230 lines
8.4 KiB
Text
230 lines
8.4 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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Authors: Floris van Doorn
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Declaration of the pushout
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-/
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import .quotient cubical.square types.sigma
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open quotient eq sum equiv equiv.ops is_trunc
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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 := quotient R -- TODO: define this in root namespace
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parameters {f g}
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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), Pinl (f x) =[glue x] Pinr (g x))
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(y : pushout) : P y :=
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begin
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induction y,
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{ cases a,
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apply Pinl,
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apply Pinr},
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{ 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), Pinl (f x) =[glue 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), Pinl (f x) =[glue x] Pinr (g x))
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(x : TL) : apdo (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, pathover_of_eq (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 eq_of_fn_eq_fn_inv !(pathover_constant (glue x)),
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rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑pushout.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)) (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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protected definition rec_hprop {P : pushout → Type} [H : Πx, is_hprop (P x)]
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(Pinl : Π(x : BL), P (inl x)) (Pinr : Π(x : TR), P (inr x)) (y : pushout) :=
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rec Pinl Pinr (λx, !is_hprop.elimo) y
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protected definition elim_hprop {P : Type} [H : is_hprop P] (Pinl : BL → P) (Pinr : TR → P)
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(y : pushout) : P :=
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elim Pinl Pinr (λa, !is_hprop.elim) y
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end
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end pushout
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attribute pushout.inl pushout.inr [constructor]
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attribute pushout.rec pushout.elim [unfold 10] [recursor 10]
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attribute pushout.elim_type [unfold 9]
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attribute pushout.rec_on pushout.elim_on [unfold 7]
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attribute pushout.elim_type_on [unfold 6]
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open sigma
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namespace pushout
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variables {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
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/- The non-dependent universal property -/
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definition pushout_arrow_equiv (C : Type)
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: (pushout f g → C) ≃ (Σ(i : BL → C) (j : TR → C), Πc, i (f c) = j (g c)) :=
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begin
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fapply equiv.MK,
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{ intro f, exact ⟨λx, f (inl x), λx, f (inr x), λx, ap f (glue x)⟩},
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{ intro v x, induction v with i w, induction w with j p, induction x,
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exact (i a), exact (j a), exact (p x)},
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{ intro v, induction v with i w, induction w with j p, esimp,
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apply ap (λp, ⟨i, j, p⟩), apply eq_of_homotopy, intro x, apply elim_glue},
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{ intro f, apply eq_of_homotopy, intro x, induction x: esimp,
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apply eq_pathover, apply hdeg_square, esimp, apply elim_glue},
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end
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end pushout
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open function sigma.ops
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namespace pushout
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/- The flattening lemma -/
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section
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universe variable u
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parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
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(Pinl : BL → Type.{u}) (Pinr : TR → Type.{u})
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(Pglue : Π(x : TL), Pinl (f x) ≃ Pinr (g x))
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include Pglue
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local abbreviation A := BL + TR
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local abbreviation R : A → A → Type := pushout_rel f g
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local abbreviation P [unfold 5] := pushout.elim_type Pinl Pinr Pglue
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local abbreviation F : sigma (Pinl ∘ f) → sigma Pinl :=
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λz, ⟨ f z.1 , z.2 ⟩
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local abbreviation G : sigma (Pinl ∘ f) → sigma Pinr :=
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λz, ⟨ g z.1 , Pglue z.1 z.2 ⟩
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local abbreviation Pglue' : Π ⦃a a' : A⦄,
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R a a' → sum.rec Pinl Pinr a ≃ sum.rec Pinl Pinr a' :=
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@pushout_rel.rec TL BL TR f g
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(λ ⦃a a' ⦄ (r : R a a'),
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(sum.rec Pinl Pinr a) ≃ (sum.rec Pinl Pinr a')) Pglue
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protected definition flattening : sigma P ≃ pushout F G :=
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begin
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assert H : Πz, P z ≃ quotient.elim_type (sum.rec Pinl Pinr) Pglue' z,
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{ intro z, apply equiv_of_eq,
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assert H1 : pushout.elim_type Pinl Pinr Pglue
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= quotient.elim_type (sum.rec Pinl Pinr) Pglue',
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{ change
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quotient.rec (sum.rec Pinl Pinr)
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(λa a' r, pushout_rel.cases_on r (λx, pathover_of_eq (ua (Pglue x))))
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= quotient.rec (sum.rec Pinl Pinr)
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(λa a' r, pathover_of_eq (ua (pushout_rel.cases_on r Pglue))),
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assert H2 : Π⦃a a'⦄ r : pushout_rel f g a a',
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pushout_rel.cases_on r (λx, pathover_of_eq (ua (Pglue x)))
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= pathover_of_eq (ua (pushout_rel.cases_on r Pglue))
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:> sum.rec Pinl Pinr a =[eq_of_rel (pushout_rel f g) r]
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sum.rec Pinl Pinr a',
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{ intros a a' r, cases r, reflexivity },
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rewrite (eq_of_homotopy3 H2) },
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apply ap10 H1 },
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apply equiv.trans (sigma_equiv_sigma_id H),
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apply equiv.trans (quotient.flattening.flattening_lemma R (sum.rec Pinl Pinr) Pglue'),
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fapply equiv.MK,
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{ intro q, induction q with z z z' fr,
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{ induction z with a p, induction a with x x,
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{ exact inl ⟨x, p⟩ },
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{ exact inr ⟨x, p⟩ } },
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{ induction fr with a a' r p, induction r with x,
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exact glue ⟨x, p⟩ } },
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{ intro q, induction q with xp xp xp,
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{ exact class_of _ ⟨sum.inl xp.1, xp.2⟩ },
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{ exact class_of _ ⟨sum.inr xp.1, xp.2⟩ },
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{ apply eq_of_rel, constructor } },
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{ intro q, induction q with xp xp xp: induction xp with x p,
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{ apply ap inl, reflexivity },
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{ apply ap inr, reflexivity },
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{ unfold F, unfold G, apply eq_pathover,
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rewrite [ap_id,ap_compose' (quotient.elim _ _)],
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krewrite elim_glue, krewrite elim_eq_of_rel, apply hrefl } },
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{ intro q, induction q with z z z' fr,
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{ induction z with a p, induction a with x x,
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{ reflexivity },
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{ reflexivity } },
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{ induction fr with a a' r p, induction r with x,
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esimp, apply eq_pathover,
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rewrite [ap_id,ap_compose' (pushout.elim _ _ _)],
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krewrite elim_eq_of_rel, krewrite elim_glue, apply hrefl } }
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end
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end
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-- Commutativity of pushouts
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section
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variables {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
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protected definition transpose [constructor] : pushout f g → pushout g f :=
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begin
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intro x, induction x, apply inr a, apply inl a, apply !glue⁻¹
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end
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--TODO prove without krewrite?
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protected definition transpose_involutive (x : pushout f g) :
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pushout.transpose g f (pushout.transpose f g x) = x :=
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begin
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induction x, apply idp, apply idp,
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apply eq_pathover, refine _ ⬝hp !ap_id⁻¹,
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refine !(ap_compose (pushout.transpose _ _)) ⬝ph _, esimp[pushout.transpose],
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krewrite [elim_glue, ap_inv, elim_glue, inv_inv], apply hrfl
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end
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protected definition symm : pushout f g ≃ pushout g f :=
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begin
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fapply equiv.MK, do 2 exact !pushout.transpose,
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do 2 (intro x; apply pushout.transpose_involutive),
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end
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end
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end pushout
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