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feat(hott): rename definition and cleanup in HoTT library 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
2015-02-21 00:30:32 +00:00
/-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: types.pointed
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
Authors: Jakob von Raumer, Floris van Doorn
feat(hott): rename definition and cleanup in HoTT library 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
2015-02-21 00:30:32 +00:00
Ported from Coq HoTT
-/
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
open core is_trunc
chore(hott) try to move library
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feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
structure pointed [class] (A : Type) :=
chore(hott) try to move library
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(point : A)
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
namespace pointed
variables {A B : Type}
definition pt [H : pointed A] := point A
chore(hott) try to move library
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-- Any contractible type is pointed
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
definition pointed_of_is_contr [instance] (A : Type) [H : is_contr A] : pointed A :=
pointed.mk !center
chore(hott) try to move library
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-- A pi type with a pointed target is pointed
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
definition pointed_pi [instance] (P : A → Type) [H : Πx, pointed (P x)]
: pointed (Πx, P x) :=
pointed.mk (λx, pt)
chore(hott) try to move library
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-- A sigma type of pointed components is pointed
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
definition pointed_sigma [instance] (P : A → Type) [G : pointed A]
[H : pointed (P pt)] : pointed (Σx, P x) :=
pointed.mk ⟨pt,pt⟩
chore(hott) try to move library
2014-12-12 04:14:53 +00:00
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
definition pointed_prod [instance] (A B : Type) [H1 : pointed A] [H2 : pointed B]
: pointed (A × B) :=
pointed.mk (pt,pt)
chore(hott) try to move library
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feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
definition pointed_loop [instance] (a : A) : pointed (a = a) :=
pointed.mk idp
chore(hott) try to move library
2014-12-12 04:14:53 +00:00
feat(hott.circle): prove that the fundamental group of the circle is equal to the integers, as groups Also many minor fixes at various places
2015-05-14 02:01:48 +00:00
definition pointed_fun_closed (f : A → B) [H : pointed A] : pointed B :=
pointed.mk (f pt)
end pointed
structure Pointed :=
{carrier : Type}
(Point : carrier)
open pointed Pointed
namespace Pointed
attribute carrier [coercion]
protected definition mk' (A : Type) [H : pointed A] : Pointed := Pointed.mk (point A)
definition pointed_carrier [instance] (A : Pointed) : pointed (carrier A) :=
pointed.mk (Point A)
definition Loop_space [reducible] (A : Pointed) : Pointed :=
Pointed.mk' (point A = point A)
definition Iterated_loop_space (A : Pointed) (n : ℕ) : Pointed :=
nat.rec_on n A (λn X, Loop_space X)
prefix `Ω`:95 := Loop_space
notation `Ω[`:95 n:0 `]`:0 A:95 := Iterated_loop_space A n
end Pointed
namespace pointed
export Pointed (hiding cases_on destruct mk)
abbreviation Cases_on := @Pointed.cases_on
abbreviation Destruct := @Pointed.destruct
abbreviation Mk := @Pointed.mk
definition iterated_loop_space (A : Type) [H : pointed A] (n : ℕ) : Type :=
carrier (Iterated_loop_space (Pointed.mk' A) n)
end pointed
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