40 lines
1.3 KiB
Text
40 lines
1.3 KiB
Text
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-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Jakob von Raumer
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-- Ported from Coq HoTT
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import hott.path hott.trunc data.sigma data.prod
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open path prod
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inductive is_pointed [class] (A : Type) :=
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pointed_mk : Π(a : A), is_pointed A
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namespace is_pointed
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variables {A B : Type} (f : A → B)
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definition point (A : Type) [H : is_pointed A] : A :=
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is_pointed.rec (λinv, inv) H
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-- Any contractible type is pointed
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protected definition contr [instance] (H : Contr A) : is_pointed A :=
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pointed_mk (center H)
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-- A pi type with a pointed target is pointed
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protected definition pi [instance] {P : A → Type} [H : Πx, is_pointed (P x)]
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: is_pointed (Πx, P x) :=
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pointed_mk (λx, point (P x))
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-- A sigma type of pointed components is pointed
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protected definition sigma [instance] {P : A → Type} [G : is_pointed A]
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[H : is_pointed (P (point A))] : is_pointed (Σx, P x) :=
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pointed_mk (sigma.dpair (point A) (point (P (point A))))
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protected definition prod [H1 : is_pointed A] [H2 : is_pointed B]
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: is_pointed (A × B) :=
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pointed_mk (prod.mk (point A) (point B))
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protected definition loop_space (a : A) : is_pointed (a ≈ a) :=
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pointed_mk idp
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end is_pointed
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