59 lines
1.9 KiB
Text
59 lines
1.9 KiB
Text
-- 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.core.eq logic.classes.decidable
|
||
|
||
-- data.prod
|
||
-- =========
|
||
|
||
open inhabited decidable
|
||
|
||
-- The cartesian product.
|
||
inductive prod (A B : Type) : Type :=
|
||
mk : A → B → prod A B
|
||
|
||
abbreviation pair := @prod.mk
|
||
infixr `×` := prod
|
||
|
||
-- notation for n-ary tuples
|
||
notation `(` h `,` t:(foldl `,` (e r, prod.mk r e) h) `)` := t
|
||
|
||
namespace prod
|
||
section
|
||
parameters {A B : Type}
|
||
|
||
abbreviation pr1 (p : prod A B) := rec (λ x y, x) p
|
||
abbreviation pr2 (p : prod A B) := rec (λ x y, y) p
|
||
|
||
theorem pr1_pair (a : A) (b : B) : pr1 (a, b) = a :=
|
||
rfl
|
||
|
||
theorem pr2_pair (a : A) (b : B) : pr2 (a, b) = b :=
|
||
rfl
|
||
|
||
theorem destruct [protected] {P : A × B → Prop} (p : A × B) (H : ∀a b, P (a, b)) : P p :=
|
||
rec H p
|
||
|
||
theorem prod_ext (p : prod A B) : pair (pr1 p) (pr2 p) = p :=
|
||
destruct p (λx y, eq.refl (x, y))
|
||
|
||
open eq_ops
|
||
|
||
theorem pair_eq {a1 a2 : A} {b1 b2 : B} (H1 : a1 = a2) (H2 : b1 = b2) : (a1, b1) = (a2, b2) :=
|
||
H1 ▸ H2 ▸ rfl
|
||
|
||
theorem equal [protected] {p1 p2 : prod A B} : ∀ (H1 : pr1 p1 = pr1 p2) (H2 : pr2 p1 = pr2 p2), p1 = p2 :=
|
||
destruct p1 (take a1 b1, destruct p2 (take a2 b2 H1 H2, pair_eq H1 H2))
|
||
|
||
theorem is_inhabited [protected] [instance] (H1 : inhabited A) (H2 : inhabited B) : inhabited (prod A B) :=
|
||
inhabited.destruct H1 (λa, inhabited.destruct H2 (λb, inhabited.mk (pair a b)))
|
||
|
||
theorem has_decidable_eq [protected] [instance] (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, H ▸ and.intro rfl rfl)
|
||
(assume H, and.elim H (assume H4 H5, equal H4 H5)),
|
||
decidable_iff_equiv _ (iff.symm H3)
|
||
end
|
||
end prod
|