module Pandora.Paradigm.Primary.Functor.Fix where import Pandora.Core.Functor (type (<-|), type (|->)) import Pandora.Pattern.Category ((.)) import Pandora.Pattern.Functor.Covariant (Covariant (comap)) newtype Fix t = Fix { Fix t -> t (Fix t) unfix :: t (Fix t) } cata :: Covariant t => (a <-| t) -> Fix t -> a cata :: (a <-| t) -> Fix t -> a cata a <-| t f = a <-| t f (a <-| t) -> (Fix t -> t a) -> Fix t -> a forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . (Fix t -> a) -> t (Fix t) -> t a forall (t :: * -> *) a b. Covariant t => (a -> b) -> t a -> t b comap ((a <-| t) -> Fix t -> a forall (t :: * -> *) a. Covariant t => (a <-| t) -> Fix t -> a cata a <-| t f) (t (Fix t) -> t a) -> (Fix t -> t (Fix t)) -> Fix t -> t a forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . Fix t -> t (Fix t) forall (t :: * -> *). Fix t -> t (Fix t) unfix ana :: Covariant t => (a |-> t) -> a -> Fix t ana :: (a |-> t) -> a -> Fix t ana a |-> t f = t (Fix t) -> Fix t forall (t :: * -> *). t (Fix t) -> Fix t Fix (t (Fix t) -> Fix t) -> (a -> t (Fix t)) -> a -> Fix t forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . (a -> Fix t) -> t a -> t (Fix t) forall (t :: * -> *) a b. Covariant t => (a -> b) -> t a -> t b comap ((a |-> t) -> a -> Fix t forall (t :: * -> *) a. Covariant t => (a |-> t) -> a -> Fix t ana a |-> t f) (t a -> t (Fix t)) -> (a |-> t) -> a -> t (Fix t) forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . a |-> t f hylo :: Covariant t => (b <-| t) -> (a |-> t) -> (a -> b) hylo :: (b <-| t) -> (a |-> t) -> a -> b hylo b <-| t phi a |-> t psi = (b <-| t) -> Fix t -> b forall (t :: * -> *) a. Covariant t => (a <-| t) -> Fix t -> a cata b <-| t phi (Fix t -> b) -> (a -> Fix t) -> a -> b forall (m :: * -> * -> *) b c a. Category m => m b c -> m a b -> m a c . (a |-> t) -> a -> Fix t forall (t :: * -> *) a. Covariant t => (a |-> t) -> a -> Fix t ana a |-> t psi