61901cff81
also add more definitions in types.pi, types.path, algebra.precategory the (pre)category library still needs cleanup authors of this commit: @avigad, @javra, @fpvandoorn
40 lines
1.1 KiB
Text
40 lines
1.1 KiB
Text
/-
|
||
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: types.pointed
|
||
Author: Jakob von Raumer
|
||
|
||
Ported from Coq HoTT
|
||
-/
|
||
|
||
open eq prod is_trunc sigma
|
||
|
||
structure is_pointed [class] (A : Type) :=
|
||
(point : A)
|
||
|
||
namespace is_pointed
|
||
variables {A B : Type} (f : A → B)
|
||
|
||
-- Any contractible type is pointed
|
||
protected definition contr [instance] [H : is_contr A] : is_pointed A :=
|
||
is_pointed.mk !center
|
||
|
||
-- A pi type with a pointed target is pointed
|
||
protected definition pi [instance] {P : A → Type} [H : Πx, is_pointed (P x)]
|
||
: is_pointed (Πx, P x) :=
|
||
is_pointed.mk (λx, point (P x))
|
||
|
||
-- A sigma type of pointed components is pointed
|
||
protected definition sigma [instance] {P : A → Type} [G : is_pointed A]
|
||
[H : is_pointed (P !point)] : is_pointed (Σx, P x) :=
|
||
is_pointed.mk ⟨!point,!point⟩
|
||
|
||
protected definition prod [H1 : is_pointed A] [H2 : is_pointed B]
|
||
: is_pointed (A × B) :=
|
||
is_pointed.mk (!point,!point)
|
||
|
||
protected definition loop_space (a : A) : is_pointed (a = a) :=
|
||
is_pointed.mk idp
|
||
|
||
end is_pointed
|