Theorem: Lean4Lean.VEnv.IsDefEqStrong.forallE_inv'
Proof state before:
case beta
env : VEnv
henv : Ordered env
envIH : OnTypes env (EnvStrong env)
U : Nat
Γ : List VExpr
e1 e2 V : VExpr
Γ✝ : List VExpr
A✝ : VExpr
u✝ : VLevel
B✝ : VExpr
v✝ : VLevel
e e'✝ : VExpr
hu : VLevel.WF U u✝
a✝² : VLevel.WF U v✝
h1 : IsDefEqStrong env U Γ✝ A✝ A✝ (sort u✝)
h2 : IsDefEqStrong env U (A✝ :: Γ✝) B✝ B✝ (sort v✝)
h3 : IsDefEqStrong env U (A✝ :: Γ✝) e e B✝
h4 : IsDefEqStrong env U Γ✝ e'✝ e'✝ A✝
a✝¹ : IsDefEqStrong env U Γ✝ (inst B✝ e'✝) (inst B✝ e'✝) (sort v✝)
a✝ : IsDefEqStrong env U Γ✝ (inst e e'✝) (inst e e'✝) (inst B✝ e'✝)
a_ih✝³ :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      A✝ = forallE A B ∨ A✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
a_ih✝² :
  ∀ {A B : VExpr},
    CtxStrong env U (A✝ :: Γ✝) →
      B✝ = forallE A B ∨ B✝ = forallE A B →
        (∃ u, IsDefEqStrong env U (A✝ :: Γ✝) A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: A✝ :: Γ✝) B B (sort v)
ih3 :
  ∀ {A B : VExpr},
    CtxStrong env U (A✝ :: Γ✝) →
      e = forallE A B ∨ e = forallE A B →
        (∃ u, IsDefEqStrong env U (A✝ :: Γ✝) A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: A✝ :: Γ✝) B B (sort v)
ih4 :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      e'✝ = forallE A B ∨ e'✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
a_ih✝¹ :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      inst B✝ e'✝ = forallE A B ∨ inst B✝ e'✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
a_ih✝ :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      inst e e'✝ = forallE A B ∨ inst e e'✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
A B : VExpr
hΓ : CtxStrong env U Γ✝
eq : app (lam A✝ e) e'✝ = forallE A B ∨ inst e e'✝ = forallE A B
⊢ (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
Tactic:
obtain ⟨⟨⟩⟩ | eq := eq
Proof state after:
case beta.inr
env : VEnv
henv : Ordered env
envIH : OnTypes env (EnvStrong env)
U : Nat
Γ : List VExpr
e1 e2 V : VExpr
Γ✝ : List VExpr
A✝ : VExpr
u✝ : VLevel
B✝ : VExpr
v✝ : VLevel
e e'✝ : VExpr
hu : VLevel.WF U u✝
a✝² : VLevel.WF U v✝
h1 : IsDefEqStrong env U Γ✝ A✝ A✝ (sort u✝)
h2 : IsDefEqStrong env U (A✝ :: Γ✝) B✝ B✝ (sort v✝)
h3 : IsDefEqStrong env U (A✝ :: Γ✝) e e B✝
h4 : IsDefEqStrong env U Γ✝ e'✝ e'✝ A✝
a✝¹ : IsDefEqStrong env U Γ✝ (inst B✝ e'✝) (inst B✝ e'✝) (sort v✝)
a✝ : IsDefEqStrong env U Γ✝ (inst e e'✝) (inst e e'✝) (inst B✝ e'✝)
a_ih✝³ :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      A✝ = forallE A B ∨ A✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
a_ih✝² :
  ∀ {A B : VExpr},
    CtxStrong env U (A✝ :: Γ✝) →
      B✝ = forallE A B ∨ B✝ = forallE A B →
        (∃ u, IsDefEqStrong env U (A✝ :: Γ✝) A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: A✝ :: Γ✝) B B (sort v)
ih3 :
  ∀ {A B : VExpr},
    CtxStrong env U (A✝ :: Γ✝) →
      e = forallE A B ∨ e = forallE A B →
        (∃ u, IsDefEqStrong env U (A✝ :: Γ✝) A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: A✝ :: Γ✝) B B (sort v)
ih4 :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      e'✝ = forallE A B ∨ e'✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
a_ih✝¹ :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      inst B✝ e'✝ = forallE A B ∨ inst B✝ e'✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
a_ih✝ :
  ∀ {A B : VExpr},
    CtxStrong env U Γ✝ →
      inst e e'✝ = forallE A B ∨ inst e e'✝ = forallE A B →
        (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)
A B : VExpr
hΓ : CtxStrong env U Γ✝
eq : inst e e'✝ = forallE A B
⊢ (∃ u, IsDefEqStrong env U Γ✝ A A (sort u)) ∧ ∃ v, IsDefEqStrong env U (A :: Γ✝) B B (sort v)