43 lines
1.2 KiB
Text
43 lines
1.2 KiB
Text
-- 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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prelude
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import ..path ..equiv
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open eq equiv
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--Ensure that the types compared are in the same universe
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section
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universe variable l
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variables {A B : Type.{l}}
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definition isequiv_path (H : A = B) :=
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(@is_equiv.transport Type (λX, X) A B H)
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definition equiv_path (H : A = B) : A ≃ B :=
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equiv.mk _ (isequiv_path H)
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end
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axiom ua_is_equiv (A B : Type) : is_equiv (@equiv_path A B)
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-- Make the Equivalence given by the axiom an instance
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protected definition inst [instance] (A B : Type) : is_equiv (@equiv_path A B) :=
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ua_is_equiv A B
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-- This is the version of univalence axiom we will probably use most often
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definition ua {A B : Type} : A ≃ B → A = B :=
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@is_equiv.inv _ _ (@equiv_path A B) (inst A B)
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-- One consequence of UA is that we can transport along equivalencies of types
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namespace Equiv
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universe variable l
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protected definition subst (P : Type → Type) {A B : Type.{l}} (H : A ≃ B)
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: P A → P B :=
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eq.transport P (ua H)
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-- We can use this for calculation evironments
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calc_subst subst
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end Equiv
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