250 lines
11 KiB
Text
250 lines
11 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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-/
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import types.trunc types.pi arity
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open eq is_trunc pi
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namespace category
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/-
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Just as in Coq-HoTT we add two redundant fields to precategories: assoc' and id_id.
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The first is to make (Cᵒᵖ)ᵒᵖ = C definitionally when C is a constructor.
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The second is to ensure that the functor from the terminal category 1 ⇒ Cᵒᵖ is
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opposite to the functor 1 ⇒ C.
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-/
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structure precategory [class] (ob : Type) : Type :=
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mk' ::
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(hom : ob → ob → Type)
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(comp : Π⦃a b c : ob⦄, hom b c → hom a b → hom a c)
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(ID : Π (a : ob), hom a a)
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(assoc : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
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comp h (comp g f) = comp (comp h g) f)
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(assoc' : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
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comp (comp h g) f = comp h (comp g f))
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(id_left : Π ⦃a b : ob⦄ (f : hom a b), comp !ID f = f)
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(id_right : Π ⦃a b : ob⦄ (f : hom a b), comp f !ID = f)
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(id_id : Π (a : ob), comp !ID !ID = ID a)
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(is_hset_hom : Π(a b : ob), is_hset (hom a b))
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attribute precategory [multiple-instances]
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attribute precategory.is_hset_hom [instance]
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infixr `∘` := precategory.comp
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-- input ⟶ using \--> (this is a different arrow than \-> (→))
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infixl [parsing-only] `⟶`:25 := precategory.hom
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namespace hom
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infixl `⟶`:25 := precategory.hom -- if you open this namespace, hom a b is printed as a ⟶ b
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end hom
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abbreviation hom [unfold 2] := @precategory.hom
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abbreviation comp [unfold 2] := @precategory.comp
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abbreviation ID [unfold 2] := @precategory.ID
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abbreviation assoc [unfold 2] := @precategory.assoc
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abbreviation assoc' [unfold 2] := @precategory.assoc'
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abbreviation id_left [unfold 2] := @precategory.id_left
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abbreviation id_right [unfold 2] := @precategory.id_right
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abbreviation id_id [unfold 2] := @precategory.id_id
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abbreviation is_hset_hom [unfold 2] := @precategory.is_hset_hom
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-- the constructor you want to use in practice
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protected definition precategory.mk [constructor] {ob : Type} (hom : ob → ob → Type)
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[hset : Π (a b : ob), is_hset (hom a b)]
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(comp : Π ⦃a b c : ob⦄, hom b c → hom a b → hom a c) (ID : Π (a : ob), hom a a)
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(ass : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
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comp h (comp g f) = comp (comp h g) f)
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(idl : Π ⦃a b : ob⦄ (f : hom a b), comp (ID b) f = f)
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(idr : Π ⦃a b : ob⦄ (f : hom a b), comp f (ID a) = f) : precategory ob :=
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precategory.mk' hom comp ID ass (λa b c d h g f, !ass⁻¹) idl idr (λa, !idl) hset
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section basic_lemmas
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variables {ob : Type} [C : precategory ob]
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variables {a b c d : ob} {h : c ⟶ d} {g : hom b c} {f f' : hom a b} {i : a ⟶ a}
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include C
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definition id [reducible] [unfold 2] := ID a
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definition id_leftright (f : hom a b) : id ∘ f ∘ id = f := !id_left ⬝ !id_right
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definition comp_id_eq_id_comp (f : hom a b) : f ∘ id = id ∘ f := !id_right ⬝ !id_left⁻¹
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definition id_comp_eq_comp_id (f : hom a b) : id ∘ f = f ∘ id := !id_left ⬝ !id_right⁻¹
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definition left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
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calc i = i ∘ id : by rewrite id_right
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... = id : by rewrite H
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definition right_id_unique (H : Π{b} {f : hom a b}, f ∘ i = f) : i = id :=
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calc i = id ∘ i : by rewrite id_left
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... = id : by rewrite H
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definition homset [reducible] [constructor] (x y : ob) : hset :=
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hset.mk (hom x y) _
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end basic_lemmas
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section squares
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parameters {ob : Type} [C : precategory ob]
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local infixl `⟶`:25 := @precategory.hom ob C
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local infixr `∘` := @precategory.comp ob C _ _ _
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definition compose_squares {xa xb xc ya yb yc : ob}
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{xg : xb ⟶ xc} {xf : xa ⟶ xb} {yg : yb ⟶ yc} {yf : ya ⟶ yb}
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{wa : xa ⟶ ya} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(xyab : wb ∘ xf = yf ∘ wa) (xybc : wc ∘ xg = yg ∘ wb)
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: wc ∘ (xg ∘ xf) = (yg ∘ yf) ∘ wa :=
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calc
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wc ∘ (xg ∘ xf) = (wc ∘ xg) ∘ xf : by rewrite assoc
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... = (yg ∘ wb) ∘ xf : by rewrite xybc
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... = yg ∘ (wb ∘ xf) : by rewrite assoc
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... = yg ∘ (yf ∘ wa) : by rewrite xyab
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... = (yg ∘ yf) ∘ wa : by rewrite assoc
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definition compose_squares_2x2 {xa xb xc ya yb yc za zb zc : ob}
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{xg : xb ⟶ xc} {xf : xa ⟶ xb} {yg : yb ⟶ yc} {yf : ya ⟶ yb} {zg : zb ⟶ zc} {zf : za ⟶ zb}
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{va : ya ⟶ za} {vb : yb ⟶ zb} {vc : yc ⟶ zc} {wa : xa ⟶ ya} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(xyab : wb ∘ xf = yf ∘ wa) (xybc : wc ∘ xg = yg ∘ wb)
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(yzab : vb ∘ yf = zf ∘ va) (yzbc : vc ∘ yg = zg ∘ vb)
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: (vc ∘ wc) ∘ (xg ∘ xf) = (zg ∘ zf) ∘ (va ∘ wa) :=
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calc
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(vc ∘ wc) ∘ (xg ∘ xf) = vc ∘ (wc ∘ (xg ∘ xf)) : by rewrite (assoc vc wc _)
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... = vc ∘ ((yg ∘ yf) ∘ wa) : by rewrite (compose_squares xyab xybc)
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... = (vc ∘ (yg ∘ yf)) ∘ wa : by rewrite assoc
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... = ((zg ∘ zf) ∘ va) ∘ wa : by rewrite (compose_squares yzab yzbc)
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... = (zg ∘ zf) ∘ (va ∘ wa) : by rewrite assoc
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definition square_precompose {xa xb xc yb yc : ob}
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{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(H : wc ∘ xg = yg ∘ wb) (xf : xa ⟶ xb) : wc ∘ xg ∘ xf = yg ∘ wb ∘ xf :=
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calc
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wc ∘ xg ∘ xf = (wc ∘ xg) ∘ xf : by rewrite assoc
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... = (yg ∘ wb) ∘ xf : by rewrite H
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... = yg ∘ wb ∘ xf : by rewrite assoc
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definition square_postcompose {xb xc yb yc yd : ob}
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{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(H : wc ∘ xg = yg ∘ wb) (yh : yc ⟶ yd) : (yh ∘ wc) ∘ xg = (yh ∘ yg) ∘ wb :=
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calc
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(yh ∘ wc) ∘ xg = yh ∘ wc ∘ xg : by rewrite assoc
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... = yh ∘ yg ∘ wb : by rewrite H
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... = (yh ∘ yg) ∘ wb : by rewrite assoc
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definition square_prepostcompose {xa xb xc yb yc yd : ob}
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{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(H : wc ∘ xg = yg ∘ wb) (yh : yc ⟶ yd) (xf : xa ⟶ xb)
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: (yh ∘ wc) ∘ (xg ∘ xf) = (yh ∘ yg) ∘ (wb ∘ xf) :=
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square_precompose (square_postcompose H yh) xf
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end squares
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structure Precategory : Type :=
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(carrier : Type)
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(struct : precategory carrier)
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definition precategory.Mk [reducible] [constructor] {ob} (C) : Precategory := Precategory.mk ob C
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definition precategory.MK [reducible] [constructor] (a b c d e f g h) : Precategory :=
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Precategory.mk a (@precategory.mk a b c d e f g h)
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abbreviation carrier := @Precategory.carrier
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attribute Precategory.carrier [coercion]
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attribute Precategory.struct [instance] [priority 10000] [coercion]
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-- definition precategory.carrier [coercion] [reducible] := Precategory.carrier
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-- definition precategory.struct [instance] [coercion] [reducible] := Precategory.struct
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notation g `∘[`:60 C:0 `]`:0 f:60 :=
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@comp (Precategory.carrier C) (Precategory.struct C) _ _ _ g f
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-- TODO: make this left associative
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-- TODO: change this notation?
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definition Precategory.eta (C : Precategory) : Precategory.mk C C = C :=
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Precategory.rec (λob c, idp) C
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/-Characterization of paths between precategories-/
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definition precategory_mk'_eq (ob : Type)
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(hom1 : ob → ob → Type)
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(hom2 : ob → ob → Type)
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(homH1 : Π(a b : ob), is_hset (hom1 a b))
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(homH2 : Π(a b : ob), is_hset (hom2 a b))
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(comp1 : Π⦃a b c : ob⦄, hom1 b c → hom1 a b → hom1 a c)
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(comp2 : Π⦃a b c : ob⦄, hom2 b c → hom2 a b → hom2 a c)
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(ID1 : Π (a : ob), hom1 a a)
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(ID2 : Π (a : ob), hom2 a a)
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(assoc1 : Π ⦃a b c d : ob⦄ (h : hom1 c d) (g : hom1 b c) (f : hom1 a b),
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comp1 h (comp1 g f) = comp1 (comp1 h g) f)
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(assoc2 : Π ⦃a b c d : ob⦄ (h : hom2 c d) (g : hom2 b c) (f : hom2 a b),
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comp2 h (comp2 g f) = comp2 (comp2 h g) f)
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(assoc1' : Π ⦃a b c d : ob⦄ (h : hom1 c d) (g : hom1 b c) (f : hom1 a b),
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comp1 (comp1 h g) f = comp1 h (comp1 g f))
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(assoc2' : Π ⦃a b c d : ob⦄ (h : hom2 c d) (g : hom2 b c) (f : hom2 a b),
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comp2 (comp2 h g) f = comp2 h (comp2 g f))
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(id_left1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 !ID1 f = f)
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(id_left2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 !ID2 f = f)
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(id_right1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 f !ID1 = f)
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(id_right2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 f !ID2 = f)
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(id_id1 : Π (a : ob), comp1 !ID1 !ID1 = ID1 a)
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(id_id2 : Π (a : ob), comp2 !ID2 !ID2 = ID2 a)
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(p : hom1 = hom2)
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(q : p ▸ comp1 = comp2)
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(r : p ▸ ID1 = ID2) :
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precategory.mk' hom1 comp1 ID1 assoc1 assoc1' id_left1 id_right1 id_id1 homH1
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= precategory.mk' hom2 comp2 ID2 assoc2 assoc2' id_left2 id_right2 id_id2 homH2 :=
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begin
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cases p, cases q, cases r,
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apply (ap0111111 (precategory.mk' hom2 comp2 ID2)),
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repeat (apply is_hprop.elim),
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end
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definition precategory_eq (ob : Type)
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(C D : precategory ob)
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(p : @hom ob C = @hom ob D)
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(q : transport (λ x, Πa b c, x b c → x a b → x a c) p
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(@comp ob C) = @comp ob D)
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(r : transport (λ x, Πa, x a a) p (@ID ob C) = @ID ob D) : C = D :=
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begin
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cases C, cases D,
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apply precategory_mk'_eq, apply q, apply r,
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end
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definition precategory_mk_eq (ob : Type)
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(hom1 : ob → ob → Type)
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(hom2 : ob → ob → Type)
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(homH1 : Π(a b : ob), is_hset (hom1 a b))
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(homH2 : Π(a b : ob), is_hset (hom2 a b))
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(comp1 : Π⦃a b c : ob⦄, hom1 b c → hom1 a b → hom1 a c)
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(comp2 : Π⦃a b c : ob⦄, hom2 b c → hom2 a b → hom2 a c)
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(ID1 : Π (a : ob), hom1 a a)
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(ID2 : Π (a : ob), hom2 a a)
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(assoc1 : Π ⦃a b c d : ob⦄ (h : hom1 c d) (g : hom1 b c) (f : hom1 a b),
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comp1 h (comp1 g f) = comp1 (comp1 h g) f)
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(assoc2 : Π ⦃a b c d : ob⦄ (h : hom2 c d) (g : hom2 b c) (f : hom2 a b),
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comp2 h (comp2 g f) = comp2 (comp2 h g) f)
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(id_left1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 !ID1 f = f)
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(id_left2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 !ID2 f = f)
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(id_right1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 f !ID1 = f)
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(id_right2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 f !ID2 = f)
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(p : Π (a b : ob), hom1 a b = hom2 a b)
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(q : transport (λ x, Π a b c, x b c → x a b → x a c)
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(eq_of_homotopy (λ a, eq_of_homotopy (λ b, p a b))) @comp1 = @comp2)
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(r : transport (λ x, Π a, x a a)
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(eq_of_homotopy (λ (x : ob), eq_of_homotopy (λ (x_1 : ob), p x x_1)))
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ID1 = ID2) :
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precategory.mk hom1 comp1 ID1 assoc1 id_left1 id_right1
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= precategory.mk hom2 comp2 ID2 assoc2 id_left2 id_right2 :=
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begin
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fapply precategory_eq,
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apply eq_of_homotopy, intros,
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apply eq_of_homotopy, intros,
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exact (p _ _),
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exact q,
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exact r,
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end
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definition Precategory_eq (C D : Precategory)
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(p : carrier C = carrier D)
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(q : p ▸ (Precategory.struct C) = Precategory.struct D) : C = D :=
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begin
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cases C, cases D,
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cases p, cases q,
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apply idp,
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end
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end category
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