Theorem: FOL.NV.Holds_coincide_Var
Proof state before:
case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (pred_const_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (pred_const_ a✝¹ a✝)

case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (pred_var_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (pred_var_ a✝¹ a✝)

case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (eq_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (eq_ a✝¹ a✝)

case cons.true_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v true_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) true_ ↔ Holds D I V' (head✝ :: tail✝) true_

case cons.false_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v false_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) false_ ↔ Holds D I V' (head✝ :: tail✝) false_

case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝ : Formula
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝.not_ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝.not_ ↔ Holds D I V' (head✝ :: tail✝) a✝.not_

case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.imp_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.imp_ a✝)

case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.and_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.and_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.and_ a✝)

case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.or_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.or_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.or_ a✝)

case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (a✝¹.iff_ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔ Holds D I V' (head✝ :: tail✝) (a✝¹.iff_ a✝)

case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (forall_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (forall_ a✝¹ a✝)

case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (exists_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (exists_ a✝¹ a✝)

case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v (def_ a✝¹ a✝) → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) (def_ a✝¹ a✝) ↔ Holds D I V' (head✝ :: tail✝) (def_ a✝¹ a✝)
Tactic:
all_goals
  simp only [isFreeIn] at h1
  simp only [Holds]
Proof state after:
case cons.pred_const_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v = V' v
⊢ I.pred_const_ a✝¹ (List.map V a✝) ↔ I.pred_const_ a✝¹ (List.map V' a✝)

case cons.pred_var_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : PredName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v = V' v
⊢ I.pred_var_ a✝¹ (List.map V a✝) ↔ I.pred_var_ a✝¹ (List.map V' a✝)

case cons.eq_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : VarName
V V' : VarAssignment D
h1 : ∀ (v : VarName), v = a✝¹ ∨ v = a✝ → V v = V' v
⊢ V a✝¹ = V a✝ ↔ V' a✝¹ = V' a✝

case cons.not_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝ : Formula
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝ → V v = V' v
⊢ ¬Holds D I V (head✝ :: tail✝) a✝ ↔ ¬Holds D I V' (head✝ :: tail✝) a✝

case cons.imp_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ → Holds D I V (head✝ :: tail✝) a✝ ↔
    Holds D I V' (head✝ :: tail✝) a✝¹ → Holds D I V' (head✝ :: tail✝) a✝

case cons.and_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∧ Holds D I V (head✝ :: tail✝) a✝ ↔
    Holds D I V' (head✝ :: tail✝) a✝¹ ∧ Holds D I V' (head✝ :: tail✝) a✝

case cons.or_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ Holds D I V (head✝ :: tail✝) a✝¹ ∨ Holds D I V (head✝ :: tail✝) a✝ ↔
    Holds D I V' (head✝ :: tail✝) a✝¹ ∨ Holds D I V' (head✝ :: tail✝) a✝

case cons.iff_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝¹ → V v = V' v) →
      (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝¹)
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), isFreeIn v a✝¹ ∨ isFreeIn v a✝ → V v = V' v
⊢ (Holds D I V (head✝ :: tail✝) a✝¹ ↔ Holds D I V (head✝ :: tail✝) a✝) ↔
    (Holds D I V' (head✝ :: tail✝) a✝¹ ↔ Holds D I V' (head✝ :: tail✝) a✝)

case cons.forall_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → V v = V' v
⊢ (∀ (d : D), Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
    ∀ (d : D), Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) a✝

case cons.exists_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  ∀ (V V' : VarAssignment D),
    (∀ (v : VarName), isFreeIn v a✝ → V v = V' v) → (Holds D I V (head✝ :: tail✝) a✝ ↔ Holds D I V' (head✝ :: tail✝) a✝)
V V' : VarAssignment D
h1 : ∀ (v : VarName), ¬v = a✝¹ ∧ isFreeIn v a✝ → V v = V' v
⊢ (∃ d, Holds D I (Function.updateITE V a✝¹ d) (head✝ :: tail✝) a✝) ↔
    ∃ d, Holds D I (Function.updateITE V' a✝¹ d) (head✝ :: tail✝) a✝

case cons.def_
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : VarAssignment D) (F : Formula),
    (∀ (v : VarName), isFreeIn v F → V v = V' v) → (Holds D I V tail✝ F ↔ Holds D I V' tail✝ F)
a✝¹ : DefName
a✝ : List VarName
V V' : VarAssignment D
h1 : ∀ v ∈ a✝, V v = V' v
⊢ (if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
      Holds D I (Function.updateListITE V head✝.args (List.map V a✝)) tail✝ head✝.q
    else Holds D I V tail✝ (def_ a✝¹ a✝)) ↔
    if a✝¹ = head✝.name ∧ a✝.length = head✝.args.length then
      Holds D I (Function.updateListITE V' head✝.args (List.map V' a✝)) tail✝ head✝.q
    else Holds D I V' tail✝ (def_ a✝¹ a✝)