Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
case s2.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x (def_ X xs) → V' x = V (σ x)) →
      (∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E_tl τ) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map σ xs)))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 : ∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
c2 : X = E_hd.name ∧ (List.map σ xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
⊢ Function.updateListITE V' E_hd.args (List.map (V ∘ σ) xs) x =
    Function.updateListITE V E_hd.args (List.map (V ∘ σ) xs) x

D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x (def_ X xs) → V' x = V (σ x)) →
      (∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E_tl τ) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map σ xs)))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 : ∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
c2 : X = E_hd.name ∧ (List.map σ xs).length = E_hd.args.length
s2 :
  Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ σ) xs)) E_tl E_hd.q ↔
    Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ σ) xs)) E_tl E_hd.q
⊢ Holds D (I' D I V'' (E_hd :: E_tl) τ) (Function.updateListITE V' E_hd.args (List.map (V ∘ σ) xs)) E_tl E_hd.q ↔
    Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map σ xs))) E_tl E_hd.q
Tactic:
apply Function.updateListITE_map_mem_ext
Proof state after:
case s2.h1.h1
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x (def_ X xs) → V' x = V (σ x)) →
      (∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E_tl τ) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map σ xs)))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 : ∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
c2 : X = E_hd.name ∧ (List.map σ xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
⊢ ∀ y ∈ xs, (V ∘ σ) y = (V ∘ σ) y

case s2.h1.h2
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x (def_ X xs) → V' x = V (σ x)) →
      (∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E_tl τ) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map σ xs)))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 : ∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
c2 : X = E_hd.name ∧ (List.map σ xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
⊢ E_hd.args.length = xs.length

case s2.h1.h3
D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x (def_ X xs) → V' x = V (σ x)) →
      (∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E_tl τ) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map σ xs)))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 : ∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
c2 : X = E_hd.name ∧ (List.map σ xs).length = E_hd.args.length
x : VarName
a1 : isFreeIn x E_hd.q
⊢ x ∈ E_hd.args

D : Type
I : Interpretation D
V'' : VarAssignment D
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
X : DefName
xs : List VarName
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : VarAssignment D) (σ : VarName → VarName),
    (∀ (x : VarName), isFreeIn x (def_ X xs) → V' x = V (σ x)) →
      (∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x) →
        (Holds D (I' D I V'' E_tl τ) V' E_tl (def_ X xs) ↔ Holds D I V E_tl (def_ X (List.map σ xs)))
V V' : VarAssignment D
σ : VarName → VarName
h1 : ∀ x ∈ xs, V' x = V (σ x)
h2 : ∀ x ∈ (def_ X xs).predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
c2 : X = E_hd.name ∧ (List.map σ xs).length = E_hd.args.length
s2 :
  Holds D I (Function.updateListITE V' E_hd.args (List.map (V ∘ σ) xs)) E_tl E_hd.q ↔
    Holds D I (Function.updateListITE V E_hd.args (List.map (V ∘ σ) xs)) E_tl E_hd.q
⊢ Holds D (I' D I V'' (E_hd :: E_tl) τ) (Function.updateListITE V' E_hd.args (List.map (V ∘ σ) xs)) E_tl E_hd.q ↔
    Holds D I (Function.updateListITE V E_hd.args (List.map V (List.map σ xs))) E_tl E_hd.q