3367c20f9d
There is one proof in realprojective which I couldn't quite fix, so for now I left a sorry
70 lines
2.2 KiB
Text
70 lines
2.2 KiB
Text
-- Authors: Floris van Doorn
|
||
|
||
import homotopy.wedge
|
||
|
||
open wedge pushout eq prod sum pointed equiv is_equiv unit lift
|
||
|
||
namespace wedge
|
||
|
||
definition wedge_flip' [unfold 3] {A B : Type*} (x : A ∨ B) : B ∨ A :=
|
||
begin
|
||
induction x,
|
||
{ exact inr a },
|
||
{ exact inl a },
|
||
{ exact (glue ⋆)⁻¹ }
|
||
end
|
||
|
||
-- TODO: fix precedences
|
||
definition wedge_flip [constructor] (A B : Type*) : A ∨ B →* B ∨ A :=
|
||
pmap.mk wedge_flip' (glue ⋆)⁻¹
|
||
|
||
definition wedge_flip'_wedge_flip' [unfold 3] {A B : Type*} (x : A ∨ B) : wedge_flip' (wedge_flip' x) = x :=
|
||
begin
|
||
induction x,
|
||
{ reflexivity },
|
||
{ reflexivity },
|
||
{ apply eq_pathover_id_right,
|
||
apply hdeg_square,
|
||
exact ap_compose wedge_flip' _ _ ⬝ ap02 _ !elim_glue ⬝ !ap_inv ⬝ !elim_glue⁻² ⬝ !inv_inv }
|
||
end
|
||
|
||
definition wedge_flip_wedge_flip (A B : Type*) : wedge_flip B A ∘* wedge_flip A B ~* pid (A ∨ B) :=
|
||
phomotopy.mk wedge_flip'_wedge_flip' (whisker_right _ (!ap_inv ⬝ !wedge.elim_glue⁻²) ⬝ !con.left_inv)⁻¹
|
||
|
||
definition wedge_comm [constructor] (A B : Type*) : A ∨ B ≃* B ∨ A :=
|
||
begin
|
||
fapply pequiv.MK,
|
||
{ exact wedge_flip A B },
|
||
{ exact wedge_flip B A },
|
||
{ exact wedge_flip_wedge_flip A B },
|
||
{ exact wedge_flip_wedge_flip B A }
|
||
end
|
||
|
||
-- TODO: wedge is associative
|
||
|
||
definition wedge_shift [unfold 3] {A B C : Type*} (x : (A ∨ B) ∨ C) : (A ∨ (B ∨ C)) :=
|
||
begin
|
||
induction x with l,
|
||
induction l with a,
|
||
exact inl a,
|
||
exact inr (inl a),
|
||
exact (glue ⋆),
|
||
exact inr (inr a),
|
||
-- exact elim_glue _ _ _,
|
||
exact sorry
|
||
|
||
end
|
||
|
||
|
||
definition wedge_pequiv [constructor] {A A' B B' : Type*} (a : A ≃* A') (b : B ≃* B') : A ∨ B ≃* A' ∨ B' :=
|
||
begin
|
||
fapply pequiv_of_equiv,
|
||
exact pushout.equiv !pconst !pconst !pconst !pconst !pequiv.refl a b (λdummy, respect_pt a) (λdummy, respect_pt b),
|
||
exact ap pushout.inl (respect_pt a)
|
||
end
|
||
|
||
definition plift_wedge.{u v} (A B : Type*) : plift.{u v} (A ∨ B) ≃* plift.{u v} A ∨ plift.{u v} B :=
|
||
calc plift.{u v} (A ∨ B) ≃* A ∨ B : by exact !pequiv_plift⁻¹ᵉ*
|
||
... ≃* plift.{u v} A ∨ plift.{u v} B : by exact wedge_pequiv !pequiv_plift !pequiv_plift
|
||
|
||
end wedge
|