23 lines
751 B
Text
23 lines
751 B
Text
prelude
|
|
-- category
|
|
|
|
definition Prop := Type.{0}
|
|
constant eq {A : Type} : A → A → Prop
|
|
infix `=`:50 := eq
|
|
|
|
inductive category (ob : Type) (mor : ob → ob → Type) : Type
|
|
| mk : Π (id : Π (A : ob), mor A A),
|
|
(Π (A B : ob) (f : mor A A), id A = f) → category
|
|
|
|
definition id (ob : Type) (mor : ob → ob → Type) (Cat : category ob mor) := category.rec (λ id idl, id) Cat
|
|
constant ob : Type.{1}
|
|
constant mor : ob → ob → Type.{1}
|
|
constant Cat : category ob mor
|
|
|
|
attribute id [reducible]
|
|
theorem id_left (A : ob) (f : mor A A) : @eq (mor A A) (id ob mor Cat A) f :=
|
|
@category.rec ob mor (λ (C : category ob mor), @eq (mor A A) (id ob mor C A) f)
|
|
(λ (id : Π (T : ob), mor T T)
|
|
(idl : Π (T : ob), _),
|
|
idl A A f)
|
|
Cat
|