2014-09-09 16:25:35 +00:00
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-- category
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2014-09-17 21:39:05 +00:00
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definition Prop := Type.{0}
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2014-09-09 16:25:35 +00:00
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variable eq {A : Type} : A → A → Prop
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infix `=`:50 := eq
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inductive category (ob : Type) (mor : ob → ob → Type) : Type :=
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mk : Π (id : Π (A : ob), mor A A),
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(Π (A B : ob) (f : mor A A), id A = f) → category ob mor
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2014-09-17 21:39:05 +00:00
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definition id (ob : Type) (mor : ob → ob → Type) (Cat : category ob mor) := category.rec (λ id idl, id) Cat
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2014-09-09 16:25:35 +00:00
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variable ob : Type.{1}
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variable mor : ob → ob → Type.{1}
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variable Cat : category ob mor
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theorem id_left (A : ob) (f : mor A A) : @eq (mor A A) (id ob mor Cat A) f :=
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@category.rec ob mor (λ (C : category ob mor), @eq (mor A A) (id ob mor C A) f)
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(λ (id : Π (T : ob), mor T T)
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(idl : Π (T : ob), _),
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idl A A f)
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Cat
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