michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

feat(init/ua): add ua_symm and ua_trans

Browse source
This commit is contained in:
Floris van Doorn 2016-03-19 11:10:33 -04:00 committed by Leonardo de Moura
parent 05b59aecf8
commit 80a2e285cb
1 changed files with 16 additions and 2 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

18
hott/init/ua.hlean
View file

@ -61,8 +61,6 @@ definition eq_of_equiv_lift {A B : Type} (f : A ≃ B) : A = lift B :=
ua (f ⬝e !equiv_lift)
namespace equiv
definition ua_refl (A : Type) : ua erfl = idpath A :=
eq_of_fn_eq_fn !eq_equiv_equiv (right_inv !eq_equiv_equiv erfl)
-- One consequence of UA is that we can transport along equivalencies of types
-- We can use this for calculation evironments
@ -90,4 +88,20 @@ namespace equiv
(f : A ≃ B) (H : P equiv.refl idp) : P f (ua f) :=
rec_on_ua' f (λq, eq.rec_on q H)
definition ua_refl (A : Type) : ua erfl = idpath A :=
eq_of_fn_eq_fn !eq_equiv_equiv (right_inv !eq_equiv_equiv erfl)
definition ua_symm {A B : Type} (f : A ≃ B) : ua f⁻¹ᵉ = (ua f)⁻¹ :=
begin
apply rec_on_ua_idp f,
refine !ua_refl ⬝ inverse2 !ua_refl⁻¹
end
definition ua_trans {A B C : Type} (f : A ≃ B) (g : B ≃ C) : ua (f ⬝e g) = ua f ⬝ ua g :=
begin
apply rec_on_ua_idp g, apply rec_on_ua_idp f,
refine !ua_refl ⬝ concat2 !ua_refl⁻¹ !ua_refl⁻¹
end
end equiv