/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura, Jeremy Avigad
-/
prelude
import init.num init.relation

definition pair [constructor] := @prod.mk
notation A × B := prod A B
-- notation for n-ary tuples
notation `(` h `, ` t:(foldl `, ` (e r, prod.mk r e) h) `)` := t

namespace prod
  notation `pr₁` := pr1
  notation `pr₂` := pr2
  postfix `.1`:(max+1) := pr1
  postfix `.2`:(max+1) := pr2

end prod

protected definition prod.is_inhabited [instance] {A B : Type} [inhabited A] [inhabited B] : inhabited (prod A B) :=
inhabited.mk (default A, default B)

open decidable

protected definition prod.has_decidable_eq [instance] {A B : Type} [h₁ : decidable_eq A] [h₂ : decidable_eq B] : ∀ p₁ p₂ : A × B, decidable (p₁ = p₂)
| (a, b) (a', b') :=
  match h₁ a a' with
  | tt e₁ :=
    match h₂ b b' with
    | tt e₂ := tt (eq.rec_on e₁ (eq.rec_on e₂ rfl))
    | ff n₂ := ff (assume h, prod.no_confusion h (λ e₁' e₂', absurd e₂' n₂))
    end
  | ff n₁ := ff (assume h, prod.no_confusion h (λ e₁' e₂', absurd e₁' n₁))
  end