Theorem: FOL.NV.Sub.Var.All.Ind.substitution_theorem_aux
Proof state before:
case s1
D : Type
I : Interpretation D
E : Env
σ : VarName → VarName
binders : Finset VarName
F F' : Formula
σ' : VarName → VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi'
V V' : VarAssignment D
h2 : ∀ v ∈ binders', V v = V' (σ' v)
h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)
h4 : ∀ v ∈ binders', v = σ' v
ih_2 :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) →
      (∀ (v : VarName),
          Function.updateITE σ' x x v ∉ binders' →
            ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) →
        (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) →
          (Holds D I V E phi ↔ Holds D I V' E phi')
v : VarName
a1 : ¬v = x
a2 : v ∈ binders'
contra : σ' v = x
⊢ False

D : Type
I : Interpretation D
E : Env
σ : VarName → VarName
binders : Finset VarName
F F' : Formula
σ' : VarName → VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi'
V V' : VarAssignment D
h2 : ∀ v ∈ binders', V v = V' (σ' v)
h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)
h4 : ∀ v ∈ binders', v = σ' v
ih_2 :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) →
      (∀ (v : VarName),
          Function.updateITE σ' x x v ∉ binders' →
            ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) →
        (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) →
          (Holds D I V E phi ↔ Holds D I V' E phi')
s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x
⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')
Tactic:
apply a1
Proof state after:
case s1
D : Type
I : Interpretation D
E : Env
σ : VarName → VarName
binders : Finset VarName
F F' : Formula
σ' : VarName → VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi'
V V' : VarAssignment D
h2 : ∀ v ∈ binders', V v = V' (σ' v)
h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)
h4 : ∀ v ∈ binders', v = σ' v
ih_2 :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) →
      (∀ (v : VarName),
          Function.updateITE σ' x x v ∉ binders' →
            ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) →
        (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) →
          (Holds D I V E phi ↔ Holds D I V' E phi')
v : VarName
a1 : ¬v = x
a2 : v ∈ binders'
contra : σ' v = x
⊢ v = x

D : Type
I : Interpretation D
E : Env
σ : VarName → VarName
binders : Finset VarName
F F' : Formula
σ' : VarName → VarName
binders' : Finset VarName
x : VarName
phi phi' : Formula
a✝ : IsSubAux (Function.updateITE σ' x x) (binders' ∪ {x}) phi phi'
V V' : VarAssignment D
h2 : ∀ v ∈ binders', V v = V' (σ' v)
h3 : ∀ (v : VarName), σ' v ∉ binders' → V v = V' (σ' v)
h4 : ∀ v ∈ binders', v = σ' v
ih_2 :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), v ∈ binders' ∨ v = x → V v = V' (Function.updateITE σ' x x v)) →
      (∀ (v : VarName),
          Function.updateITE σ' x x v ∉ binders' →
            ¬Function.updateITE σ' x x v = x → V v = V' (Function.updateITE σ' x x v)) →
        (∀ (v : VarName), v ∈ binders' ∨ v = x → v = Function.updateITE σ' x x v) →
          (Holds D I V E phi ↔ Holds D I V' E phi')
s1 : ∀ (v : VarName), ¬v = x → v ∈ binders' → ¬σ' v = x
⊢ Holds D I V E (exists_ x phi) ↔ Holds D I V' E (exists_ x phi')