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| -- X = presheaf ^ profunctor | |
| -- Yoneda = X (->) | |
| newtype X p f a = X { runX :: forall b. p a b -> f b } | |
| hoistX :: forall p f g. (f ~> g) -> (X p f ~> X p g) | |
| hoistX phi (X f) = X (\g -> phi (f g)) | |
| withX :: forall w p f a. (Category p, Functor f, Comonad w, Sieve p w) => (X p f a -> X p f a) -> (f a -> f a) | |
| withX phi = lowerX @p @f . phi . liftX @w @p @f | |
| withX' :: forall w p f g a. (Category p, Functor f, Comonad w, Sieve p w) => (X p f a -> X p g a) -> (f a -> g a) | |
| withX' phi = lowerX @p @g . phi . liftX @w @p @f | |
| arr :: forall a b p. Category p => Profunctor p => (a -> b) -> p a b | |
| arr f = rmap f id | |
| unarr :: forall w p a b. (Comonad w, Sieve p w) => p a b -> a -> b | |
| unarr pab a = extract (sieve pab a) | |
| liftX :: forall w p f. (Comonad w, Sieve p w, Functor f) => f ~> X p f | |
| liftX m = X (\g -> unarr g <$> m) | |
| lowerX :: forall p f. (Category p, Profunctor p) => X p f ~> f | |
| lowerX (X k) = k (arr id) | |
| instance Profunctor p => Functor (X p f) where | |
| fmap f (X m) = X (\k -> m (lmap f k)) | |
| transform :: forall f. Functor f => f Int -> f Int | |
| transform = fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) . fmap (*2) | |
| -- Yoneda optimization | |
| {- | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) . fmap (*2) . liftX @(Identity) @(->) @[] $ m = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) . fmap (*2) $ X (\g -> unarr g <$> m) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) . fmap (*2) $ X (\g -> (\x -> runIdentity (Identity $ g x)) <$> m) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) . fmap (*2) $ X (\g -> g <$> m) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) $ X (\k -> (\g -> g <$> m) (lmap (*2) k)) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) $ X (\k -> (lmap (*2) k) <$> m) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) . fmap (+1) $ X (\k -> (k . (*2)) <$> m) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) . fmap (*2) $ X (\k -> (k . (+1) . (*2)) <$> m) = | |
| lowerX @(->) @[] . fmap (+1) . fmap (*2) $ X (\k -> (k . (*2) . (+1) . (*2)) <$> m) = | |
| lowerX @(->) @[] . fmap (+1) $ X (\k -> (k . (*2) . (*2) . (+1) . (*2)) <$> m) = | |
| lowerX @(->) @[] $ X (\k -> (k . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m) = | |
| k (arr id) $ (\k -> (k . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m) | |
| (\k -> (k . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m) (arr id) | |
| ((arr id) . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m | |
| ((rmap id id) . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m | |
| ((id . id) . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m | |
| (id . id . (+1) . (*2) . (*2) . (+1) . (*2)) <$> m | |
| ((+1) . (*2) . (*2) . (+1) . (*2)) <$> m | |
| -} | |
| res :: [Int] | |
| res = withX @(Identity) @(->) transform [1,2,3] |
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