182 lines
6.1 KiB
Text
182 lines
6.1 KiB
Text
/-
|
||
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
||
Module: init.hit
|
||
Authors: Floris van Doorn
|
||
|
||
Declaration of hits
|
||
-/
|
||
|
||
structure diagram [class] :=
|
||
(Iob : Type)
|
||
(Ihom : Type)
|
||
(ob : Iob → Type)
|
||
(dom cod : Ihom → Iob)
|
||
(hom : Π(j : Ihom), ob (dom j) → ob (cod j))
|
||
|
||
open eq diagram
|
||
|
||
-- structure col (D : diagram) :=
|
||
-- (incl : Π{i : Iob}, ob i)
|
||
-- (eq_endpoint : Π{j : Ihom} (x : ob (dom j)), ob (cod j))
|
||
-- set_option pp.universes true
|
||
-- check @diagram
|
||
-- check @col
|
||
|
||
constant colimit.{u v w} : diagram.{u v w} → Type.{max u v w}
|
||
|
||
namespace colimit
|
||
|
||
constant inclusion : Π [D : diagram] {i : Iob}, ob i → colimit D
|
||
abbreviation ι := @inclusion
|
||
|
||
constant cglue : Π [D : diagram] (j : Ihom) (x : ob (dom j)), ι (hom j x) = ι x
|
||
|
||
/-protected-/ constant rec : Π [D : diagram] {P : colimit D → Type}
|
||
(Pincl : Π⦃i : Iob⦄ (x : ob i), P (ι x))
|
||
(Pglue : Π(j : Ihom) (x : ob (dom j)), cglue j x ▸ Pincl (hom j x) = Pincl x)
|
||
(y : colimit D), P y
|
||
|
||
-- {P : my_colim f → Type} (Hinc : Π⦃n : ℕ⦄ (a : A n), P (inc f a))
|
||
-- (Heq : Π(n : ℕ) (a : A n), inc_eq f a ▸ Hinc (f a) = Hinc a) : Πaa, P aa
|
||
-- init_hit
|
||
|
||
definition comp_incl [D : diagram] {P : colimit D → Type}
|
||
(Pincl : Π⦃i : Iob⦄ (x : ob i), P (ι x))
|
||
(Pglue : Π(j : Ihom) (x : ob (dom j)), cglue j x ▸ Pincl (hom j x) = Pincl x)
|
||
{i : Iob} (x : ob i) : rec Pincl Pglue (ι x) = Pincl x :=
|
||
sorry --idp
|
||
|
||
--set_option pp.notation false
|
||
definition comp_cglue [D : diagram] {P : colimit D → Type}
|
||
(Pincl : Π⦃i : Iob⦄ (x : ob i), P (ι x))
|
||
(Pglue : Π(j : Ihom) (x : ob (dom j)), cglue j x ▸ Pincl (hom j x) = Pincl x)
|
||
{j : Ihom} (x : ob (dom j)) : apdt (rec Pincl Pglue) (cglue j x) = sorry ⬝ Pglue j x ⬝ sorry :=
|
||
--the sorry's in the statement can be removed when comp_incl is definitional
|
||
sorry --idp
|
||
|
||
protected definition rec_on [D : diagram] {P : colimit D → Type} (y : colimit D)
|
||
(Pincl : Π⦃i : Iob⦄ (x : ob i), P (ι x))
|
||
(Pglue : Π(j : Ihom) (x : ob (dom j)), cglue j x ▸ Pincl (hom j x) = Pincl x) : P y :=
|
||
colimit.rec Pincl Pglue y
|
||
|
||
end colimit
|
||
|
||
open colimit bool
|
||
|
||
namespace pushout
|
||
section
|
||
|
||
universe u
|
||
parameters {TL BL TR : Type.{u}} (f : TL → BL) (g : TL → TR)
|
||
|
||
inductive pushout_ob :=
|
||
| tl : pushout_ob
|
||
| bl : pushout_ob
|
||
| tr : pushout_ob
|
||
|
||
open pushout_ob
|
||
|
||
definition pushout_diag [reducible] : diagram :=
|
||
diagram.mk pushout_ob
|
||
bool
|
||
(λi, pushout_ob.rec_on i TL BL TR)
|
||
(λj, bool.rec_on j tl tl)
|
||
(λj, bool.rec_on j bl tr)
|
||
(λj, bool.rec_on j f g)
|
||
|
||
local notation `D` := pushout_diag
|
||
-- open bool
|
||
-- definition pushout_diag : diagram :=
|
||
-- diagram.mk pushout_ob
|
||
-- bool
|
||
-- (λi, match i with | tl := TL | tr := TR | bl := BL end)
|
||
-- (λj, match j with | tt := tl | ff := tl end)
|
||
-- (λj, match j with | tt := bl | ff := tr end)
|
||
-- (λj, match j with | tt := f | ff := g end)
|
||
|
||
definition pushout := colimit pushout_diag
|
||
local attribute pushout_diag [instance]
|
||
|
||
definition inl (x : BL) : pushout :=
|
||
@ι _ _ x
|
||
|
||
definition inr (x : TR) : pushout :=
|
||
@ι _ _ x
|
||
|
||
definition coherence (x : TL) : inl (f x) = @ι _ _ x :=
|
||
@cglue _ _ x
|
||
|
||
definition glue (x : TL) : inl (f x) = inr (g x) :=
|
||
@cglue _ _ x ⬝ (@cglue _ _ x)⁻¹
|
||
|
||
set_option pp.notation false
|
||
set_option pp.implicit true
|
||
set_option pp.beta false
|
||
-- set_option pp.universes true
|
||
|
||
protected theorem rec {P : pushout → Type} --make def
|
||
(Pinl : Π(x : BL), P (inl x))
|
||
(Pinr : Π(x : TR), P (inr x))
|
||
(Pglue : Π(x : TL), glue x ▸ Pinl (f x) = Pinr (g x))
|
||
(y : pushout) : P y :=
|
||
begin
|
||
fapply (@colimit.rec_on _ _ y),
|
||
{ intros [i, x], cases i,
|
||
exact (coherence x ▸ Pinl (f x)),
|
||
apply Pinl,
|
||
apply Pinr},
|
||
{ intros [j, x],
|
||
cases j,
|
||
exact idp,
|
||
change (transport P (@cglue _ tt x) (Pinr (g x)) = transport P (coherence x) (Pinl (f x))),
|
||
--(@cglue _ tt x ▸ (Pinr (g x)) = (coherence x ▸ Pinl (f x))),
|
||
apply concat;rotate 1;apply (idpath (coherence x ▸ Pinl (f x))),
|
||
apply concat;apply (ap (transport _ _));apply (idpath (Pinr (g x))),
|
||
apply tr_eq_of_eq_inv_tr,
|
||
-- rewrite -{(transport (λ (x : pushout), P x) (inverse (coherence x)) (transport P (@cglue _ tt x) (Pinr (g x))))}tr_con,
|
||
apply concat, rotate 1, apply con_tr,
|
||
rewrite -Pglue}
|
||
end
|
||
|
||
example
|
||
{P : pushout → Type}
|
||
(Pinl : Π (x : BL), P (inl x))
|
||
(Pinr : Π (x : TR), P (inr x))
|
||
(Pglue :
|
||
Π (x : TL),
|
||
@eq (P (inr (g x))) (@transport pushout (λ (x : pushout), P x) (inl (f x)) (inr (g x)) (glue x) (Pinl (f x)))
|
||
(Pinr (g x)))
|
||
(y : pushout)
|
||
(x : @ob pushout_diag (@dom pushout_diag tt)) :
|
||
@eq ((λ (x : colimit pushout_diag), P x) (@ι pushout_diag (@dom pushout_diag tt) x))
|
||
(@transport (colimit pushout_diag) (λ (x : colimit pushout_diag), P x)
|
||
(@ι pushout_diag (@cod pushout_diag tt) (@hom pushout_diag tt x))
|
||
(@ι pushout_diag (@dom pushout_diag tt) x)
|
||
(@cglue pushout_diag tt x)
|
||
(@pushout_ob.cases_on (λ (n : pushout_ob), Π (x : @ob pushout_diag n), P (@ι pushout_diag n x))
|
||
(@cod pushout_diag tt)
|
||
(λ (x : @ob pushout_diag tl),
|
||
@transport pushout (λ (x : pushout), P x) (inl (f x)) (@ι pushout_diag (@dom pushout_diag ff) x)
|
||
(coherence x)
|
||
(Pinl (f x)))
|
||
(λ (x : @ob pushout_diag bl), Pinl x)
|
||
(λ (x : @ob pushout_diag tr), Pinr x)
|
||
(@hom pushout_diag tt x)))
|
||
(@pushout_ob.cases_on (λ (n : pushout_ob), Π (x : @ob pushout_diag n), P (@ι pushout_diag n x))
|
||
(@dom pushout_diag tt)
|
||
(λ (x : @ob pushout_diag tl),
|
||
@transport pushout (λ (x : pushout), P x) (inl (f x)) (@ι pushout_diag (@dom pushout_diag ff) x)
|
||
(coherence x)
|
||
(Pinl (f x)))
|
||
(λ (x : @ob pushout_diag bl), Pinl x)
|
||
(λ (x : @ob pushout_diag tr), Pinr x)
|
||
x)
|
||
:=
|
||
begin
|
||
change (transport P (@cglue _ tt x) (Pinr (g x)) = transport P (coherence x) (Pinl (f x))),
|
||
apply sorry
|
||
end
|
||
|
||
exit
|