Theorem: FOL.NV.Sub.Var.One.Rec.replaceFreeAux_admitsAux
Proof state before:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_const_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_const_ a✝¹ a✝))

case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (pred_var_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (pred_var_ a✝¹ a✝))

case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬occursIn t (eq_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (eq_ a✝¹ a✝))

case true_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t true_
⊢ admitsAux t v binders (replaceFreeAux v t binders true_)

case false_
v t : VarName
binders : Finset VarName
h1 : ¬occursIn t false_
⊢ admitsAux t v binders (replaceFreeAux v t binders false_)

case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝.not_
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝.not_)

case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.imp_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.imp_ a✝))

case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.and_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.and_ a✝))

case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.or_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.or_ a✝))

case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (a✝¹.iff_ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (a✝¹.iff_ a✝))

case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (forall_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (forall_ a✝¹ a✝))

case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t (exists_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (exists_ a✝¹ a✝))

case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬occursIn t (def_ a✝¹ a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders (def_ a✝¹ a✝))
Tactic:
all_goals
  simp only [occursIn] at h1
  simp only [replaceFreeAux]
  simp only [admitsAux]
Proof state after:
case pred_const_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders

case pred_var_
v t : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders

case eq_
v t a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ t = a✝)
⊢ ((t = if v = a✝¹ ∧ a✝¹ ∉ binders then t else a✝¹) ∨ t = if v = a✝ ∧ a✝ ∉ binders then t else a✝) ∧ t ∉ binders →
    v ∉ binders

case not_
v t : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬occursIn t a✝
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝)

case imp_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)

case and_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)

case or_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)

case iff_
v t : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬occursIn t a✝¹ → admitsAux t v binders (replaceFreeAux v t binders a✝¹)
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(occursIn t a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v binders (replaceFreeAux v t binders a✝¹) ∧ admitsAux t v binders (replaceFreeAux v t binders a✝)

case forall_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)

case exists_
v t a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬occursIn t a✝ → admitsAux t v binders (replaceFreeAux v t binders a✝)
binders : Finset VarName
h1 : ¬(t = a✝¹ ∨ occursIn t a✝)
⊢ admitsAux t v (binders ∪ {a✝¹}) (replaceFreeAux v t (binders ∪ {a✝¹}) a✝)

case def_
v t : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : t ∉ a✝
⊢ t ∈ List.map (fun x => if v = x ∧ x ∉ binders then t else x) a✝ ∧ t ∉ binders → v ∉ binders