lean2/library/standard/data/pair.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura
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import logic.classes.inhabited logic.connectives.eq

namespace pair
inductive pair (A : Type) (B : Type) : Type :=
| mk_pair : A → B → pair A B

section thms
  parameter {A : Type}
  parameter {B : Type}

  definition fst [inline] (p : pair A B) := pair_rec (λ x y, x) p
  definition snd [inline] (p : pair A B) := pair_rec (λ x y, y) p

  theorem pair_inhabited (H1 : inhabited A) (H2 : inhabited B) : inhabited (pair A B) :=
  inhabited_elim H1 (λ a, inhabited_elim H2 (λ b, inhabited_intro (mk_pair a b)))

  theorem fst_mk_pair (a : A) (b : B) : fst (mk_pair a b) = a :=
  refl a

  theorem snd_mk_pair (a : A) (b : B) : snd (mk_pair a b) = b :=
  refl b

  theorem pair_ext (p : pair A B) : mk_pair (fst p) (snd p) = p :=
  pair_rec (λ x y, refl (mk_pair x y)) p

  theorem pair_ext_eq {p1 p2 : pair A B} (H1 : fst p1 = fst p2) (H2 : snd p1 = snd p2) : p1 = p2 :=
  calc p1  = mk_pair (fst p1) (snd p1) : symm (pair_ext p1)
       ... = mk_pair (fst p2) (snd p1) : {H1}
       ... = mk_pair (fst p2) (snd p2) : {H2}
       ... = p2                        : pair_ext p2
end thms

instance pair_inhabited

precedence `×`:30
infixr × := pair

-- notation for n-ary tuples
notation `(` h `,` t:(foldl `,` (e r, mk_pair r e) h) `)` := t
end pair