lean2/library/hott/axioms/ua.lean

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-- Copyright (c) 2014 Jakob von Raumer. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- 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
section
universe variable l
variables {A B : Type.{l}}
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 H)
end
inductive ua_type [class] : Type :=
mk : (Π (A B : Type), IsEquiv (@equiv_path A B)) → ua_type
namespace ua_type
context
universe k
parameters [F : ua_type.{k}] {A B: Type.{k}}
-- Make the Equivalence given by the axiom an instance
protected definition inst [instance] : IsEquiv (@equiv_path.{k} A B) :=
rec_on F (λ H, H A B)
-- This is the version of univalence axiom we will probably use most often
definition ua : A ≃ B → A ≈ B :=
@IsEquiv.inv _ _ (@equiv_path A B) inst
end
end ua_type
-- One consequence of UA is that we can transport along equivalencies of types
namespace Equiv
universe variable l
protected definition subst [UA : ua_type] (P : Type → Type) {A B : Type.{l}} (H : A ≃ B)
: P A → P B :=
path.transport P (ua_type.ua H)
-- We can use this for calculation evironments
calc_subst subst
end Equiv