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feat(init.ua): add some useful consequences of ua

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Floris van Doorn 2015-04-27 17:39:23 -04:00 committed by Leonardo de Moura
parent 40086d0084
commit 4173c958f7
1 changed files with 18 additions and 5 deletions
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23
hott/init/ua.hlean
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@ -10,18 +10,18 @@ Ported from Coq HoTT
prelude
import .equiv
open eq equiv is_equiv
open eq equiv is_equiv equiv.ops
--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) :=
definition is_equiv_cast_of_eq (H : A = B) : is_equiv (cast 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)
equiv.mk _ (is_equiv_cast_of_eq H)
end
@ -30,8 +30,21 @@ 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
definition ua [reducible] {A B : Type} : A ≃ B → A = B :=
equiv_of_eq⁻¹
definition equiv_of_eq_ua [reducible] {A B : Type} (f : A ≃ B) : equiv_of_eq (ua f) = f :=
right_inv equiv_of_eq f
definition cast_ua_fn {A B : Type} (f : A ≃ B) : cast (ua f) = f :=
ap to_fun (equiv_of_eq_ua f)
definition cast_ua {A B : Type} (f : A ≃ B) (a : A) : cast (ua f) a = f a :=
ap10 (cast_ua_fn f) a
definition ua_equiv_of_eq [reducible] {A B : Type} (p : A = B) : ua (equiv_of_eq p) = p :=
left_inv equiv_of_eq p
-- One consequence of UA is that we can transport along equivalencies of types
namespace equiv