no goals
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
⊢ Ico a b \ Ico a c = Ico (max a c) b
cases le_total a c with
| inl h =>
  ext x
  rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]
  exact and_congr_left' ⟨fun hx => hx.2 hx.1, fun hx => ⟨h.trans hx, fun _ => hx⟩⟩
| inr h => rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]
case inl.a
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h : a ≤ c
x : α
⊢ x ∈ Ico a b \ Ico a c ↔ x ∈ Ico (max a c) b
case inl
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h : a ≤ c
⊢ Ico a b \ Ico a c = Ico (max a c) b
ext x
case inl.a
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h : a ≤ c
x : α
⊢ (a ≤ x ∧ (a ≤ x → c ≤ x)) ∧ x < b ↔ c ≤ x ∧ x < b
case inl.a
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h : a ≤ c
x : α
⊢ x ∈ Ico a b \ Ico a c ↔ x ∈ Ico (max a c) b
rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]
no goals
case inl.a
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h : a ≤ c
x : α
⊢ (a ≤ x ∧ (a ≤ x → c ≤ x)) ∧ x < b ↔ c ≤ x ∧ x < b
exact and_congr_left' ⟨fun hx => hx.2 hx.1, fun hx => ⟨h.trans hx, fun _ => hx⟩⟩
no goals
case inr
ι : Type ?u.155748
α : Type u_1
inst✝¹ : LinearOrder α
inst✝ : LocallyFiniteOrder α
a✝ b✝ a b c : α
h : c ≤ a
⊢ Ico a b \ Ico a c = Ico (max a c) b
rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]