Module: Mathlib.Data.Sum.Interval

Goal state before:
pp: case mp.inl.inl
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₁
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b) →
    (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inl b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inl b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inl b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inl b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

pp: case mp.inl.inr
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) →
    (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inr b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inr b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

pp: case mp.inr.inl
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₂
b : β₁
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) →
    (∃ a₁ b₁ c₁, inr a = inl a₁ ∧ inl b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inr a = inl a₁ ∧ inl b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inr a = inl a₁ ∧ inl b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inr a = inr a₂ ∧ inl b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

pp: case mp.inr.inr
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₂
b : β₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) →
    (∃ a₁ b₁ c₁, inr a = inl a₁ ∧ inr b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inr a = inl a₁ ∧ inr b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inr a = inl a₁ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inr a = inr a₂ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

Tactic:
· rw [sumLexLift, mem_map]
      rintro ⟨c, hc, rfl⟩
      exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩

Goal state after:
case mp.inl.inr
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b) →
    (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inr b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inr b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

case mp.inr.inl
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₂
b : β₁
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inr a) (inl b) →
    (∃ a₁ b₁ c₁, inr a = inl a₁ ∧ inl b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inr a = inl a₁ ∧ inl b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inr a = inl a₁ ∧ inl b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inr a = inr a₂ ∧ inl b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

case mp.inr.inr
α₁ : Type u_1
α₂ : Type u_2
β₁ : Type u_3
β₂ : Type u_4
γ₁ : Type u_5
γ₂ : Type u_6
f₁ : α₁ → β₁ → Finset γ₁
f₂ : α₂ → β₂ → Finset γ₂
g₁ : α₁ → β₂ → Finset γ₁
g₂ : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₂
b : β₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b) →
    (∃ a₁ b₁ c₁, inr a = inl a₁ ∧ inr b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, inr a = inl a₁ ∧ inr b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, inr a = inl a₁ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, inr a = inr a₂ ∧ inr b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂