contradiction
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
x : VarName
phi : Formula
ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x phi → V' x = V (σ x)) →
      (∀ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E τ) V' E phi ↔ Holds D I V E (subAux c τ σ phi))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ (x_1 : VarName), ¬x_1 = x ∧ isFreeIn x_1 phi → V' x_1 = V (σ x_1)
h2 : ∀ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
d : D
v : VarName
a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ)
c1 : x ∈ Finset.image (Function.updateITE σ x x) phi.freeVarSet ∪ phi.predVarSet.biUnion (predVarFreeVarSet τ)
s1 :
  (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet =
    Finset.image (Function.updateITE σ x x) phi.freeVarSet
c2 :
  v =
    fresh x c
      ((Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet ∪
        phi.predVarSet.biUnion (predVarFreeVarSet τ))
s2_left : v ∉ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet
s2_right : v ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = d
no goals