lean2/hott/types/pointed.hlean

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/-
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
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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
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-- A pi type with a pointed target is pointed
definition is_pointed_pi [instance] {P : A → Type} [H : Πx, is_pointed (P x)]
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: 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⟩
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definition is_pointed_prod [H1 : is_pointed A] [H2 : is_pointed B]
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: is_pointed (A × B) :=
is_pointed.mk (!point,!point)
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definition is_pointed_loop_space (a : A) : is_pointed (a = a) :=
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is_pointed.mk idp
end is_pointed