125 lines
4 KiB
Text
125 lines
4 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: hit.pushout
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Authors: Floris van Doorn
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Declaration of pushout
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-/
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open colimit bool eq
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namespace pushout
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context
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universe u
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parameters {TL BL TR : Type.{u}} (f : TL → BL) (g : TL → TR)
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inductive pushout_ob :=
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| tl : pushout_ob
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| bl : pushout_ob
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| tr : pushout_ob
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open pushout_ob
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definition pushout_diag [reducible] : diagram :=
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diagram.mk pushout_ob
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bool
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(λi, pushout_ob.rec_on i TL BL TR)
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(λj, bool.rec_on j tl tl)
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(λj, bool.rec_on j bl tr)
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(λj, bool.rec_on j f g)
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local notation `D` := pushout_diag
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-- open bool
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-- definition pushout_diag : diagram :=
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-- diagram.mk pushout_ob
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-- bool
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-- (λi, match i with | tl := TL | tr := TR | bl := BL end)
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-- (λj, match j with | tt := tl | ff := tl end)
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-- (λj, match j with | tt := bl | ff := tr end)
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-- (λj, match j with | tt := f | ff := g end)
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definition pushout := colimit pushout_diag
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local attribute pushout_diag [instance]
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definition inl (x : BL) : pushout :=
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@ι _ _ x
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definition inr (x : TR) : pushout :=
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@ι _ _ x
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definition coherence (x : TL) : inl (f x) = @ι _ _ x :=
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@cglue _ _ x
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definition glue (x : TL) : inl (f x) = inr (g x) :=
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@cglue _ _ x ⬝ (@cglue _ _ x)⁻¹
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protected definition rec {P : pushout → Type} (Pinl : Π(x : BL), P (inl x))
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(Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▹ Pinl (f x) = Pinr (g x))
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(y : pushout) : P y :=
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begin
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fapply (@colimit.rec_on _ _ y),
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{ intros [i, x], cases i,
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exact (coherence x ▹ Pinl (f x)),
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apply Pinl,
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apply Pinr},
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{ intros [j, x], cases j,
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exact idp,
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esimp [pushout_ob.cases_on, pushout_diag],
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-- change (transport P (@cglue _ tt x) (Pinr (g x)) = transport P (coherence x) (Pinl (f x))),
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-- apply concat;rotate 1;apply (idpath (coherence x ▹ Pinl (f x))),
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-- apply concat;apply (ap (transport _ _));apply (idpath (Pinr (g x))),
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apply tr_eq_of_eq_inv_tr,
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-- rewrite -{(transport (λ (x : pushout), P x) (inverse (coherence x)) (transport P (@cglue _ tt x) (Pinr (g x))))}tr_con,
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apply concat, rotate 1, apply tr_con,
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rewrite -Pglue}
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end
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protected definition rec_on {P : pushout → Type} (y : pushout) (Pinl : Π(x : BL), P (inl x))
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(Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▹ Pinl (f x) = Pinr (g x)) : P y :=
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rec Pinl Pinr Pglue y
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definition comp_inl {P : pushout → Type} (Pinl : Π(x : BL), P (inl x))
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(Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▹ Pinl (f x) = Pinr (g x))
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(x : BL) : rec Pinl Pinr Pglue (inl x) = Pinl x :=
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@colimit.comp_incl _ _ _ _ _ _ --idp
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definition comp_inr {P : pushout → Type} (Pinl : Π(x : BL), P (inl x))
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(Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▹ Pinl (f x) = Pinr (g x))
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(x : TR) : rec Pinl Pinr Pglue (inr x) = Pinr x :=
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@colimit.comp_incl _ _ _ _ _ _ --idp
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definition comp_glue {P : pushout → Type} (Pinl : Π(x : BL), P (inl x))
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(Pinr : Π(x : TR), P (inr x)) (Pglue : Π(x : TL), glue x ▹ Pinl (f x) = Pinr (g x))
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(x : TL) : apD (rec Pinl Pinr Pglue) (glue x) = sorry ⬝ Pglue x ⬝ sorry :=
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sorry
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end
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end pushout
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open pushout equiv is_equiv unit
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namespace test
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definition foo (u : empty) : unit := star
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example : pushout foo foo ≃ bool :=
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begin
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fapply equiv.MK,
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{ intro x, fapply (pushout.rec_on _ _ x),
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{ intro u, exact ff},
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{ intro u, exact tt},
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{ intro c, cases c}},
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{ intro b, cases b,
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{ exact (inl _ _ ⋆)},
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{ exact (inr _ _ ⋆)},},
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{ intro b, cases b, apply comp_inl, apply comp_inr},
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{ intro x, fapply (pushout.rec_on _ _ x),
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{ intro u, cases u, rewrite [↑function.compose,↑pushout.rec_on,comp_inl]},
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{ intro u, cases u, rewrite [↑function.compose,↑pushout.rec_on,comp_inr]},
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{ intro c, cases c}},
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end
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end test
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