internlm/Lean-Github huggingface dataset:
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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 :
  v ∉ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet ∪ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = d
Tactic:
simp only [Finset.mem_union] at s2
Proof state after:
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 :
  ¬(v ∈ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet ∨
      v ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ))
⊢ V'' v = d
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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 :
  ¬(v ∈ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet ∨
      v ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ))
⊢ V'' v = d
Tactic:
push_neg at s2
Proof state after:
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 :
  v ∉ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet ∧
    v ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = d
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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 :
  v ∉ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet ∧
    v ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = d
Tactic:
cases s2
Proof state after:
case intro
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))
left✝ : v ∉ (Var.All.Rec.Fresh.sub (Function.updateITE σ x x) c phi).freeVarSet
right✝ : v ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = d
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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
Tactic:
case _ s2_left s2_right =>
  contradiction
Proof state after:
no goals
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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
Tactic:
contradiction
Proof state after:
no goals
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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)
c2 :
  ¬v = fresh x c (Finset.image (Function.updateITE σ x x) phi.freeVarSet ∪ phi.predVarSet.biUnion (predVarFreeVarSet τ))
⊢ V'' v = V v
Tactic:
exact h2 v a1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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)
⊢ V'' v = Function.updateITE V x d v
Tactic:
simp only [Finset.mem_union] at c1
Proof state after:
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 ∨ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ))
⊢ V'' v = Function.updateITE V x d v
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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 ∨ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ))
⊢ V'' v = Function.updateITE V x d v
Tactic:
push_neg at c1
Proof state after:
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 ∧ x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = Function.updateITE V x d v
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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 ∧ x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = Function.updateITE V x d v
Tactic:
cases c1
Proof state after:
case intro
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)
left✝ : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
right✝ : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = Function.updateITE V x d v
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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_left : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
c1_right : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = Function.updateITE V x d v
Tactic:
case _ c1_left c1_right =>
  have s1 : ¬ v = x
  intro contra
  apply c1_right
  subst contra
  exact a1
  simp only [Function.updateITE]
  simp only [if_neg s1]
  exact h2 v a1
Proof state after:
no goals
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
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_left : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
c1_right : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ V'' v = Function.updateITE V x d v
Tactic:
have s1 : ¬ v = x
Proof state after:
case s1
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_left : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
c1_right : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ ¬v = x

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_left : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
c1_right : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
s1 : ¬v = x
⊢ V'' v = Function.updateITE V x d v
Theorem: FOL.NV.Sub.Pred.All.Rec.Option.Fresh.substitution_theorem_aux
Proof state before:
case s1
D : Type
I : Interpretation D
V'' : VarAssignment D
E : Env
c : Char
τ : PredName → ℕ → Option (List VarName × Formula)
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
h2 : ∀ x ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ), V'' x = V x
d : D
v : VarName
a1 : v ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ)
h1 : ∀ (x : VarName), ¬x = v ∧ isFreeIn x phi → V' x = V (σ x)
c1_left : v ∉ Finset.image (Function.updateITE σ v v) phi.freeVarSet
c1_right : v ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
⊢ v ∈ phi.predVarSet.biUnion (predVarFreeVarSet τ)

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_left : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
c1_right : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
s1 : ¬v = x
⊢ V'' v = Function.updateITE V x d v
Tactic:
exact a1
Proof state after:
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_left : x ∉ Finset.image (Function.updateITE σ x x) phi.freeVarSet
c1_right : x ∉ phi.predVarSet.biUnion (predVarFreeVarSet τ)
s1 : ¬v = x
⊢ V'' v = Function.updateITE V x d v