Theorem: FOL.NV.replace_empty_Holds
Proof state before:
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula),
    ¬isFreeIn v F →
      ¬isBoundIn v F →
        (Holds D I (Function.updateITE V u a) tail✝ F ↔
          Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D),
    ¬isFreeIn v phi →
      ¬isBoundIn v phi →
        (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
          Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
  ∀ (V : VarAssignment D),
    ¬isFreeIn v psi →
      ¬isBoundIn v psi →
        (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
          Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
      Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi) ↔
    (Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi) ↔
      Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
Tactic:
congr! 1
Proof state after:
case a.h.e'_1.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula),
    ¬isFreeIn v F →
      ¬isBoundIn v F →
        (Holds D I (Function.updateITE V u a) tail✝ F ↔
          Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D),
    ¬isFreeIn v phi →
      ¬isBoundIn v phi →
        (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
          Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
  ∀ (V : VarAssignment D),
    ¬isFreeIn v psi →
      ¬isBoundIn v psi →
        (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
          Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
    Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi)

case a.h.e'_2.a
D : Type
I : Interpretation D
u v : VarName
a : D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula),
    ¬isFreeIn v F →
      ¬isBoundIn v F →
        (Holds D I (Function.updateITE V u a) tail✝ F ↔
          Holds D I (Function.updateITE V v a) tail✝ (Sub.Var.One.Rec.fastReplaceFree u v F))
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D),
    ¬isFreeIn v phi →
      ¬isBoundIn v phi →
        (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) phi ↔
          Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v phi))
psi_ih :
  ∀ (V : VarAssignment D),
    ¬isFreeIn v psi →
      ¬isBoundIn v psi →
        (Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
          Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi))
V : VarAssignment D
h1_left : ¬isFreeIn v phi
h1_right : ¬isFreeIn v psi
h2_left : ¬isBoundIn v phi
h2_right : ¬isBoundIn v psi
⊢ Holds D I (Function.updateITE V u a) (head✝ :: tail✝) psi ↔
    Holds D I (Function.updateITE V v a) (head✝ :: tail✝) (Sub.Var.One.Rec.fastReplaceFree u v psi)