lean2/hott/types/pointed.hlean

/-
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
  definition is_pointed_of_is_contr [instance] [H : is_contr A] : is_pointed A :=
    is_pointed.mk !center

  -- A pi type with a pointed target is pointed
  definition is_pointed_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
  definition is_pointed_sigma [instance] {P : A → Type} [G : is_pointed A]
      [H : is_pointed (P !point)] : is_pointed (Σx, P x) :=
    is_pointed.mk ⟨!point,!point⟩

  definition is_pointed_prod [H1 : is_pointed A] [H2 : is_pointed B]
      : is_pointed (A × B) :=
    is_pointed.mk (!point,!point)

  definition is_pointed_loop_space (a : A) : is_pointed (a = a) :=
    is_pointed.mk idp

end is_pointed
-												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
 								Author: Jakob von Raumer
 								Ported from Coq HoTT
 								-/
 								open eq prod is_trunc sigma
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
 								structure is_pointed [class] (A : Type) :=
 								  (point : A)
 								namespace is_pointed
 								  variables {A B : Type} (f : A → B)
 								  -- Any contractible type is pointed
-												feat(hott/algebra/precategory): general cleanup in precategories, define uncurrying functor

											
										
										
											2015-02-27 05:45:21 +00:00
+								  definition is_pointed_of_is_contr [instance] [H : is_contr A] : is_pointed A :=
-												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
+								    is_pointed.mk !center
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
 								  -- A pi type with a pointed target is pointed
-												feat(hott/algebra/precategory): general cleanup in precategories, define uncurrying functor

											
										
										
											2015-02-27 05:45:21 +00:00
+								  definition is_pointed_pi [instance] {P : A → Type} [H : Πx, is_pointed (P x)]
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
+								      : is_pointed (Πx, P x) :=
 								    is_pointed.mk (λx, point (P x))
 								  -- A sigma type of pointed components is pointed
-												feat(hott/algebra/precategory): general cleanup in precategories, define uncurrying functor

											
										
										
											2015-02-27 05:45:21 +00:00
+								  definition is_pointed_sigma [instance] {P : A → Type} [G : is_pointed A]
-												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
+								      [H : is_pointed (P !point)] : is_pointed (Σx, P x) :=
 								    is_pointed.mk ⟨!point,!point⟩
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
-												feat(hott/algebra/precategory): general cleanup in precategories, define uncurrying functor

											
										
										
											2015-02-27 05:45:21 +00:00
+								  definition is_pointed_prod [H1 : is_pointed A] [H2 : is_pointed B]
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
+								      : is_pointed (A × B) :=
-												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
+								    is_pointed.mk (!point,!point)
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
-												feat(hott/algebra/precategory): general cleanup in precategories, define uncurrying functor

											
										
										
											2015-02-27 05:45:21 +00:00
+								  definition is_pointed_loop_space (a : A) : is_pointed (a = a) :=
-												chore(hott) try to move library

											
										
										
											2014-12-12 04:14:53 +00:00
+								    is_pointed.mk idp
 								end is_pointed