Theorem: FOL.NV.Sub.Var.One.Rec.not_isBoundIn_imp_admitsAux
Proof state before:
F : Formula
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u F
h2 : u ∉ binders
⊢ admitsAux v u binders F
Tactic:
induction F generalizing binders
Proof state after:
case pred_const_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_const_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_const_ a✝¹ a✝)

case pred_var_
v u : VarName
a✝¹ : PredName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (pred_var_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (pred_var_ a✝¹ a✝)

case eq_
v u a✝¹ a✝ : VarName
binders : Finset VarName
h1 : ¬isBoundIn u (eq_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (eq_ a✝¹ a✝)

case true_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u true_
h2 : u ∉ binders
⊢ admitsAux v u binders true_

case false_
v u : VarName
binders : Finset VarName
h1 : ¬isBoundIn u false_
h2 : u ∉ binders
⊢ admitsAux v u binders false_

case not_
v u : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u a✝.not_
h2 : u ∉ binders
⊢ admitsAux v u binders a✝.not_

case imp_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.imp_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.imp_ a✝)

case and_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.and_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.and_ a✝)

case or_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.or_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.or_ a✝)

case iff_
v u : VarName
a✝¹ a✝ : Formula
a_ih✝¹ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝¹ → u ∉ binders → admitsAux v u binders a✝¹
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (a✝¹.iff_ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (a✝¹.iff_ a✝)

case forall_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (forall_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (forall_ a✝¹ a✝)

case exists_
v u a✝¹ : VarName
a✝ : Formula
a_ih✝ : ∀ (binders : Finset VarName), ¬isBoundIn u a✝ → u ∉ binders → admitsAux v u binders a✝
binders : Finset VarName
h1 : ¬isBoundIn u (exists_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (exists_ a✝¹ a✝)

case def_
v u : VarName
a✝¹ : DefName
a✝ : List VarName
binders : Finset VarName
h1 : ¬isBoundIn u (def_ a✝¹ a✝)
h2 : u ∉ binders
⊢ admitsAux v u binders (def_ a✝¹ a✝)