lean2/tests/lean/531.hlean

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/-
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 : ) (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)) : apd (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
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,
esimp [pushout_ob.cases_on],
apply concat, rotate 1, apply (idpath (coherence x ▹ Pinl (f x))),
apply concat, apply (ap (transport _ _)), apply (idpath (Pinr (g x))),
apply eq_tr_of_inv_tr_eq,
rewrite -{(transport (λ (x : pushout), P x) (inverse (coherence x)) (transport P (@cglue _ tt x) (Pinr (g x))))}con_tr,
apply sorry
}
end
example
{P : pushout → Type}
{Pinl : Π (x : BL), P (inl x)}
{Pinr : Π (x : TR), P (inr x)}
{Pglue : Π (x : TL), eq (transport (λ (x : pushout), P x) (glue x) (Pinl (f x))) (Pinr (g x))}
{y : pushout}
{x : @ob _ (@dom _ tt)}
: eq (transport (λ (x : pushout), P x) (inverse (coherence x)) (transport P (@cglue _ tt x) (Pinr (g x))))
(Pinl (f x)) :=
begin
rewrite -{(transport (λ (x : pushout), P x) (inverse (coherence x)) (transport P (@cglue _ tt x) (Pinr (g x))))}con_tr,
apply sorry
end
exit