78 lines
2.2 KiB
Text
78 lines
2.2 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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Module: types.pointed
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Authors: Jakob von Raumer, Floris van Doorn
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Ported from Coq HoTT
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-/
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open core is_trunc
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structure pointed [class] (A : Type) :=
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(point : A)
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namespace pointed
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variables {A B : Type}
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definition pt [H : pointed A] := point A
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-- Any contractible type is pointed
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definition pointed_of_is_contr [instance] (A : Type) [H : is_contr A] : pointed A :=
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pointed.mk !center
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-- A pi type with a pointed target is pointed
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definition pointed_pi [instance] (P : A → Type) [H : Πx, pointed (P x)]
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: pointed (Πx, P x) :=
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pointed.mk (λx, pt)
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-- A sigma type of pointed components is pointed
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definition pointed_sigma [instance] (P : A → Type) [G : pointed A]
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[H : pointed (P pt)] : pointed (Σx, P x) :=
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pointed.mk ⟨pt,pt⟩
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definition pointed_prod [instance] (A B : Type) [H1 : pointed A] [H2 : pointed B]
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: pointed (A × B) :=
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pointed.mk (pt,pt)
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definition pointed_loop [instance] (a : A) : pointed (a = a) :=
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pointed.mk idp
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definition pointed_fun_closed (f : A → B) [H : pointed A] : pointed B :=
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pointed.mk (f pt)
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end pointed
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structure Pointed :=
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{carrier : Type}
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(Point : carrier)
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open pointed Pointed
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namespace Pointed
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attribute carrier [coercion]
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protected definition mk' (A : Type) [H : pointed A] : Pointed := Pointed.mk (point A)
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definition pointed_carrier [instance] (A : Pointed) : pointed (carrier A) :=
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pointed.mk (Point A)
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definition Loop_space [reducible] (A : Pointed) : Pointed :=
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Pointed.mk' (point A = point A)
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definition Iterated_loop_space (A : Pointed) (n : ℕ) : Pointed :=
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nat.rec_on n A (λn X, Loop_space X)
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prefix `Ω`:95 := Loop_space
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notation `Ω[`:95 n:0 `]`:0 A:95 := Iterated_loop_space A n
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end Pointed
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namespace pointed
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export Pointed (hiding cases_on destruct mk)
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abbreviation Cases_on := @Pointed.cases_on
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abbreviation Destruct := @Pointed.destruct
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abbreviation Mk := @Pointed.mk
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definition iterated_loop_space (A : Type) [H : pointed A] (n : ℕ) : Type :=
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carrier (Iterated_loop_space (Pointed.mk' A) n)
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end pointed
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