Theorem: Finset.mem_sumLexLift
File: Mathlib/Data/Sum/Interval.lean

tactic: constructor
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁ ⊕ α₂
b : β₁ ⊕ β₂
c : γ₁ ⊕ γ₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ a b ↔
    (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: case mp
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁ ⊕ α₂
b : β₁ ⊕ β₂
c : γ₁ ⊕ γ₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ a b →
    (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

case mpr
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁ ⊕ α₂
b : β₁ ⊕ β₂
c : γ₁ ⊕ γ₂
⊢ ((∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂) →
    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b

tactic: obtain a | a := a <;> obtain b | b := b
state_before: case mp
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁ ⊕ α₂
b : β₁ ⊕ β₂
c : γ₁ ⊕ γ₂
⊢ c ∈ sumLexLift f₁ f₂ g₁ g₂ a b →
    (∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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

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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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]
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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
state_after: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₁
⊢ (∃ a_1 ∈ f₁ a b, Embedding.inl a_1 = c) →
    (∃ 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

tactic: rintro ⟨c, hc, rfl⟩
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₁
⊢ (∃ a_1 ∈ f₁ a b, Embedding.inl a_1 = c) →
    (∃ 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
state_after: case mp.inl.inl.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₁
c : γ₁
hc : c ∈ f₁ a b
⊢ (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inl b = inl b₁ ∧ Embedding.inl c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inl b = inr b₂ ∧ Embedding.inl c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inl b = inr b₂ ∧ Embedding.inl c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inl b = inr b₂ ∧ Embedding.inl c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

tactic: exact Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩
state_before: case mp.inl.inl.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₁
c : γ₁
hc : c ∈ f₁ a b
⊢ (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inl b = inl b₁ ∧ Embedding.inl c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inl b = inr b₂ ∧ Embedding.inl c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inl b = inr b₂ ∧ Embedding.inl c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inl b = inr b₂ ∧ Embedding.inl c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: no goals

tactic: refine fun h ↦ (mem_disjSum.1 h).elim ?_ ?_
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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
state_after: case mp.inl.inr.refine_1
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₂
h : c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ a_1 ∈ g₁ a b, inl a_1 = c) →
    (∃ 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.inl.inr.refine_2
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₂
h : c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ b_1 ∈ g₂ a b, inr b_1 = c) →
    (∃ 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

tactic: rintro ⟨c, hc, rfl⟩
state_before: case mp.inl.inr.refine_1
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₂
h : c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ a_1 ∈ g₁ a b, inl a_1 = c) →
    (∃ 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
state_after: case mp.inl.inr.refine_1.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₁
hc : c ∈ g₁ a b
h : inl c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inr b = inl b₁ ∧ inl c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inl c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inl c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inr b = inr b₂ ∧ inl c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

tactic: exact Or.inr (Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)
state_before: case mp.inl.inr.refine_1.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₁
hc : c ∈ g₁ a b
h : inl c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inr b = inl b₁ ∧ inl c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inl c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inl c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inr b = inr b₂ ∧ inl c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: no goals

tactic: rintro ⟨c, hc, rfl⟩
state_before: case mp.inl.inr.refine_2
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₁
b : β₂
h : c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ b_1 ∈ g₂ a b, inr b_1 = c) →
    (∃ 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
state_after: case mp.inl.inr.refine_2.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₂
hc : c ∈ g₂ a b
h : inr c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inr b = inl b₁ ∧ inr c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inr c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inr c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inr b = inr b₂ ∧ inr c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

tactic: exact Or.inr (Or.inr <| Or.inl ⟨a, b, c, rfl, rfl, rfl, hc⟩)
state_before: case mp.inl.inr.refine_2.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₂
hc : c ∈ g₂ a b
h : inr c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
⊢ (∃ a₁ b₁ c₁, inl a = inl a₁ ∧ inr b = inl b₁ ∧ inr c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inr c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inl a = inl a₁ ∧ inr b = inr b₂ ∧ inr c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inl a = inr a₂ ∧ inr b = inr b₂ ∧ inr c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: no goals

tactic: exact fun h ↦ (not_mem_empty _ h).elim
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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
state_after: no goals

tactic: rw [sumLexLift, mem_map]
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ 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
state_after: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₂
b : β₂
⊢ (∃ a_1 ∈ f₂ a b, { toFun := inr, inj' := ⋯ } a_1 = c) →
    (∃ 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: rintro ⟨c, hc, rfl⟩
state_before: 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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
c : γ₁ ⊕ γ₂
a : α₂
b : β₂
⊢ (∃ a_1 ∈ f₂ a b, { toFun := inr, inj' := ⋯ } a_1 = c) →
    (∃ 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
state_after: case mp.inr.inr.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₂
b : β₂
c : γ₂
hc : c ∈ f₂ a b
⊢ (∃ a₁ b₁ c₁, inr a = inl a₁ ∧ inr b = inl b₁ ∧ { toFun := inr, inj' := ⋯ } c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inr a = inl a₁ ∧ inr b = inr b₂ ∧ { toFun := inr, inj' := ⋯ } c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inr a = inl a₁ ∧ inr b = inr b₂ ∧ { toFun := inr, inj' := ⋯ } c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inr a = inr a₂ ∧ inr b = inr b₂ ∧ { toFun := inr, inj' := ⋯ } c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂

tactic: exact Or.inr (Or.inr <| Or.inr <| ⟨a, b, c, rfl, rfl, rfl, hc⟩)
state_before: case mp.inr.inr.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₂
b : β₂
c : γ₂
hc : c ∈ f₂ a b
⊢ (∃ a₁ b₁ c₁, inr a = inl a₁ ∧ inr b = inl b₁ ∧ { toFun := inr, inj' := ⋯ } c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
    (∃ a₁ b₂ c₁, inr a = inl a₁ ∧ inr b = inr b₂ ∧ { toFun := inr, inj' := ⋯ } c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
      (∃ a₁ b₂ c₂, inr a = inl a₁ ∧ inr b = inr b₂ ∧ { toFun := inr, inj' := ⋯ } c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
        ∃ a₂ b₂ c₂, inr a = inr a₂ ∧ inr b = inr b₂ ∧ { toFun := inr, inj' := ⋯ } c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂
state_after: no goals

tactic: rintro (⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩ |
  ⟨a, b, c, rfl, rfl, rfl, hc⟩ | ⟨a, b, c, rfl, rfl, rfl, hc⟩)
state_before: case mpr
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁ ⊕ α₂
b : β₁ ⊕ β₂
c : γ₁ ⊕ γ₂
⊢ ((∃ a₁ b₁ c₁, a = inl a₁ ∧ b = inl b₁ ∧ c = inl c₁ ∧ c₁ ∈ f₁ a₁ b₁) ∨
      (∃ a₁ b₂ c₁, a = inl a₁ ∧ b = inr b₂ ∧ c = inl c₁ ∧ c₁ ∈ g₁ a₁ b₂) ∨
        (∃ a₁ b₂ c₂, a = inl a₁ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ g₂ a₁ b₂) ∨
          ∃ a₂ b₂ c₂, a = inr a₂ ∧ b = inr b₂ ∧ c = inr c₂ ∧ c₂ ∈ f₂ a₂ b₂) →
    c ∈ sumLexLift f₁ f₂ g₁ g₂ a b
state_after: case mpr.inl.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₁
c : γ₁
hc : c ∈ f₁ a b
⊢ inl c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b)

case mpr.inr.inl.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₁
hc : c ∈ g₁ a b
⊢ inl c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)

case mpr.inr.inr.inl.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₂
hc : c ∈ g₂ a b
⊢ inr c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)

case mpr.inr.inr.inr.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₂
b : β₂
c : γ₂
hc : c ∈ f₂ a b
⊢ inr c ∈ sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b)

tactic: exact mem_map_of_mem _ hc
state_before: case mpr.inl.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₁
c : γ₁
hc : c ∈ f₁ a b
⊢ inl c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inl b)
state_after: no goals

tactic: exact inl_mem_disjSum.2 hc
state_before: case mpr.inr.inl.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₁
hc : c ∈ g₁ a b
⊢ inl c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
state_after: no goals

tactic: exact inr_mem_disjSum.2 hc
state_before: case mpr.inr.inr.inl.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₁
b : β₂
c : γ₂
hc : c ∈ g₂ a b
⊢ inr c ∈ sumLexLift f₁ f₂ g₁ g₂ (inl a) (inr b)
state_after: no goals

tactic: exact mem_map_of_mem _ hc
state_before: case mpr.inr.inr.inr.intro.intro.intro.intro.intro.intro
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 f₁' : α₁ → β₁ → Finset γ₁
f₂ f₂' : α₂ → β₂ → Finset γ₂
g₁ g₁' : α₁ → β₂ → Finset γ₁
g₂ g₂' : α₁ → β₂ → Finset γ₂
a : α₂
b : β₂
c : γ₂
hc : c ∈ f₂ a b
⊢ inr c ∈ sumLexLift f₁ f₂ g₁ g₂ (inr a) (inr b)
state_after: no goals