lean2/library/standard/data/prod.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, Jeremy Avigad
import logic.classes.inhabited logic.connectives.eq logic.classes.decidable
using inhabited decidable
inductive prod (A B : Type) : Type :=
| pair : A → B → prod A B
precedence `×`:30
infixr × := prod
-- notation for n-ary tuples
notation `(` h `,` t:(foldl `,` (e r, pair r e) h) `)` := t
namespace prod
section
parameters {A B : Type}
abbreviation pr1 (p : prod A B) := prod_rec (λ x y, x) p
abbreviation pr2 (p : prod A B) := prod_rec (λ x y, y) p
theorem pr1_pair (a : A) (b : B) : pr1 (a, b) = a := refl a
theorem pr2_pair (a : A) (b : B) : pr2 (a, b) = b := refl b
-- TODO: remove prefix when we can protect it
theorem pair_destruct {P : A × B → Prop} (p : A × B) (H : ∀a b, P (a, b)) : P p :=
prod_rec H p
theorem prod_ext (p : prod A B) : pair (pr1 p) (pr2 p) = p :=
pair_destruct p (λx y, refl (x, y))
theorem pair_eq {a1 a2 : A} {b1 b2 : B} (H1 : a1 = a2) (H2 : b1 = b2) : (a1, b1) = (a2, b2) :=
subst H1 (subst H2 (refl _))
theorem prod_eq {p1 p2 : prod A B} : ∀ (H1 : pr1 p1 = pr1 p2) (H2 : pr2 p1 = pr2 p2), p1 = p2 :=
pair_destruct p1 (take a1 b1, pair_destruct p2 (take a2 b2 H1 H2, pair_eq H1 H2))
theorem prod_inhabited (H1 : inhabited A) (H2 : inhabited B) : inhabited (prod A B) :=
inhabited_destruct H1 (λa, inhabited_destruct H2 (λb, inhabited_mk (pair a b)))
theorem prod_eq_decidable (u v : A × B) (H1 : decidable (pr1 u = pr1 v))
(H2 : decidable (pr2 u = pr2 v)) : decidable (u = v) :=
have H3 : u = v ↔ (pr1 u = pr1 v) ∧ (pr2 u = pr2 v), from
iff_intro
(assume H, subst H (and_intro (refl _) (refl _)))
(assume H, and_elim H (assume H4 H5, prod_eq H4 H5)),
decidable_iff_equiv _ (iff_symm H3)
end
instance prod_inhabited
instance prod_eq_decidable
end prod