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November 30, 2019 15:25
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Yoneda as a CPS transformer
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(* http://m-hiyama.hatenablog.com/entry/20080116/1200468797 *) | |
Module cps. | |
(* CPS transformation *) | |
Definition yoneda (X Y A: Type) (f: X -> Y) : (Y -> A) -> X -> A := | |
fun cont: Y -> A => fun x: X => cont (f x). | |
Definition yoneda_objpart (X A: Type) (x: X) : (X -> A) -> A := | |
fun cont: X -> A => cont(x). | |
Definition f (x: nat) : nat := x + 1. | |
Definition g (x: nat) : nat := x * 2. | |
Example yoneda_nat: forall A: Type, forall cont: nat -> A, forall a: nat, | |
((yoneda nat nat A f) ((yoneda nat nat A g) cont)) a = (yoneda nat nat A (fun x: nat => g (f x))) cont a. | |
Proof. | |
intros A cont a. | |
reflexivity. | |
Qed. | |
Theorem yoneda_is_contravariant_functor : forall X Y Z A: Type, forall f: X -> Y, forall g: Y -> Z, forall cont: Z -> A, | |
(yoneda X Y A f) ((yoneda Y Z A g) cont) = (yoneda X Z A (fun x: X => g (f x))) cont. | |
Proof. | |
intros X Y Z A f0 g0 cont. | |
reflexivity. | |
Qed. | |
Definition id (X: Type) x: X := x. | |
Theorem yoneda_is_contravariant_functor2 : forall X A: Type, forall cont: X -> A, forall x: X, | |
yoneda X X A (id X) cont x = (id A (yoneda_objpart X A x cont)). | |
Proof. | |
intros X A cont x. | |
reflexivity. | |
Qed. |
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