/-
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

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

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

end prod

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

open decidable

attribute [instance]
protected definition prod.has_decidable_eq {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