feat(pushout/susp): change definition of elim_type, so that flattening is easier to prove
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2 changed files with 8 additions and 36 deletions
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@ -71,8 +71,9 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
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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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(Pglue : Π(x : TL), Pinl (f x) ≃ Pinr (g x)) : pushout → Type :=
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quotient.elim_type (sum.rec Pinl Pinr)
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begin intro v v' r, induction r, apply Pglue end
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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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@ -81,7 +82,7 @@ parameters {TL BL TR : Type} (f : TL → BL) (g : TL → TR)
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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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!elim_type_eq_of_rel_fn
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protected definition rec_prop {P : pushout → Type} [H : Πx, is_prop (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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@ -150,37 +151,9 @@ namespace pushout
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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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have H : Πz, P z ≃ quotient.elim_type (sum.rec Pinl Pinr) Pglue' z,
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begin
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intro z, apply equiv_of_eq,
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have H1 : pushout.elim_type Pinl Pinr Pglue
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= quotient.elim_type (sum.rec Pinl Pinr) Pglue',
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begin
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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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have 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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begin intros a a' r, cases r, reflexivity end,
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rewrite (eq_of_homotopy3 H2)
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end,
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apply ap10 H1
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end,
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apply equiv.trans (sigma_equiv_sigma_right H),
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apply equiv.trans (quotient.flattening.flattening_lemma R (sum.rec Pinl Pinr) Pglue'),
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apply equiv.trans !quotient.flattening.flattening_lemma,
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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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@ -59,7 +59,7 @@ namespace susp
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protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(x : susp A) : Type :=
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susp.elim PN PS (λa, ua (Pm a)) x
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pushout.elim_type (λx, PN) (λx, PS) Pm x
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protected definition elim_type_on [reducible] (x : susp A)
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(PN : Type) (PS : Type) (Pm : A → PN ≃ PS) : Type :=
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@ -67,7 +67,7 @@ namespace susp
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theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(a : A) : transport (susp.elim_type PN PS Pm) (merid a) = Pm a :=
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by rewrite [tr_eq_cast_ap_fn,↑susp.elim_type,elim_merid];apply cast_ua_fn
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!elim_type_glue
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protected definition merid_square {a a' : A} (p : a = a')
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: square (merid a) (merid a') idp idp :=
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@ -134,8 +134,7 @@ namespace susp
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protected definition flattening : sigma P ≃ pushout F G :=
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begin
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apply equiv.trans (pushout.flattening (λ(a : A), star) (λ(a : A), star)
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(λx, unit.cases_on x PN) (λx, unit.cases_on x PS) Pm),
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apply equiv.trans !pushout.flattening,
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fapply pushout.equiv,
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{ exact sigma.equiv_prod A PN },
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{ apply sigma.sigma_unit_left },
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