61901cff81
also add more definitions in types.pi, types.path, algebra.precategory the (pre)category library still needs cleanup authors of this commit: @avigad, @javra, @fpvandoorn
41 lines
1.1 KiB
Text
41 lines
1.1 KiB
Text
-- 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
|
|
prelude
|
|
import ..path ..equiv
|
|
open eq equiv is_equiv
|
|
|
|
--Ensure that the types compared are in the same universe
|
|
section
|
|
universe variable l
|
|
variables {A B : Type.{l}}
|
|
|
|
definition is_equiv_tr_of_eq (H : A = B) : is_equiv (transport (λX:Type, X) H) :=
|
|
(@is_equiv_tr Type (λX, X) A B H)
|
|
|
|
definition equiv_of_eq (H : A = B) : A ≃ B :=
|
|
equiv.mk _ (is_equiv_tr_of_eq H)
|
|
|
|
end
|
|
|
|
axiom univalence (A B : Type) : is_equiv (@equiv_of_eq A B)
|
|
|
|
attribute univalence [instance]
|
|
|
|
-- This is the version of univalence axiom we will probably use most often
|
|
definition ua {A B : Type} : A ≃ B → A = B :=
|
|
(@equiv_of_eq A B)⁻¹
|
|
|
|
-- One consequence of UA is that we can transport along equivalencies of types
|
|
namespace equiv
|
|
universe variable l
|
|
|
|
protected definition transport_of_equiv (P : Type → Type) {A B : Type.{l}} (H : A ≃ B)
|
|
: P A → P B :=
|
|
eq.transport P (ua H)
|
|
|
|
-- We can use this for calculation evironments
|
|
calc_subst transport_of_equiv
|
|
|
|
end equiv
|