| /- | |
| Copyright (c) 2022 Microsoft Corporation. All rights reserved. | |
| Released under Apache 2.0 license as described in the file LICENSE. | |
| Author: Leonardo de Moura | |
| -/ | |
| prelude | |
| import Init.SimpLemmas | |
| import Init.NotationExtra | |
| instance [BEq α] [BEq β] [LawfulBEq α] [LawfulBEq β] : LawfulBEq (α × β) where | |
| eq_of_beq {a b} (h : a.1 == b.1 && a.2 == b.2) := by | |
| cases a; cases b | |
| refine congr (congrArg _ (eq_of_beq ?_)) (eq_of_beq ?_) <;> simp_all | |
| rfl {a} := by cases a; simp [BEq.beq, LawfulBEq.rfl] | |
| @[simp] | |
| protected theorem Prod.forall {p : α × β → Prop} : (∀ x, p x) ↔ ∀ a b, p (a, b) := | |
| ⟨fun h a b ↦ h (a, b), fun h ⟨a, b⟩ ↦ h a b⟩ | |
| @[simp] | |
| protected theorem Prod.exists {p : α × β → Prop} : (∃ x, p x) ↔ ∃ a b, p (a, b) := | |
| ⟨fun ⟨⟨a, b⟩, h⟩ ↦ ⟨a, b, h⟩, fun ⟨a, b, h⟩ ↦ ⟨⟨a, b⟩, h⟩⟩ |