internlm/Lean-Github huggingface dataset:
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case forallEDF
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ : VExpr
v✝ : VLevel
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
ih1 :
  CtxClosed Γ✝ → ClosedN A✝ (List.length Γ✝) ∧ ClosedN A'✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort u✝) (List.length Γ✝)
ih2 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN body✝ (List.length (A✝ :: Γ✝)) ∧
      ClosedN body'✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
hΓ : CtxClosed Γ✝
⊢ ClosedN (VExpr.forallE A✝ body✝) (List.length Γ✝) ∧
    ClosedN (VExpr.forallE A'✝ body'✝) (List.length Γ✝) ∧ ClosedN (VExpr.sort (VLevel.imax u✝ v✝)) (List.length Γ✝)
Tactic:
let ⟨hA, hA', _⟩ := ih1 hΓ
Proof state after:
case forallEDF
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ : VExpr
v✝ : VLevel
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
ih1 :
  CtxClosed Γ✝ → ClosedN A✝ (List.length Γ✝) ∧ ClosedN A'✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort u✝) (List.length Γ✝)
ih2 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN body✝ (List.length (A✝ :: Γ✝)) ∧
      ClosedN body'✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
hΓ : CtxClosed Γ✝
hA : ClosedN A✝ (List.length Γ✝)
hA' : ClosedN A'✝ (List.length Γ✝)
right✝ : ClosedN (VExpr.sort u✝) (List.length Γ✝)
⊢ ClosedN (VExpr.forallE A✝ body✝) (List.length Γ✝) ∧
    ClosedN (VExpr.forallE A'✝ body'✝) (List.length Γ✝) ∧ ClosedN (VExpr.sort (VLevel.imax u✝ v✝)) (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case forallEDF
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ : VExpr
v✝ : VLevel
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
ih1 :
  CtxClosed Γ✝ → ClosedN A✝ (List.length Γ✝) ∧ ClosedN A'✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort u✝) (List.length Γ✝)
ih2 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN body✝ (List.length (A✝ :: Γ✝)) ∧
      ClosedN body'✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
hΓ : CtxClosed Γ✝
hA : ClosedN A✝ (List.length Γ✝)
hA' : ClosedN A'✝ (List.length Γ✝)
right✝ : ClosedN (VExpr.sort u✝) (List.length Γ✝)
⊢ ClosedN (VExpr.forallE A✝ body✝) (List.length Γ✝) ∧
    ClosedN (VExpr.forallE A'✝ body'✝) (List.length Γ✝) ∧ ClosedN (VExpr.sort (VLevel.imax u✝ v✝)) (List.length Γ✝)
Tactic:
let ⟨hb, hb', _⟩ := ih2 ⟨hΓ, hA⟩
Proof state after:
case forallEDF
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ : VExpr
v✝ : VLevel
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
ih1 :
  CtxClosed Γ✝ → ClosedN A✝ (List.length Γ✝) ∧ ClosedN A'✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort u✝) (List.length Γ✝)
ih2 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN body✝ (List.length (A✝ :: Γ✝)) ∧
      ClosedN body'✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
hΓ : CtxClosed Γ✝
hA : ClosedN A✝ (List.length Γ✝)
hA' : ClosedN A'✝ (List.length Γ✝)
right✝¹ : ClosedN (VExpr.sort u✝) (List.length Γ✝)
hb : ClosedN body✝ (List.length (A✝ :: Γ✝))
hb' : ClosedN body'✝ (List.length (A✝ :: Γ✝))
right✝ : ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
⊢ ClosedN (VExpr.forallE A✝ body✝) (List.length Γ✝) ∧
    ClosedN (VExpr.forallE A'✝ body'✝) (List.length Γ✝) ∧ ClosedN (VExpr.sort (VLevel.imax u✝ v✝)) (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case forallEDF
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ : VExpr
v✝ : VLevel
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
ih1 :
  CtxClosed Γ✝ → ClosedN A✝ (List.length Γ✝) ∧ ClosedN A'✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort u✝) (List.length Γ✝)
ih2 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN body✝ (List.length (A✝ :: Γ✝)) ∧
      ClosedN body'✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
hΓ : CtxClosed Γ✝
hA : ClosedN A✝ (List.length Γ✝)
hA' : ClosedN A'✝ (List.length Γ✝)
right✝¹ : ClosedN (VExpr.sort u✝) (List.length Γ✝)
hb : ClosedN body✝ (List.length (A✝ :: Γ✝))
hb' : ClosedN body'✝ (List.length (A✝ :: Γ✝))
right✝ : ClosedN (VExpr.sort v✝) (List.length (A✝ :: Γ✝))
⊢ ClosedN (VExpr.forallE A✝ body✝) (List.length Γ✝) ∧
    ClosedN (VExpr.forallE A'✝ body'✝) (List.length Γ✝) ∧ ClosedN (VExpr.sort (VLevel.imax u✝ v✝)) (List.length Γ✝)
Tactic:
exact ⟨⟨hA, hb⟩, ⟨hA', hb'⟩, trivial⟩
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case defeqDF
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ B✝ : VExpr
u✝ : VLevel
e1✝ e2✝ : VExpr
a✝¹ : IsDefEq env U Γ✝ A✝ B✝ (VExpr.sort u✝)
a✝ : IsDefEq env U Γ✝ e1✝ e2✝ A✝
ih1 :
  CtxClosed Γ✝ → ClosedN A✝ (List.length Γ✝) ∧ ClosedN B✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort u✝) (List.length Γ✝)
ih2 : CtxClosed Γ✝ → ClosedN e1✝ (List.length Γ✝) ∧ ClosedN e2✝ (List.length Γ✝) ∧ ClosedN A✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
⊢ ClosedN e1✝ (List.length Γ✝) ∧ ClosedN e2✝ (List.length Γ✝) ∧ ClosedN B✝ (List.length Γ✝)
Tactic:
exact ⟨(ih2 hΓ).1, (ih2 hΓ).2.1, (ih1 hΓ).2.1⟩
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case beta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A A✝ : VExpr
Γ✝ : List VExpr
e✝ B✝ e'✝ : VExpr
a✝¹ : IsDefEq env U (A✝ :: Γ✝) e✝ e✝ B✝
a✝ : IsDefEq env U Γ✝ e'✝ e'✝ A✝
ih1 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN B✝ (List.length (A✝ :: Γ✝))
ih2 : CtxClosed Γ✝ → ClosedN e'✝ (List.length Γ✝) ∧ ClosedN e'✝ (List.length Γ✝) ∧ ClosedN A✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
⊢ ClosedN (app (lam A✝ e✝) e'✝) (List.length Γ✝) ∧
    ClosedN (inst e✝ e'✝) (List.length Γ✝) ∧ ClosedN (inst B✝ e'✝) (List.length Γ✝)
Tactic:
let ⟨he', _, hA⟩ := ih2 hΓ
Proof state after:
case beta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A A✝ : VExpr
Γ✝ : List VExpr
e✝ B✝ e'✝ : VExpr
a✝¹ : IsDefEq env U (A✝ :: Γ✝) e✝ e✝ B✝
a✝ : IsDefEq env U Γ✝ e'✝ e'✝ A✝
ih1 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN B✝ (List.length (A✝ :: Γ✝))
ih2 : CtxClosed Γ✝ → ClosedN e'✝ (List.length Γ✝) ∧ ClosedN e'✝ (List.length Γ✝) ∧ ClosedN A✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
he' left✝ : ClosedN e'✝ (List.length Γ✝)
hA : ClosedN A✝ (List.length Γ✝)
⊢ ClosedN (app (lam A✝ e✝) e'✝) (List.length Γ✝) ∧
    ClosedN (inst e✝ e'✝) (List.length Γ✝) ∧ ClosedN (inst B✝ e'✝) (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case beta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A A✝ : VExpr
Γ✝ : List VExpr
e✝ B✝ e'✝ : VExpr
a✝¹ : IsDefEq env U (A✝ :: Γ✝) e✝ e✝ B✝
a✝ : IsDefEq env U Γ✝ e'✝ e'✝ A✝
ih1 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN B✝ (List.length (A✝ :: Γ✝))
ih2 : CtxClosed Γ✝ → ClosedN e'✝ (List.length Γ✝) ∧ ClosedN e'✝ (List.length Γ✝) ∧ ClosedN A✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
he' left✝ : ClosedN e'✝ (List.length Γ✝)
hA : ClosedN A✝ (List.length Γ✝)
⊢ ClosedN (app (lam A✝ e✝) e'✝) (List.length Γ✝) ∧
    ClosedN (inst e✝ e'✝) (List.length Γ✝) ∧ ClosedN (inst B✝ e'✝) (List.length Γ✝)
Tactic:
let ⟨he, _, hB⟩ := ih1 ⟨hΓ, hA⟩
Proof state after:
case beta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A A✝ : VExpr
Γ✝ : List VExpr
e✝ B✝ e'✝ : VExpr
a✝¹ : IsDefEq env U (A✝ :: Γ✝) e✝ e✝ B✝
a✝ : IsDefEq env U Γ✝ e'✝ e'✝ A✝
ih1 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN B✝ (List.length (A✝ :: Γ✝))
ih2 : CtxClosed Γ✝ → ClosedN e'✝ (List.length Γ✝) ∧ ClosedN e'✝ (List.length Γ✝) ∧ ClosedN A✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
he' left✝¹ : ClosedN e'✝ (List.length Γ✝)
hA : ClosedN A✝ (List.length Γ✝)
he left✝ : ClosedN e✝ (List.length (A✝ :: Γ✝))
hB : ClosedN B✝ (List.length (A✝ :: Γ✝))
⊢ ClosedN (app (lam A✝ e✝) e'✝) (List.length Γ✝) ∧
    ClosedN (inst e✝ e'✝) (List.length Γ✝) ∧ ClosedN (inst B✝ e'✝) (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case beta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A A✝ : VExpr
Γ✝ : List VExpr
e✝ B✝ e'✝ : VExpr
a✝¹ : IsDefEq env U (A✝ :: Γ✝) e✝ e✝ B✝
a✝ : IsDefEq env U Γ✝ e'✝ e'✝ A✝
ih1 :
  CtxClosed (A✝ :: Γ✝) →
    ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN e✝ (List.length (A✝ :: Γ✝)) ∧ ClosedN B✝ (List.length (A✝ :: Γ✝))
ih2 : CtxClosed Γ✝ → ClosedN e'✝ (List.length Γ✝) ∧ ClosedN e'✝ (List.length Γ✝) ∧ ClosedN A✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
he' left✝¹ : ClosedN e'✝ (List.length Γ✝)
hA : ClosedN A✝ (List.length Γ✝)
he left✝ : ClosedN e✝ (List.length (A✝ :: Γ✝))
hB : ClosedN B✝ (List.length (A✝ :: Γ✝))
⊢ ClosedN (app (lam A✝ e✝) e'✝) (List.length Γ✝) ∧
    ClosedN (inst e✝ e'✝) (List.length Γ✝) ∧ ClosedN (inst B✝ e'✝) (List.length Γ✝)
Tactic:
exact ⟨⟨⟨hA, he⟩, he'⟩, he.inst he', hB.inst he'⟩
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case eta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
e✝ A✝ B✝ : VExpr
a✝ : IsDefEq env U Γ✝ e✝ e✝ (VExpr.forallE A✝ B✝)
ih :
  CtxClosed Γ✝ →
    ClosedN e✝ (List.length Γ✝) ∧ ClosedN e✝ (List.length Γ✝) ∧ ClosedN (VExpr.forallE A✝ B✝) (List.length Γ✝)
hΓ : CtxClosed Γ✝
⊢ ClosedN (lam A✝ (app (lift e✝) (VExpr.bvar 0))) (List.length Γ✝) ∧
    ClosedN e✝ (List.length Γ✝) ∧ ClosedN (VExpr.forallE A✝ B✝) (List.length Γ✝)
Tactic:
let ⟨he, _, hA, hB⟩ := ih hΓ
Proof state after:
case eta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
e✝ A✝ B✝ : VExpr
a✝ : IsDefEq env U Γ✝ e✝ e✝ (VExpr.forallE A✝ B✝)
ih :
  CtxClosed Γ✝ →
    ClosedN e✝ (List.length Γ✝) ∧ ClosedN e✝ (List.length Γ✝) ∧ ClosedN (VExpr.forallE A✝ B✝) (List.length Γ✝)
hΓ : CtxClosed Γ✝
he left✝ : ClosedN e✝ (List.length Γ✝)
hA : ClosedN A✝ (List.length Γ✝)
hB : ClosedN B✝ (List.length Γ✝ + 1)
⊢ ClosedN (lam A✝ (app (lift e✝) (VExpr.bvar 0))) (List.length Γ✝) ∧
    ClosedN e✝ (List.length Γ✝) ∧ ClosedN (VExpr.forallE A✝ B✝) (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case eta
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
e✝ A✝ B✝ : VExpr
a✝ : IsDefEq env U Γ✝ e✝ e✝ (VExpr.forallE A✝ B✝)
ih :
  CtxClosed Γ✝ →
    ClosedN e✝ (List.length Γ✝) ∧ ClosedN e✝ (List.length Γ✝) ∧ ClosedN (VExpr.forallE A✝ B✝) (List.length Γ✝)
hΓ : CtxClosed Γ✝
he left✝ : ClosedN e✝ (List.length Γ✝)
hA : ClosedN A✝ (List.length Γ✝)
hB : ClosedN B✝ (List.length Γ✝ + 1)
⊢ ClosedN (lam A✝ (app (lift e✝) (VExpr.bvar 0))) (List.length Γ✝) ∧
    ClosedN e✝ (List.length Γ✝) ∧ ClosedN (VExpr.forallE A✝ B✝) (List.length Γ✝)
Tactic:
exact ⟨⟨hA, he.liftN, Nat.succ_pos _⟩, he, hA, hB⟩
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case proofIrrel
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
p✝ h✝ h'✝ : VExpr
a✝² : IsDefEq env U Γ✝ p✝ p✝ (VExpr.sort VLevel.zero)
a✝¹ : IsDefEq env U Γ✝ h✝ h✝ p✝
a✝ : IsDefEq env U Γ✝ h'✝ h'✝ p✝
a_ih✝ :
  CtxClosed Γ✝ →
    ClosedN p✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort VLevel.zero) (List.length Γ✝)
ih2 : CtxClosed Γ✝ → ClosedN h✝ (List.length Γ✝) ∧ ClosedN h✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
ih3 : CtxClosed Γ✝ → ClosedN h'✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
⊢ ClosedN h✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
Tactic:
let ⟨hh, _, _⟩ := ih2 hΓ
Proof state after:
case proofIrrel
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
p✝ h✝ h'✝ : VExpr
a✝² : IsDefEq env U Γ✝ p✝ p✝ (VExpr.sort VLevel.zero)
a✝¹ : IsDefEq env U Γ✝ h✝ h✝ p✝
a✝ : IsDefEq env U Γ✝ h'✝ h'✝ p✝
a_ih✝ :
  CtxClosed Γ✝ →
    ClosedN p✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort VLevel.zero) (List.length Γ✝)
ih2 : CtxClosed Γ✝ → ClosedN h✝ (List.length Γ✝) ∧ ClosedN h✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
ih3 : CtxClosed Γ✝ → ClosedN h'✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
hh left✝ : ClosedN h✝ (List.length Γ✝)
right✝ : ClosedN p✝ (List.length Γ✝)
⊢ ClosedN h✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case proofIrrel
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
p✝ h✝ h'✝ : VExpr
a✝² : IsDefEq env U Γ✝ p✝ p✝ (VExpr.sort VLevel.zero)
a✝¹ : IsDefEq env U Γ✝ h✝ h✝ p✝
a✝ : IsDefEq env U Γ✝ h'✝ h'✝ p✝
a_ih✝ :
  CtxClosed Γ✝ →
    ClosedN p✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort VLevel.zero) (List.length Γ✝)
ih2 : CtxClosed Γ✝ → ClosedN h✝ (List.length Γ✝) ∧ ClosedN h✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
ih3 : CtxClosed Γ✝ → ClosedN h'✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
hh left✝ : ClosedN h✝ (List.length Γ✝)
right✝ : ClosedN p✝ (List.length Γ✝)
⊢ ClosedN h✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
Tactic:
let ⟨hh', _, hp⟩ := ih3 hΓ
Proof state after:
case proofIrrel
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
p✝ h✝ h'✝ : VExpr
a✝² : IsDefEq env U Γ✝ p✝ p✝ (VExpr.sort VLevel.zero)
a✝¹ : IsDefEq env U Γ✝ h✝ h✝ p✝
a✝ : IsDefEq env U Γ✝ h'✝ h'✝ p✝
a_ih✝ :
  CtxClosed Γ✝ →
    ClosedN p✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort VLevel.zero) (List.length Γ✝)
ih2 : CtxClosed Γ✝ → ClosedN h✝ (List.length Γ✝) ∧ ClosedN h✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
ih3 : CtxClosed Γ✝ → ClosedN h'✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
hh left✝¹ : ClosedN h✝ (List.length Γ✝)
right✝ : ClosedN p✝ (List.length Γ✝)
hh' left✝ : ClosedN h'✝ (List.length Γ✝)
hp : ClosedN p✝ (List.length Γ✝)
⊢ ClosedN h✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case proofIrrel
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
p✝ h✝ h'✝ : VExpr
a✝² : IsDefEq env U Γ✝ p✝ p✝ (VExpr.sort VLevel.zero)
a✝¹ : IsDefEq env U Γ✝ h✝ h✝ p✝
a✝ : IsDefEq env U Γ✝ h'✝ h'✝ p✝
a_ih✝ :
  CtxClosed Γ✝ →
    ClosedN p✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝) ∧ ClosedN (VExpr.sort VLevel.zero) (List.length Γ✝)
ih2 : CtxClosed Γ✝ → ClosedN h✝ (List.length Γ✝) ∧ ClosedN h✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
ih3 : CtxClosed Γ✝ → ClosedN h'✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
hΓ : CtxClosed Γ✝
hh left✝¹ : ClosedN h✝ (List.length Γ✝)
right✝ : ClosedN p✝ (List.length Γ✝)
hh' left✝ : ClosedN h'✝ (List.length Γ✝)
hp : ClosedN p✝ (List.length Γ✝)
⊢ ClosedN h✝ (List.length Γ✝) ∧ ClosedN h'✝ (List.length Γ✝) ∧ ClosedN p✝ (List.length Γ✝)
Tactic:
exact ⟨hh, hh', hp⟩
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case extra
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
df✝ : VDefEq
ls✝ : List VLevel
Γ✝ : List VExpr
h1 : env.defeqs df✝
a✝¹ : ∀ (l : VLevel), l ∈ ls✝ → VLevel.WF U l
a✝ : List.length ls✝ = df✝.uvars
hΓ : CtxClosed Γ✝
⊢ ClosedN (instL ls✝ df✝.lhs) (List.length Γ✝) ∧
    ClosedN (instL ls✝ df✝.rhs) (List.length Γ✝) ∧ ClosedN (instL ls✝ df✝.type) (List.length Γ✝)
Tactic:
let ⟨⟨hl, _⟩, ⟨hr, hA⟩⟩ := henv.2 h1
Proof state after:
case extra
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
df✝ : VDefEq
ls✝ : List VLevel
Γ✝ : List VExpr
h1 : env.defeqs df✝
a✝¹ : ∀ (l : VLevel), l ∈ ls✝ → VLevel.WF U l
a✝ : List.length ls✝ = df✝.uvars
hΓ : CtxClosed Γ✝
hl : ClosedN df✝.lhs
right✝ : ClosedN df✝.type
hr : ClosedN df✝.rhs
hA : ClosedN df✝.type
⊢ ClosedN (instL ls✝ df✝.lhs) (List.length Γ✝) ∧
    ClosedN (instL ls✝ df✝.rhs) (List.length Γ✝) ∧ ClosedN (instL ls✝ df✝.type) (List.length Γ✝)
Theorem: Lean4Lean.VEnv.IsDefEq.closedN'
Proof state before:
case extra
env : VEnv
henv : OnTypes env fun x e A => ClosedN e ∧ ClosedN A
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
df✝ : VDefEq
ls✝ : List VLevel
Γ✝ : List VExpr
h1 : env.defeqs df✝
a✝¹ : ∀ (l : VLevel), l ∈ ls✝ → VLevel.WF U l
a✝ : List.length ls✝ = df✝.uvars
hΓ : CtxClosed Γ✝
hl : ClosedN df✝.lhs
right✝ : ClosedN df✝.type
hr : ClosedN df✝.rhs
hA : ClosedN df✝.type
⊢ ClosedN (instL ls✝ df✝.lhs) (List.length Γ✝) ∧
    ClosedN (instL ls✝ df✝.rhs) (List.length Γ✝) ∧ ClosedN (instL ls✝ df✝.type) (List.length Γ✝)
Tactic:
exact ⟨
  hl.instL.mono (Nat.zero_le _),
  hr.instL.mono (Nat.zero_le _),
  hA.instL.mono (Nat.zero_le _)⟩
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEqCtx.closed
Proof state before:
env : VEnv
henv : Ordered env
Γ₀ : List VExpr
U : Nat
Γ₁ Γ₂ : List VExpr
H : CtxClosed Γ₀
Γ₁✝ Γ₂✝ : List VExpr
A₁✝ A₂✝ : VExpr
u✝ : VLevel
h1 : IsDefEqCtx env U Γ₀ Γ₁✝ Γ₂✝
h2 : IsDefEq env U Γ₁✝ A₁✝ A₂✝ (sort u✝)
c1 : CtxClosed Γ₁✝
c2 : CtxClosed Γ₂✝
⊢ (fun Γ A => ClosedN A (List.length Γ)) Γ₂✝ A₂✝
Tactic:
simpa [h1.length_eq] using h2.symm.closedN henv c1
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
H : IsDefEq env U Γ e1 e2 A
⊢ IsDefEq env' U Γ e1 e2 A
Tactic:
induction H with
| bvar h => exact .bvar h
| constDF h1 h2 h3 h4 h5 => exact .constDF (henv.1 _ _ h1) h2 h3 h4 h5
| sortDF h1 h2 h3 => exact .sortDF h1 h2 h3
| symm _ ih => exact .symm ih
| trans _ _ ih1 ih2 => exact .trans ih1 ih2
| appDF _ _ ih1 ih2 => exact .appDF ih1 ih2
| lamDF _ _ ih1 ih2 => exact .lamDF ih1 ih2
| forallEDF _ _ ih1 ih2 => exact .forallEDF ih1 ih2
| defeqDF _ _ ih1 ih2 => exact .defeqDF ih1 ih2
| beta _ _ ih1 ih2 => exact .beta ih1 ih2
| eta _ ih => exact .eta ih
| proofIrrel _ _ _ ih1 ih2 ih3 => exact .proofIrrel ih1 ih2 ih3
| extra h1 h2 h3 => exact .extra (henv.2 _ h1) h2 h3
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case bvar
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
i✝ : Nat
A✝ : VExpr
h : Lookup Γ✝ i✝ A✝
⊢ IsDefEq env' U Γ✝ (VExpr.bvar i✝) (VExpr.bvar i✝) A✝
Tactic:
exact .bvar h
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case constDF
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
c✝ : Name
ci✝ : VConstant
ls✝ ls'✝ : List VLevel
Γ✝ : List VExpr
h1 : env.constants c✝ = some (some ci✝)
h2 : ∀ (l : VLevel), l ∈ ls✝ → VLevel.WF U l
h3 : ∀ (l : VLevel), l ∈ ls'✝ → VLevel.WF U l
h4 : List.length ls✝ = ci✝.uvars
h5 : List.Forall₂ (fun x x_1 => x ≈ x_1) ls✝ ls'✝
⊢ IsDefEq env' U Γ✝ (VExpr.const c✝ ls✝) (VExpr.const c✝ ls'✝) (instL ls✝ ci✝.type)
Tactic:
exact .constDF (henv.1 _ _ h1) h2 h3 h4 h5
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case sortDF
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
l✝ l'✝ : VLevel
Γ✝ : List VExpr
h1 : VLevel.WF U l✝
h2 : VLevel.WF U l'✝
h3 : l✝ ≈ l'✝
⊢ IsDefEq env' U Γ✝ (VExpr.sort l✝) (VExpr.sort l'✝) (VExpr.sort (VLevel.succ l✝))
Tactic:
exact .sortDF h1 h2 h3
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case symm
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
e✝ e'✝ A✝ : VExpr
a✝ : IsDefEq env U Γ✝ e✝ e'✝ A✝
ih : IsDefEq env' U Γ✝ e✝ e'✝ A✝
⊢ IsDefEq env' U Γ✝ e'✝ e✝ A✝
Tactic:
exact .symm ih
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case trans
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
e₁✝ e₂✝ A✝ e₃✝ : VExpr
a✝¹ : IsDefEq env U Γ✝ e₁✝ e₂✝ A✝
a✝ : IsDefEq env U Γ✝ e₂✝ e₃✝ A✝
ih1 : IsDefEq env' U Γ✝ e₁✝ e₂✝ A✝
ih2 : IsDefEq env' U Γ✝ e₂✝ e₃✝ A✝
⊢ IsDefEq env' U Γ✝ e₁✝ e₃✝ A✝
Tactic:
exact .trans ih1 ih2
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case appDF
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
f✝ f'✝ A✝ B✝ a✝² a'✝ : VExpr
a✝¹ : IsDefEq env U Γ✝ f✝ f'✝ (VExpr.forallE A✝ B✝)
a✝ : IsDefEq env U Γ✝ a✝² a'✝ A✝
ih1 : IsDefEq env' U Γ✝ f✝ f'✝ (VExpr.forallE A✝ B✝)
ih2 : IsDefEq env' U Γ✝ a✝² a'✝ A✝
⊢ IsDefEq env' U Γ✝ (app f✝ a✝²) (app f'✝ a'✝) (inst B✝ a✝²)
Tactic:
exact .appDF ih1 ih2
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case lamDF
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ B✝ : VExpr
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ B✝
ih1 : IsDefEq env' U Γ✝ A✝ A'✝ (VExpr.sort u✝)
ih2 : IsDefEq env' U (A✝ :: Γ✝) body✝ body'✝ B✝
⊢ IsDefEq env' U Γ✝ (lam A✝ body✝) (lam A'✝ body'✝) (VExpr.forallE A✝ B✝)
Tactic:
exact .lamDF ih1 ih2
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case forallEDF
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ A'✝ : VExpr
u✝ : VLevel
body✝ body'✝ : VExpr
v✝ : VLevel
a✝¹ : IsDefEq env U Γ✝ A✝ A'✝ (VExpr.sort u✝)
a✝ : IsDefEq env U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
ih1 : IsDefEq env' U Γ✝ A✝ A'✝ (VExpr.sort u✝)
ih2 : IsDefEq env' U (A✝ :: Γ✝) body✝ body'✝ (VExpr.sort v✝)
⊢ IsDefEq env' U Γ✝ (VExpr.forallE A✝ body✝) (VExpr.forallE A'✝ body'✝) (VExpr.sort (VLevel.imax u✝ v✝))
Tactic:
exact .forallEDF ih1 ih2
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case defeqDF
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
A✝ B✝ : VExpr
u✝ : VLevel
e1✝ e2✝ : VExpr
a✝¹ : IsDefEq env U Γ✝ A✝ B✝ (VExpr.sort u✝)
a✝ : IsDefEq env U Γ✝ e1✝ e2✝ A✝
ih1 : IsDefEq env' U Γ✝ A✝ B✝ (VExpr.sort u✝)
ih2 : IsDefEq env' U Γ✝ e1✝ e2✝ A✝
⊢ IsDefEq env' U Γ✝ e1✝ e2✝ B✝
Tactic:
exact .defeqDF ih1 ih2
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case beta
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A A✝ : VExpr
Γ✝ : List VExpr
e✝ B✝ e'✝ : VExpr
a✝¹ : IsDefEq env U (A✝ :: Γ✝) e✝ e✝ B✝
a✝ : IsDefEq env U Γ✝ e'✝ e'✝ A✝
ih1 : IsDefEq env' U (A✝ :: Γ✝) e✝ e✝ B✝
ih2 : IsDefEq env' U Γ✝ e'✝ e'✝ A✝
⊢ IsDefEq env' U Γ✝ (app (lam A✝ e✝) e'✝) (inst e✝ e'✝) (inst B✝ e'✝)
Tactic:
exact .beta ih1 ih2
Proof state after:
no goals
Theorem: Lean4Lean.VEnv.IsDefEq.mono
Proof state before:
case eta
env env' : VEnv
henv : env ≤ env'
U : Nat
Γ : List VExpr
e1 e2 A : VExpr
Γ✝ : List VExpr
e✝ A✝ B✝ : VExpr
a✝ : IsDefEq env U Γ✝ e✝ e✝ (VExpr.forallE A✝ B✝)
ih : IsDefEq env' U Γ✝ e✝ e✝ (VExpr.forallE A✝ B✝)
⊢ IsDefEq env' U Γ✝ (lam A✝ (app (lift e✝) (VExpr.bvar 0))) e✝ (VExpr.forallE A✝ B✝)
Tactic:
exact .eta ih
Proof state after:
no goals