internlm/Lean-Github huggingface dataset:
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔ Holds D I V E_ref (replace P zs H (phi.iff_ psi))
Tactic:
simp only [replace]
Proof state after:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔
    Holds D I V E_ref ((replace P zs H phi).iff_ (replace P zs H psi))
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (phi.iff_ psi) ↔
    Holds D I V E_ref ((replace P zs H phi).iff_ (replace P zs H psi))
Tactic:
simp only [Holds]
Proof state after:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
    (Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi))
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders phi ∧ admitsAux P zs H binders psi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
    (Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi))
Tactic:
cases h1
Proof state after:
case intro
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
left✝ : admitsAux P zs H binders phi
right✝ : admitsAux P zs H binders psi
⊢ (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D (I' D I V' E_ref P zs H) V E_ref psi) ↔
    (Holds D I V E_ref (replace P zs H phi) ↔ Holds D I V E_ref (replace P zs H psi))
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
case a.h.e'_1.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi)
Tactic:
exact phi_ih V binders h1_left h2
Proof state after:
no goals
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
case a.h.e'_2.a
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
phi psi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
psi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders psi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi))
V : VarAssignment D
binders : Finset VarName
h2 : ∀ x ∉ binders, V x = V' x
h1_left : admitsAux P zs H binders phi
h1_right : admitsAux P zs H binders psi
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref psi ↔ Holds D I V E_ref (replace P zs H psi)
Tactic:
exact psi_ih V binders h1_right h2
Proof state after:
no goals
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H binders (exists_ x phi)
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi))
Tactic:
simp only [admitsAux] at h1
Proof state after:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi))
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (replace P zs H (exists_ x phi))
Tactic:
simp only [replace]
Proof state after:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (exists_ x (replace P zs H phi))
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ Holds D (I' D I V' E_ref P zs H) V E_ref (exists_ x phi) ↔ Holds D I V E_ref (exists_ x (replace P zs H phi))
Tactic:
simp only [Holds]
Proof state after:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d, Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi) ↔
    ∃ d, Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi)
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ (∃ d, Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi) ↔
    ∃ d, Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi)
Tactic:
first | apply forall_congr' | apply exists_congr
Proof state after:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
    Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x a) E_ref phi ↔
      Holds D I (Function.updateITE V x a) E_ref (replace P zs H phi)
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
⊢ ∀ (a : D),
    Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x a) E_ref phi ↔
      Holds D I (Function.updateITE V x a) E_ref (replace P zs H phi)
Tactic:
intro d
Proof state after:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi ↔
    Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi)
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ Holds D (I' D I V' E_ref P zs H) (Function.updateITE V x d) E_ref phi ↔
    Holds D I (Function.updateITE V x d) E_ref (replace P zs H phi)
Tactic:
apply phi_ih (Function.updateITE V x d) (binders ∪ {x}) h1
Proof state after:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ ∀ x_1 ∉ binders ∪ {x}, Function.updateITE V x d x_1 = V' x_1
Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_theorem_aux
Proof state before:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
⊢ ∀ x_1 ∉ binders ∪ {x}, Function.updateITE V x d x_1 = V' x_1
Tactic:
intro v a1
Proof state after:
case h
D : Type
I : Interpretation D
V' : VarAssignment D
P : PredName
zs : List VarName
H : Formula
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula) (binders : Finset VarName),
    admitsAux P zs H binders F →
      (∀ x ∉ binders, V x = V' x) →
        let E_ref := tail✝;
        Holds D (I' D I V' E_ref P zs H) V E_ref F ↔ Holds D I V E_ref (replace P zs H F)
E_ref : Env := head✝ :: tail✝
x : VarName
phi : Formula
phi_ih :
  ∀ (V : VarAssignment D) (binders : Finset VarName),
    admitsAux P zs H binders phi →
      (∀ x ∉ binders, V x = V' x) →
        (Holds D (I' D I V' E_ref P zs H) V E_ref phi ↔ Holds D I V E_ref (replace P zs H phi))
V : VarAssignment D
binders : Finset VarName
h1 : admitsAux P zs H (binders ∪ {x}) phi
h2 : ∀ x ∉ binders, V x = V' x
d : D
v : VarName
a1 : v ∉ binders ∪ {x}
⊢ Function.updateITE V x d v = V' v