Theorem: FOL.NV.Sub.Var.One.Ind.fastAdmitsAux_and_fastReplaceFree_imp_isFreeSub
Proof state before:
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✝))
Tactic:
all_goals
  simp only [Rec.fastAdmitsAux] at h1
  simp only [Rec.fastReplaceFree]
Proof state after:
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✝))