37 lines
1.1 KiB
Text
37 lines
1.1 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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import hott.path hott.equiv
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open path Equiv
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--Ensure that the types compared are in the same universe
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universe l
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variables (A B : Type.{l})
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private definition isequiv_path (H : A ≈ B) :=
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(@IsEquiv.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 A B H)
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axiom ua_equiv (A B : Type) : IsEquiv (equiv_path A B)
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-- Make the Equivalence given by the axiom an instance
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definition ua_inst [instance] {A B : Type} := (@ua_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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IsEquiv.inv (@equiv_path 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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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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path.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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