internlm/Lean-Github huggingface dataset:
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M tail✝ F ↔ Holds D I V' M tail✝ F)
x : VarName
phi : Formula
phi_ih :
  ∀ (V V' : Valuation D) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs phi →
      (∀ v ∈ vs, V v = V' v) → (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V' M (head✝ :: tail✝) phi)
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : NoMetaVarAndAllFreeInList (x :: vs) phi
a : D
v : VarName
c1 : ¬v = x
a1 : v = x ∨ v ∈ vs
⊢ V v = V' v
Tactic:
tauto
Proof state after:
no goals
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I V' M [] (def_ X xs)
Tactic:
simp only [Holds]
Proof state after:
no goals
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h1 : NoMetaVarAndAllFreeInList vs (def_ X xs)
h2 : ∀ v ∈ vs, V v = V' v
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V' M (E_hd :: E_tl) (def_ X xs)
Tactic:
unfold NoMetaVarAndAllFreeInList at h1
Proof state after:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V' M (E_hd :: E_tl) (def_ X xs)
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ Holds D I V M (E_hd :: E_tl) (def_ X xs) ↔ Holds D I V' M (E_hd :: E_tl) (def_ X xs)
Tactic:
simp only [Holds]
Proof state after:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then
      Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
    else Holds D I V M E_tl (def_ X xs)) ↔
    if X = E_hd.name ∧ xs.length = E_hd.args.length then
      Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F
    else Holds D I V' M E_tl (def_ X xs)
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
⊢ (if X = E_hd.name ∧ xs.length = E_hd.args.length then
      Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F
    else Holds D I V M E_tl (def_ X xs)) ↔
    if X = E_hd.name ∧ xs.length = E_hd.args.length then
      Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F
    else Holds D I V' M E_tl (def_ X xs)
Tactic:
split_ifs
Proof state after:
case pos
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
h✝ : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔
    Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F

case neg
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
h✝ : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I V' M E_tl (def_ X xs)
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ Holds D I (Function.updateListITE V E_hd.args (List.map V xs)) M E_tl E_hd.F ↔
    Holds D I (Function.updateListITE V' E_hd.args (List.map V' xs)) M E_tl E_hd.F
Tactic:
apply E_ih (Function.updateListITE V E_hd.args (List.map V xs)) (Function.updateListITE V' E_hd.args (List.map V' xs)) E_hd.F E_hd.args E_hd.nf
Proof state after:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ v ∈ E_hd.args,
    Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
⊢ ∀ v ∈ E_hd.args,
    Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v
Tactic:
intro v a1
Proof state after:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ Function.updateListITE V E_hd.args (List.map V xs) v = Function.updateListITE V' E_hd.args (List.map V' xs) v
Tactic:
apply Function.updateListITE_fun_coincide_mem_eq_len
Proof state after:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ ∀ v ∈ xs, V v = V' v

case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ v ∈ E_hd.args

case h3
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ E_hd.args.length = xs.length
Theorem: MM0.holds_coincide_var
Proof state before:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ ∀ v ∈ xs, V v = V' v
Tactic:
tauto
Proof state after:
no goals
Theorem: MM0.holds_coincide_var
Proof state before:
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ v ∈ E_hd.args
Tactic:
exact a1
Proof state after:
no goals
Theorem: MM0.holds_coincide_var
Proof state before:
case h3
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : X = E_hd.name ∧ xs.length = E_hd.args.length
v : VarName
a1 : v ∈ E_hd.args
⊢ E_hd.args.length = xs.length
Tactic:
tauto
Proof state after:
no goals
Theorem: MM0.holds_coincide_var
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ Holds D I V M E_tl (def_ X xs) ↔ Holds D I V' M E_tl (def_ X xs)
Tactic:
apply E_ih V V' (def_ X xs) vs
Proof state after:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ NoMetaVarAndAllFreeInList vs (def_ X xs)

case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ v ∈ vs, V v = V' v
Theorem: MM0.holds_coincide_var
Proof state before:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ NoMetaVarAndAllFreeInList vs (def_ X xs)
Tactic:
unfold NoMetaVarAndAllFreeInList
Proof state after:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ xs ⊆ vs
Theorem: MM0.holds_coincide_var
Proof state before:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ xs ⊆ vs
Tactic:
exact h1
Proof state after:
no goals
Theorem: MM0.holds_coincide_var
Proof state before:
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E_hd : Definition
E_tl : List Definition
E_ih :
  ∀ (V V' : Valuation D) (F : Formula) (vs : List VarName),
    NoMetaVarAndAllFreeInList vs F → (∀ v ∈ vs, V v = V' v) → (Holds D I V M E_tl F ↔ Holds D I V' M E_tl F)
X : DefName
xs : List VarName
V V' : Valuation D
vs : List VarName
h2 : ∀ v ∈ vs, V v = V' v
h1 : xs ⊆ vs
c1 : ¬(X = E_hd.name ∧ xs.length = E_hd.args.length)
⊢ ∀ v ∈ vs, V v = V' v
Tactic:
exact h2
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
E : Env
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M E F ↔ Holds D I V M' E F
Tactic:
induction E generalizing F V M M'
Proof state after:
case nil
D : Type
I : Interpretation D
V : Valuation D
M M' : MetaValuation D
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M [] F ↔ Holds D I V M' [] F

case cons
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
F : Formula
h1 : ∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) F ↔ Holds D I V M' (head✝ :: tail✝) F
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
X : DefName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (def_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M [] (def_ X xs) ↔ Holds D I V M' [] (def_ X xs)
Tactic:
case nil.def_ X xs =>
  simp only [Holds]
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (meta_var_ X).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X)
Tactic:
case meta_var_ X =>
  unfold Formula.metaVarSet at h1
  simp at h1
  simp only [Holds]
  exact h1 V
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : PredName
xs : List VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (pred_ X xs).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (pred_ X xs) ↔ Holds D I V M' (head✝ :: tail✝) (pred_ X xs)
Tactic:
case pred_ X xs =>
  simp only [Holds]
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x y : VarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (eq_ x y).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (eq_ x y) ↔ Holds D I V M' (head✝ :: tail✝) (eq_ x y)
Tactic:
case eq_ x y =>
  simp only [Holds]
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ true_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) true_ ↔ Holds D I V M' (head✝ :: tail✝) true_
Tactic:
case true_ =>
  simp only [Holds]
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
phi : Formula
phi_ih :
  ∀ (V : Valuation D) (M M' : MetaValuation D),
    (∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
      (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ phi.not_.metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) phi.not_ ↔ Holds D I V M' (head✝ :: tail✝) phi.not_
Tactic:
case not_ phi phi_ih =>
  unfold Formula.metaVarSet at h1
  simp only [Holds]
  congr! 1
  exact phi_ih V M M' h1
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
x : VarName
phi : Formula
phi_ih :
  ∀ (V : Valuation D) (M M' : MetaValuation D),
    (∀ (V' : Valuation D), ∀ X ∈ phi.metaVarSet, M X V' ↔ M' X V') →
      (Holds D I V M (head✝ :: tail✝) phi ↔ Holds D I V M' (head✝ :: tail✝) phi)
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X ∈ (forall_ x phi).metaVarSet, M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (forall_ x phi) ↔ Holds D I V M' (head✝ :: tail✝) (forall_ x phi)
Tactic:
case forall_ x phi phi_ih =>
  unfold Formula.metaVarSet at h1
  simp only [Holds]
  apply forall_congr'
  intro a
  exact phi_ih (Function.updateITE V x a) M M' h1
Proof state after:
no goals
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ (meta_var_ X).metaVarSet, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X)
Tactic:
unfold Formula.metaVarSet at h1
Proof state after:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ {X}, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X)
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), ∀ X_1 ∈ {X}, M X_1 V' ↔ M' X_1 V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X)
Tactic:
simp at h1
Proof state after:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X)
Theorem: MM0.holds_coincide_meta_var
Proof state before:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ Holds D I V M (head✝ :: tail✝) (meta_var_ X) ↔ Holds D I V M' (head✝ :: tail✝) (meta_var_ X)
Tactic:
simp only [Holds]
Proof state after:
D : Type
I : Interpretation D
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : Valuation D) (M M' : MetaValuation D) (F : Formula),
    (∀ (V' : Valuation D), ∀ X ∈ F.metaVarSet, M X V' ↔ M' X V') → (Holds D I V M tail✝ F ↔ Holds D I V M' tail✝ F)
X : MetaVarName
V : Valuation D
M M' : MetaValuation D
h1 : ∀ (V' : Valuation D), M X V' ↔ M' X V'
⊢ M X V ↔ M' X V