case inl h1 =>
  contradiction
D : Type
I : Interpretation D
V' : VarAssignment D
v t : VarName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName) (F : Formula),
    fastAdmitsAux v t binders F →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) tail✝ F ↔ Holds D I V tail✝ (fastReplaceFree v t F))
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders phi →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) phi ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t phi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ v ∉ binders, V' v = V v
c1 : ¬v = x
d : D
h1 : v = x
⊢ Holds D I (Function.updateITE (Function.updateITE V v (V' t)) x d) (head✝ :: tail✝) phi ↔
    Holds D I (Function.updateITE V x d) (head✝ :: tail✝) (fastReplaceFree v t phi)
no goals