lean2/library/hott/axioms/ua.lean

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2014-11-07 00:41:08 +00:00
-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
2014-11-07 00:41:08 +00:00
-- Author: Jakob von Raumer
-- Ported from Coq HoTT
import hott.path hott.equiv
open path Equiv
--Ensure that the types compared are in the same universe
universe variable l
variables (A B : Type.{l})
private definition isequiv_path (H : A ≈ B) :=
(@IsEquiv.transport Type (λX, X) A B H)
definition equiv_path (H : A ≈ B) : A ≃ B :=
Equiv.mk _ (isequiv_path A B H)
axiom ua_equiv (A B : Type) : IsEquiv (equiv_path A B)
-- Make the Equivalence given by the axiom an instance
definition ua_inst [instance] {A B : Type} := (@ua_equiv A B)
-- This is the version of univalence axiom we will probably use most often
definition ua {A B : Type} : A ≃ B → A ≈ B :=
IsEquiv.inv (@equiv_path A B)
-- One consequence of UA is that we can transport along equivalencies of types
namespace Equiv
protected definition subst (P : Type → Type) {A B : Type.{l}} (H : A ≃ B)
: P A → P B :=
path.transport P (ua H)
-- We can use this for calculation evironments
calc_subst subst
end Equiv