113 lines
3.7 KiB
Text
113 lines
3.7 KiB
Text
/-
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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 the pushout
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-/
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import .type_quotient
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open type_quotient eq sum equiv equiv.ops
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namespace pushout
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section
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parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
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local abbreviation A := BL + TR
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inductive pushout_rel : A → A → Type :=
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| Rmk : Π(x : TL), pushout_rel (inl (f x)) (inr (g x))
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open pushout_rel
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local abbreviation R := pushout_rel
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definition pushout : Type := type_quotient pushout_rel -- TODO: define this in root namespace
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definition inl (x : BL) : pushout :=
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class_of R (inl x)
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definition inr (x : TR) : pushout :=
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class_of R (inr x)
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definition glue (x : TL) : inl (f x) = inr (g x) :=
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eq_of_rel pushout_rel (Rmk f g x)
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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 (type_quotient.rec_on y),
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{ intro a, cases a,
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apply Pinl,
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apply Pinr},
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{ intros [a, a', H], cases H, apply Pglue}
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end
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protected definition rec_on [reducible] {P : pushout → Type} (y : pushout)
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(Pinl : Π(x : BL), P (inl x)) (Pinr : Π(x : TR), P (inr x))
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(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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theorem rec_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) = Pglue x :=
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!rec_eq_of_rel
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protected definition elim {P : Type} (Pinl : BL → P) (Pinr : TR → P)
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(Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) (y : pushout) : P :=
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rec Pinl Pinr (λx, !tr_constant ⬝ Pglue x) y
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protected definition elim_on [reducible] {P : Type} (y : pushout) (Pinl : BL → P)
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(Pinr : TR → P) (Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) : P :=
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elim Pinl Pinr Pglue y
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theorem elim_glue {P : Type} (Pinl : BL → P) (Pinr : TR → P)
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(Pglue : Π(x : TL), Pinl (f x) = Pinr (g x)) (x : TL)
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: ap (elim Pinl Pinr Pglue) (glue x) = Pglue x :=
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begin
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apply (@cancel_left _ _ _ _ (tr_constant (glue x) (elim Pinl Pinr Pglue (inl (f x))))),
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rewrite [-apd_eq_tr_constant_con_ap,↑elim,rec_glue],
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end
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protected definition elim_type (Pinl : BL → Type) (Pinr : TR → Type)
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(Pglue : Π(x : TL), Pinl (f x) ≃ Pinr (g x)) (y : pushout) : Type :=
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elim Pinl Pinr (λx, ua (Pglue x)) y
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protected definition elim_type_on [reducible] (y : pushout) (Pinl : BL → Type)
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(Pinr : TR → Type) (Pglue : Π(x : TL), Pinl (f x) ≃ Pinr (g x)) : Type :=
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elim_type Pinl Pinr Pglue y
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theorem elim_type_glue (Pinl : BL → Type) (Pinr : TR → Type)
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(Pglue : Π(x : TL), Pinl (f x) ≃ Pinr (g x)) (y : pushout) (x : TL)
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: transport (elim_type Pinl Pinr Pglue) (glue x) = Pglue x :=
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by rewrite [tr_eq_cast_ap_fn,↑elim_type,elim_glue];apply cast_ua_fn
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end
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namespace test
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open pushout equiv is_equiv unit bool
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private definition unit_of_empty (u : empty) : unit := star
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example : pushout unit_of_empty unit_of_empty ≃ 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, esimp, esimp},
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{ intro x, fapply (pushout.rec_on _ _ x),
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intro u, cases u, esimp,
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intro u, cases u, esimp,
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intro c, cases c},
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end
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end test
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end pushout
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