u = x ⊢ ¬x = u
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	tauto	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi s1 : u ∉ binders ∧ ¬u = x ⊢ ¬x = u	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact phi_ih (binders ∪ {x}) h1	case pos.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : isFreeIn v phi ⊢ IsSub phi v u (Rec.fastReplaceFree v u phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	have s1 : Rec.fastReplaceFree v u phi = phi	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))	case s1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi ⊢ Rec.fastReplaceFree v u phi = phi case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	exact Rec.not_free_in_fastReplaceFree_self phi v u c2	case s1 v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi ⊢ Rec.fastReplaceFree v u phi = phi case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [s1]	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x (Rec.fastReplaceFree v u phi))	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	first \| apply IsSub.forall_not_free_in \| apply IsSub.exists_not_free_in	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	case neg.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ ¬isFreeIn v (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	simp only [isFreeIn]	case neg.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ ¬isFreeIn v (exists_ x phi)	case neg.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ ¬(¬v = x ∧ isFreeIn v phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	tauto	case neg.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ ¬(¬v = x ∧ isFreeIn v phi)	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.forall_not_free_in	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (forall_ x phi) v u (forall_ x phi)	case neg.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ ¬isFreeIn v (forall_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.exists_not_free_in	case neg v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ IsSub (exists_ x phi) v u (exists_ x phi)	case neg.a v u x : VarName phi : Formula phi_ih : ∀ (binders : Finset VarName), Rec.fastAdmitsAux v u binders phi → IsSub phi v u (Rec.fastReplaceFree v u phi) binders : Finset VarName h1 : Rec.fastAdmitsAux v u (binders ∪ {x}) phi c1 : ¬v = x c2 : ¬isFreeIn v phi s1 : Rec.fastReplaceFree v u phi = phi ⊢ ¬isFreeIn v (exists_ x phi)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub	[125, 1]	[194, 21]	apply IsSub.def_	v u : VarName X : DefName xs : List VarName binders : Finset VarName h1 : v ∈ xs → u ∉ binders ⊢ IsSub (def_ X xs) v u (def_ X (List.map (fun x => if v = x then u else x) xs))	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	apply Exists.elim h1	F : Formula v u : VarName binders : Finset VarName h1 : ∃ F', IsSub F v u F' h2 : u ∉ binders ⊢ Rec.fastAdmitsAux v u binders F	F : Formula v u : VarName binders : Finset VarName h1 : ∃ F', IsSub F v u F' h2 : u ∉ binders ⊢ ∀ (a : Formula), IsSub F v u a → Rec.fastAdmitsAux v u binders F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	intro F' h1_1	F : Formula v u : VarName binders : Finset VarName h1 : ∃ F', IsSub F v u F' h2 : u ∉ binders ⊢ ∀ (a : Formula), IsSub F v u a → Rec.fastAdmitsAux v u binders F	F : Formula v u : VarName binders : Finset VarName h1 : ∃ F', IsSub F v u F' h2 : u ∉ binders F' : Formula h1_1 : IsSub F v u F' ⊢ Rec.fastAdmitsAux v u binders F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	clear h1	F : Formula v u : VarName binders : Finset VarName h1 : ∃ F', IsSub F v u F' h2 : u ∉ binders F' : Formula h1_1 : IsSub F v u F' ⊢ Rec.fastAdmitsAux v u binders F	F : Formula v u : VarName binders : Finset VarName h2 : u ∉ binders F' : Formula h1_1 : IsSub F v u F' ⊢ Rec.fastAdmitsAux v u binders F
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	induction h1_1 generalizing binders	F : Formula v u : VarName binders : Finset VarName h2 : u ∉ binders F' : Formula h1_1 : IsSub F v u F' ⊢ Rec.fastAdmitsAux v u binders F	case pred_const_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_const_ X✝ xs✝) case pred_var_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_var_ X✝ xs✝) case eq_ F : Formula v u : VarName F' : Formula x✝ y✝ v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (eq_ x✝ y✝) case true_ F : Formula v u : VarName F' : Formula v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders true_ case false_ F : Formula v u : VarName F' : Formula v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders false_ case not_ F : Formula v u : VarName F' phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝.not_ case imp_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.imp_ psi✝) case and_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.and_ psi✝) case or_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.or_ psi✝) case iff_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.iff_ psi✝) case forall_not_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝) binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝) case forall_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝² : isFreeIn v✝ (forall_ x✝ phi✝) a✝¹ : ¬x✝ = t✝ a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝) case exists_not_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝) binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝) case exists_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝² : isFreeIn v✝ (exists_ x✝ phi✝) a✝¹ : ¬x✝ = t✝ a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝) case def_ F : Formula v u : VarName F' : Formula X✝ : DefName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (def_ X✝ xs✝)
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	all_goals simp only [Rec.fastAdmitsAux]	case pred_const_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_const_ X✝ xs✝) case pred_var_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (pred_var_ X✝ xs✝) case eq_ F : Formula v u : VarName F' : Formula x✝ y✝ v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (eq_ x✝ y✝) case true_ F : Formula v u : VarName F' : Formula v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders true_ case false_ F : Formula v u : VarName F' : Formula v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders false_ case not_ F : Formula v u : VarName F' phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝.not_ case imp_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.imp_ psi✝) case and_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.and_ psi✝) case or_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.or_ psi✝) case iff_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (phi✝.iff_ psi✝) case forall_not_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝) binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝) case forall_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝² : isFreeIn v✝ (forall_ x✝ phi✝) a✝¹ : ¬x✝ = t✝ a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (forall_ x✝ phi✝) case exists_not_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝) binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝) case exists_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝² : isFreeIn v✝ (exists_ x✝ phi✝) a✝¹ : ¬x✝ = t✝ a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (exists_ x✝ phi✝) case def_ F : Formula v u : VarName F' : Formula X✝ : DefName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (def_ X✝ xs✝)	case pred_const_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders case pred_var_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders case eq_ F : Formula v u : VarName F' : Formula x✝ y✝ v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ = x✝ ∨ v✝ = y✝ → t✝ ∉ binders case not_ F : Formula v u : VarName F' phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ case imp_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case and_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case or_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case iff_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case forall_not_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName a✝ : ¬isFreeIn v✝ (forall_ x✝ phi✝) binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝ case forall_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝² : isFreeIn v✝ (forall_ x✝ phi✝) a✝¹ : ¬x✝ = t✝ a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝ case exists_not_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName a✝ : ¬isFreeIn v✝ (exists_ x✝ phi✝) binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝ case exists_free_in F : Formula v u : VarName F' : Formula x✝ : VarName phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝² : isFreeIn v✝ (exists_ x✝ phi✝) a✝¹ : ¬x✝ = t✝ a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ = x✝ ∨ Rec.fastAdmitsAux v✝ t✝ (binders ∪ {x✝}) phi✝ case def_ F : Formula v u : VarName F' : Formula X✝ : DefName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	case forall_free_in h1_1_x h1_1_phi h1_1_v h1_1_t _ _ h1_1_2 _ h1_1_ih \| exists_free_in h1_1_x h1_1_phi h1_1_v h1_1_t _ _ h1_1_2 _ h1_1_ih => right apply h1_1_ih simp tauto	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	all_goals tauto	case pred_const_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders case pred_var_ F : Formula v u : VarName F' : Formula X✝ : PredName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders case eq_ F : Formula v u : VarName F' : Formula x✝ y✝ v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ = x✝ ∨ v✝ = y✝ → t✝ ∉ binders case not_ F : Formula v u : VarName F' phi✝ : Formula v✝ t✝ : VarName phi'✝ : Formula a✝ : IsSub phi✝ v✝ t✝ phi'✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ case imp_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case and_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case or_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case iff_ F : Formula v u : VarName F' phi✝ psi✝ : Formula v✝ t✝ : VarName phi'✝ psi'✝ : Formula a✝¹ : IsSub phi✝ v✝ t✝ phi'✝ a✝ : IsSub psi✝ v✝ t✝ psi'✝ a_ih✝¹ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders phi✝ a_ih✝ : ∀ (binders : Finset VarName), t✝ ∉ binders → Rec.fastAdmitsAux v✝ t✝ binders psi✝ binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders phi✝ ∧ Rec.fastAdmitsAux v✝ t✝ binders psi✝ case def_ F : Formula v u : VarName F' : Formula X✝ : DefName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	simp only [Rec.fastAdmitsAux]	case def_ F : Formula v u : VarName F' : Formula X✝ : DefName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ Rec.fastAdmitsAux v✝ t✝ binders (def_ X✝ xs✝)	case def_ F : Formula v u : VarName F' : Formula X✝ : DefName xs✝ : List VarName v✝ t✝ : VarName binders : Finset VarName h2 : t✝ ∉ binders ⊢ v✝ ∈ xs✝ → t✝ ∉ binders
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	simp only [isFreeIn] at h1_1_1	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName h1_1_1 : ¬isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName h1_1_1 : ¬(¬h1_1_v = h1_1_x ∧ isFreeIn h1_1_v h1_1_phi) binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	simp at h1_1_1	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName h1_1_1 : ¬(¬h1_1_v = h1_1_x ∧ isFreeIn h1_1_v h1_1_phi) binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	by_cases c1 : h1_1_v = h1_1_x	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	case pos F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : h1_1_v = h1_1_x ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi case neg F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : ¬h1_1_v = h1_1_x ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	left	case pos F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : h1_1_v = h1_1_x ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	case pos.h F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : h1_1_v = h1_1_x ⊢ h1_1_v = h1_1_x
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	exact c1	case pos.h F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : h1_1_v = h1_1_x ⊢ h1_1_v = h1_1_x	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	right	case neg F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : ¬h1_1_v = h1_1_x ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	case neg.h F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : ¬h1_1_v = h1_1_x ⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	apply Rec.not_isFreeIn_imp_fastAdmitsAux	case neg.h F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : ¬h1_1_v = h1_1_x ⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	case neg.h.h1 F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : ¬h1_1_v = h1_1_x ⊢ ¬isFreeIn h1_1_v h1_1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	exact h1_1_1 c1	case neg.h.h1 F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName binders : Finset VarName h2 : h1_1_t ∉ binders h1_1_1 : ¬h1_1_v = h1_1_x → ¬isFreeIn h1_1_v h1_1_phi c1 : ¬h1_1_v = h1_1_x ⊢ ¬isFreeIn h1_1_v h1_1_phi	no goals
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	right	F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_v = h1_1_x ∨ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	case h F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	apply h1_1_ih	case h F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ Rec.fastAdmitsAux h1_1_v h1_1_t (binders ∪ {h1_1_x}) h1_1_phi	case h.h2 F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_t ∉ binders ∪ {h1_1_x}
https://github.com/pthomas505/FOL.git	097a4abea51b641d144539b9a0f7516f3b9d818c	FOL/NV/Sub/Var/One/Ind/Sub.lean	FOL.NV.Sub.Var.One.Ind.isFreeSub_imp_fastAdmitsAux	[197, 1]	[231, 10]	simp	case h.h2 F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_t ∉ binders ∪ {h1_1_x}	case h.h2 F : Formula v u : VarName F' : Formula h1_1_x : VarName h1_1_phi : Formula h1_1_v h1_1_t : VarName phi'✝ : Formula a✝¹ : isFreeIn h1_1_v (exists_ h1_1_x h1_1_phi) h1_1_2 : ¬h1_1_x = h1_1_t a✝ : IsSub h1_1_phi h1_1_v h1_1_t phi'✝ h1_1_ih : ∀ (binders : Finset VarName), h1_1_t ∉ binders → Rec.fastAdmitsAux h1_1_v h1_1_t binders h1_1_phi binders : Finset VarName h2 : h1_1_t ∉ binders ⊢ h1_1_t ∉ binders ∧ ¬h1_1_t = h1_1_x

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