Theorem: FOL.NV.Sub.Var.One.Rec.substitution_theorem_aux
Proof state before:
case cons
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))
V : VarAssignment D
binders : Finset VarName
F : Formula
h1 : fastAdmitsAux v t binders F
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) F ↔ Holds D I V (head✝ :: tail✝) (fastReplaceFree v t F)
Tactic:
induction F generalizing binders V
Proof state after:
case cons.pred_const_
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))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (pred_const_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (pred_const_ a✝¹ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (pred_const_ a✝¹ a✝))

case cons.pred_var_
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))
a✝¹ : PredName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (pred_var_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (pred_var_ a✝¹ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (pred_var_ a✝¹ a✝))

case cons.eq_
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))
a✝¹ a✝ : VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (eq_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (eq_ a✝¹ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (eq_ a✝¹ a✝))

case cons.true_
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))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders true_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) true_ ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t true_)

case cons.false_
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))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders false_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) false_ ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t false_)

case cons.not_
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))
a✝ : Formula
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders a✝.not_
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝.not_ ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝.not_)

case cons.imp_
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))
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝¹ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.imp_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.imp_ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.imp_ a✝))

case cons.and_
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))
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝¹ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.and_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.and_ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.and_ a✝))

case cons.or_
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))
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝¹ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.or_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.or_ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.or_ a✝))

case cons.iff_
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))
a✝¹ a✝ : Formula
a_ih✝¹ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝¹ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝¹ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝¹))
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (a✝¹.iff_ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (a✝¹.iff_ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (a✝¹.iff_ a✝))

case cons.forall_
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))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (forall_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (forall_ a✝¹ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (forall_ a✝¹ a✝))

case cons.exists_
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))
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    fastAdmitsAux v t binders a✝ →
      (∀ v ∉ binders, V' v = V v) →
        (Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) a✝ ↔
          Holds D I V (head✝ :: tail✝) (fastReplaceFree v t a✝))
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (exists_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (exists_ a✝¹ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (exists_ a✝¹ a✝))

case cons.def_
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))
a✝¹ : DefName
a✝ : List VarName
V : VarAssignment D
binders : Finset VarName
h1 : fastAdmitsAux v t binders (def_ a✝¹ a✝)
h2 : ∀ v ∉ binders, V' v = V v
⊢ Holds D I (Function.updateITE V v (V' t)) (head✝ :: tail✝) (def_ a✝¹ a✝) ↔
    Holds D I V (head✝ :: tail✝) (fastReplaceFree v t (def_ a✝¹ a✝))