internlm/Lean-Github huggingface dataset:
Theorem: MM0.lem_1
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
⊢ ∀ (V : Valuation D) (d : D), Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
Tactic:
intro V d
Proof state after:
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
Theorem: MM0.lem_1
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h1 : ↑σ ∘ σ' = id ∧ σ' ∘ ↑σ = id
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
Tactic:
cases h1
Proof state after:
case intro
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
left✝ : ↑σ ∘ σ' = id
right✝ : σ' ∘ ↑σ = id
⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
Theorem: MM0.lem_1
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE V v d ∘ σ') M E (τ X)
Tactic:
simp only [Function.updateITE_comp_right σ' σ.1 V v d h1_left h1_right]
Proof state after:
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE (V ∘ σ') (↑σ v) d) M E (τ X)
Theorem: MM0.lem_1
Proof state before:
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ Holds D I (V ∘ σ') M E (τ X) ↔ Holds D I (Function.updateITE (V ∘ σ') (↑σ v) d) M E (τ X)
Tactic:
apply not_free_imp_is_not_free D I M E (τ X) Γ' (σ.1 v)
Proof state after:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ NotFree Γ' (↑σ v) (τ X)

case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ ∀ (X : MetaVarName), (↑σ v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) (↑σ v)
Theorem: MM0.lem_1
Proof state before:
case h1
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ NotFree Γ' (↑σ v) (τ X)
Tactic:
exact h3 v X a1
Proof state after:
no goals
Theorem: MM0.lem_1
Proof state before:
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
⊢ ∀ (X : MetaVarName), (↑σ v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) (↑σ v)
Tactic:
intro X' a2
Proof state after:
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
X' : MetaVarName
a2 : (↑σ v, X') ∈ Γ'
⊢ IsNotFree D I M E (meta_var_ X') (↑σ v)
Theorem: MM0.lem_1
Proof state before:
case h2
D : Type
I : Interpretation D
M : MetaValuation D
E : Env
Γ Γ' : List (VarName × MetaVarName)
σ : Instantiation
σ' : VarName → VarName
τ : MetaInstantiation
h2 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ' → IsNotFree D I M E (meta_var_ X) v
h3 : ∀ (v : VarName) (X : MetaVarName), (v, X) ∈ Γ → NotFree Γ' (↑σ v) (τ X)
v : VarName
X : MetaVarName
a1 : (v, X) ∈ Γ
V : Valuation D
d : D
h1_left : ↑σ ∘ σ' = id
h1_right : σ' ∘ ↑σ = id
X' : MetaVarName
a2 : (↑σ v, X') ∈ Γ'
⊢ IsNotFree D I M E (meta_var_ X') (↑σ v)
Tactic:
exact h2 (σ.1 v) X' a2
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
F : Formula
h1 : IsMetaVarOrAllDefInEnv E F
h2 : ∀ X ∈ F.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ F)
Tactic:
induction F
Proof state after:
case meta_var_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝ : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ a✝)
h2 : ∀ X ∈ (meta_var_ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ a✝))

case pred_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝¹ : PredName
a✝ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ a✝¹ a✝)
h2 : ∀ X ∈ (pred_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (pred_ a✝¹ a✝))

case eq_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝¹ a✝ : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ a✝¹ a✝)
h2 : ∀ X ∈ (eq_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (eq_ a✝¹ a✝))

case true_
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ true_)

case not_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝ : Formula
a_ih✝ :
  IsMetaVarOrAllDefInEnv E a✝ →
    (∀ X ∈ a✝.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝)
h1 : IsMetaVarOrAllDefInEnv E a✝.not_
h2 : ∀ X ∈ a✝.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ a✝.not_)

case imp_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝¹ a✝ : Formula
a_ih✝¹ :
  IsMetaVarOrAllDefInEnv E a✝¹ →
    (∀ X ∈ a✝¹.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝¹)
a_ih✝ :
  IsMetaVarOrAllDefInEnv E a✝ →
    (∀ X ∈ a✝.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝)
h1 : IsMetaVarOrAllDefInEnv E (a✝¹.imp_ a✝)
h2 : ∀ X ∈ (a✝¹.imp_ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (a✝¹.imp_ a✝))

case forall_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝¹ : VarName
a✝ : Formula
a_ih✝ :
  IsMetaVarOrAllDefInEnv E a✝ →
    (∀ X ∈ a✝.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ a✝)
h1 : IsMetaVarOrAllDefInEnv E (forall_ a✝¹ a✝)
h2 : ∀ X ∈ (forall_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ a✝¹ a✝))

case def_
E : Env
σ : Instantiation
τ : MetaInstantiation
a✝¹ : DefName
a✝ : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ a✝¹ a✝)
h2 : ∀ X ∈ (def_ a✝¹ a✝).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ a✝¹ a✝))
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : ∀ X_1 ∈ (meta_var_ X).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
Tactic:
case meta_var_ X =>
  unfold Formula.metaVarSet at h2
  simp at h2
  unfold sub
  exact h2
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (pred_ X xs))
Tactic:
case pred_ X xs =>
  unfold sub
  unfold IsMetaVarOrAllDefInEnv
  simp only
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (eq_ x y))
Tactic:
case eq_ x y =>
  unfold sub
  unfold IsMetaVarOrAllDefInEnv
  simp only
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ true_)
Tactic:
case true_ =>
  unfold sub
  unfold IsMetaVarOrAllDefInEnv
  simp only
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi.not_
h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
Tactic:
case not_ phi phi_ih =>
  unfold IsMetaVarOrAllDefInEnv at h1
  unfold Formula.metaVarSet at h2
  unfold sub
  unfold IsMetaVarOrAllDefInEnv
  exact phi_ih h1 h2
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
x : VarName
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E (forall_ x phi)
h2 : ∀ X ∈ (forall_ x phi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (forall_ x phi))
Tactic:
case forall_ x phi phi_ih =>
  unfold IsMetaVarOrAllDefInEnv at h1
  unfold Formula.metaVarSet at h2
  unfold sub
  unfold IsMetaVarOrAllDefInEnv
  exact phi_ih h1 h2
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : DefName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (def_ X xs)
h2 : ∀ X_1 ∈ (def_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (def_ X xs))
Tactic:
case def_ X xs =>
  unfold IsMetaVarOrAllDefInEnv at h1
  unfold sub
  unfold IsMetaVarOrAllDefInEnv
  simp
  exact h1
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : ∀ X_1 ∈ (meta_var_ X).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
Tactic:
unfold Formula.metaVarSet at h2
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : ∀ X_1 ∈ {X}, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : ∀ X_1 ∈ {X}, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
Tactic:
simp at h2
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (meta_var_ X))
Tactic:
unfold sub
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (τ X)
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : MetaVarName
h1 : IsMetaVarOrAllDefInEnv E (meta_var_ X)
h2 : IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (τ X)
Tactic:
exact h2
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (pred_ X xs))
Tactic:
unfold sub
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (↑σ) xs))
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ IsMetaVarOrAllDefInEnv E (pred_ X (List.map (↑σ) xs))
Tactic:
unfold IsMetaVarOrAllDefInEnv
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ True
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
X : PredName
xs : List VarName
h1 : IsMetaVarOrAllDefInEnv E (pred_ X xs)
h2 : ∀ X_1 ∈ (pred_ X xs).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X_1)
⊢ True
Tactic:
simp only
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (eq_ x y))
Tactic:
unfold sub
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (eq_ (↑σ x) (↑σ y))
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (eq_ (↑σ x) (↑σ y))
Tactic:
unfold IsMetaVarOrAllDefInEnv
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ True
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
x y : VarName
h1 : IsMetaVarOrAllDefInEnv E (eq_ x y)
h2 : ∀ X ∈ (eq_ x y).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ True
Tactic:
simp only
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ true_)
Tactic:
unfold sub
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E true_
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E true_
Tactic:
unfold IsMetaVarOrAllDefInEnv
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ True
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
h1 : IsMetaVarOrAllDefInEnv E true_
h2 : ∀ X ∈ true_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ True
Tactic:
simp only
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi.not_
h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
Tactic:
unfold IsMetaVarOrAllDefInEnv at h1
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h2 : ∀ X ∈ phi.not_.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
h1 : IsMetaVarOrAllDefInEnv E phi
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
Tactic:
unfold Formula.metaVarSet at h2
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi.not_)
Tactic:
unfold sub
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi).not_
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi).not_
Tactic:
unfold IsMetaVarOrAllDefInEnv
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
h1 : IsMetaVarOrAllDefInEnv E phi
h2 : ∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ phi)
Tactic:
exact phi_ih h1 h2
Proof state after:
no goals
Theorem: MM0.lem_2_a
Proof state before:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
psi_ih :
  IsMetaVarOrAllDefInEnv E psi →
    (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi)
h1 : IsMetaVarOrAllDefInEnv E (phi.imp_ psi)
h2 : ∀ X ∈ (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))
Tactic:
unfold IsMetaVarOrAllDefInEnv at h1
Proof state after:
E : Env
σ : Instantiation
τ : MetaInstantiation
phi psi : Formula
phi_ih :
  IsMetaVarOrAllDefInEnv E phi →
    (∀ X ∈ phi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ phi)
psi_ih :
  IsMetaVarOrAllDefInEnv E psi →
    (∀ X ∈ psi.metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)) → IsMetaVarOrAllDefInEnv E (sub σ τ psi)
h2 : ∀ X ∈ (phi.imp_ psi).metaVarSet, IsMetaVarOrAllDefInEnv E (τ X)
h1 : IsMetaVarOrAllDefInEnv E phi ∧ IsMetaVarOrAllDefInEnv E psi
⊢ IsMetaVarOrAllDefInEnv E (sub σ τ (phi.imp_ psi))