62 lines
2.8 KiB
Text
62 lines
2.8 KiB
Text
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/-
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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, Jakob von Raumer
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Functor product precategory and (TODO) category
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-/
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import ..category ..functor types.sum
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open eq sum is_trunc functor lift
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namespace category
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--set_option pp.universes true
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definition sum_hom.{u v w x} [unfold 5 6] {obC : Type.{u}} {obD : Type.{v}}
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(C : precategory.{u w} obC) (D : precategory.{v x} obD)
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: obC + obD → obC + obD → Type.{max w x} :=
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sum.rec (λc, sum.rec (λc', lift (c ⟶ c')) (λd, lift empty))
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(λd, sum.rec (λc, lift empty) (λd', lift (d ⟶ d')))
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theorem is_hset_sum_hom {obC : Type} {obD : Type}
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(C : precategory obC) (D : precategory obD) (x y : obC + obD)
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: is_hset (sum_hom C D x y) :=
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by induction x: induction y: esimp at *: exact _
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local attribute is_hset_sum_hom [instance]
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definition precategory_sum [constructor] {obC obD : Type}
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(C : precategory obC) (D : precategory obD) : precategory (obC + obD) :=
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precategory.mk (sum_hom C D)
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(λ a b c g f, begin induction a: induction b: induction c: esimp at *;
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induction f with f; induction g with g; (contradiction | exact up (g ∘ f)) end)
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(λ a, by induction a: exact up id)
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(λ a b c d h g f,
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abstract begin induction a: induction b: induction c: induction d:
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esimp at *; induction f with f; induction g with g; induction h with h;
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esimp at *; try contradiction: apply ap up !assoc end end)
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(λ a b f, abstract begin induction a: induction b: esimp at *;
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induction f with f; esimp; try contradiction: exact ap up !id_left end end)
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(λ a b f, abstract begin induction a: induction b: esimp at *;
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induction f with f; esimp; try contradiction: exact ap up !id_right end end)
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definition Precategory_sum [constructor] (C D : Precategory) : Precategory :=
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precategory.Mk (precategory_sum C D)
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infixr `+c`:27 := Precategory_sum
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definition sum_functor [constructor] {C C' D D' : Precategory}
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(F : C ⇒ D) (G : C' ⇒ D') : C +c C' ⇒ D +c D' :=
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functor.mk (λ a, by induction a: (exact inl (F a)|exact inr (G a)))
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(λ a b f, begin induction a: induction b: esimp at *;
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induction f with f; esimp; try contradiction: (exact up (F f)|exact up (G f)) end)
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(λ a, abstract by induction a: esimp; exact ap up !respect_id end)
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(λ a b c g f, abstract begin induction a: induction b: induction c: esimp at *;
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induction f with f; induction g with g; try contradiction:
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esimp; exact ap up !respect_comp end end)
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infixr `+f`:27 := sum_functor
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end category
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