refactor(hott/algebra/precategory/yoneda): reduce compilation time to 1sec using rewrite tactic
After the latest improvements, the rewrite tactic "works" more often at yoneda.hlean
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Leonardo de Moura
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7ca882d69a
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1 changed files with 27 additions and 32 deletions
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@ -49,9 +49,9 @@ namespace functor
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(λd d' g, F (id, g))
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(λd, !respect_id)
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(λd₁ d₂ d₃ g' g, calc
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : {(id_comp c)⁻¹}
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... = F ((id,g') ∘ (id, g)) : idp
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... = F (id,g') ∘ F (id, g) : by rewrite (respect_comp F))
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F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : by rewrite id_comp
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... = F ((id,g') ∘ (id, g)) : by esimp
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... = F (id,g') ∘ F (id, g) : by rewrite (respect_comp F))
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local abbreviation Fob := @functor_curry_ob
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@ -59,10 +59,10 @@ namespace functor
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nat_trans.mk (λd, F (f, id))
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(λd d' g, calc
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F (id, g) ∘ F (f, id) = F (id ∘ f, g ∘ id) : respect_comp F
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f, g) : by rewrite id_right
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... = F (f ∘ id, g) : by rewrite id_right
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... = F (f ∘ id, id ∘ g) : by rewrite id_left
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f, g) : by rewrite id_right
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... = F (f ∘ id, g) : by rewrite id_right
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... = F (f ∘ id, id ∘ g) : by rewrite id_left
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... = F (f, id) ∘ F (id, g) : (respect_comp F (f, id) (id, g))⁻¹ᵖ)
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local abbreviation Fhom := @functor_curry_hom
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@ -77,10 +77,10 @@ namespace functor
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: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
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nat_trans_eq_mk (λd, calc
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natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : functor_curry_hom_def
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... = F (f' ∘ f, id ∘ id) : {(id_comp d)⁻¹}
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... = F ((f',id) ∘ (f, id)) : idp
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... = F (f',id) ∘ F (f, id) : respect_comp F
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : idp)
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... = F (f' ∘ f, id ∘ id) : by rewrite id_comp
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... = F ((f',id) ∘ (f, id)) : by esimp
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... = F (f',id) ∘ F (f, id) : respect_comp F
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... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : by esimp)
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definition functor_curry [reducible] : C ⇒ E ^c D :=
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functor.mk (functor_curry_ob F)
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@ -98,28 +98,28 @@ namespace functor
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theorem functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
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calc
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Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : idp
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... = id ∘ natural_map (to_fun_hom G id) p.2 : ap (λx, x ∘ _) (respect_id (to_fun_ob G p.1) p.2)
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... = id ∘ natural_map nat_trans.id p.2 : {respect_id G p.1}
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... = id : id_comp
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Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : by esimp
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... = id ∘ natural_map (to_fun_hom G id) p.2 : by rewrite respect_id
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... = id ∘ natural_map nat_trans.id p.2 : by rewrite respect_id
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... = id : id_comp
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theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
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: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
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calc
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Ghom G (f' ∘ f)
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= to_fun_hom (to_fun_ob G p''.1) (f'.2 ∘ f.2) ∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : idp
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= to_fun_hom (to_fun_ob G p''.1) (f'.2 ∘ f.2) ∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by esimp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : {respect_comp (to_fun_ob G p''.1) f'.2 f.2}
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∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : by rewrite (respect_comp (to_fun_ob G p''.1) f'.2 f.2)
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : {respect_comp G f'.1 f.1}
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∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : by rewrite (respect_comp G f'.1 f.1)
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : idp
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∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
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∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : idp
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∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : by esimp
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... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ natural_map (to_fun_hom G f'.1) p'.2)
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∘ (to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2) :
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square_prepostcompose (!naturality⁻¹ᵖ) _ _
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... = Ghom G f' ∘ Ghom G f : idp
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... = Ghom G f' ∘ Ghom G f : by esimp
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definition functor_uncurry [reducible] : C ×c D ⇒ E :=
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functor.mk (functor_uncurry_ob G)
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@ -151,10 +151,6 @@ namespace functor
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-- → F₁ = F₂ :=
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-- functor.rec_on F₁ (λO₁ H₁ id₁ comp₁, functor.rec_on F₂ (λO₂ H₂ id₂ comp₂ p, !functor_eq_mk'1))
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set_option pp.full_names true
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open tactic
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print raw id
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--set_option pp.notation false
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definition functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
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functor_eq_mk (λp, ap (to_fun_ob F) !prod.eta)
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@ -163,10 +159,10 @@ namespace functor
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cases cd with (c,d), cases cd' with (c',d'), cases fg with (f,g),
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have H : (functor_uncurry (functor_curry F)) (f, g) = F (f,g),
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from calc
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(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : idp
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(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : by esimp
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... = F (id ∘ f, g ∘ id) : respect_comp F (id,g) (f,id)
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... = F (f, g ∘ id) : {id_left f}
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... = F (f,g) : {id_right g},
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... = F (f, g ∘ id) : by rewrite id_left
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... = F (f,g) : by rewrite id_right,
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rewrite H,
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apply sorry
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end
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@ -181,12 +177,11 @@ namespace functor
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have H : to_fun_hom (functor_curry (functor_uncurry G) c) g = to_fun_hom (G c) g,
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from calc
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to_fun_hom (functor_curry (functor_uncurry G) c) g
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= to_fun_hom (G c) g ∘ natural_map (to_fun_hom G (ID c)) d : idp
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... = to_fun_hom (G c) g ∘ natural_map (ID (G c)) d
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: ap (λx, to_fun_hom (G c) g ∘ natural_map x d) (respect_id G c)
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= to_fun_hom (G c) g ∘ natural_map (to_fun_hom G (ID c)) d : by esimp
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... = to_fun_hom (G c) g ∘ natural_map (ID (G c)) d : by rewrite respect_id
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... = to_fun_hom (G c) g : id_right,
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rewrite H,
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-- esimp {idp},
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-- esimp {idp},
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apply sorry
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}
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},
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