yaya (empty) → 0.1.0.0
raw patch · 14 files changed
+1838/−0 lines, 14 filesdep +basedep +bifunctorsdep +comonad
Dependencies added: base, bifunctors, comonad, constraints, containers, deriving-compat, distributive, either, errors, free, hedgehog, kan-extensions, lens, profunctors, template-haskell, transformers, yaya, yaya-hedgehog
Files
- LICENSE +661/−0
- README.md +61/−0
- src/Yaya/Applied.hs +102/−0
- src/Yaya/Experimental/Foldable.hs +45/−0
- src/Yaya/Fold.hs +396/−0
- src/Yaya/Fold/Common.hs +108/−0
- src/Yaya/Fold/Native.hs +60/−0
- src/Yaya/Pattern.hs +36/−0
- src/Yaya/Retrofit.hs +14/−0
- src/Yaya/Zoo.hs +220/−0
- test/Test/Fold.hs +25/−0
- test/Test/Fold/Common.hs +19/−0
- test/test.hs +17/−0
- yaya.cabal +74/−0
+ LICENSE view
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If not, see <https://www.gnu.org/licenses/>.++Also add information on how to contact you by electronic and paper mail.++ If your software can interact with users remotely through a computer+network, you should also make sure that it provides a way for users to+get its source. For example, if your program is a web application, its+interface could display a "Source" link that leads users to an archive+of the code. There are many ways you could offer source, and different+solutions will be better for different programs; see section 13 for the+specific requirements.++ You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU AGPL, see+<https://www.gnu.org/licenses/>.
+ README.md view
@@ -0,0 +1,61 @@+# Yaya++Yet another … yet another recursion scheme library for Haskell.++## Overview++Recursion schemes allow you to separate _any_ recursion from your business logic, writing step-wise operations that can be applied in a way that guarantees termination (or, dually, progress).++How is this possible? You can’t have totality _and_ Turing-completeness, can you? Oh, but [you can](https://pdfs.semanticscholar.org/e291/5b546b9039a8cf8f28e0b814f6502630239f.pdf) – there is a particular type, `Partial a` (encoded with a fixed-point) that handles potential non-termination, akin to the way that `Maybe a` handles exceptional cases. It can be folded into `IO` in your main function, so that the runtime can execute a Turing-complete program that was modeled totally.++## organization++This organization is intended to make this a lightly-opinionated library. You should only need to import one module (per package) into any module of yours.++* `Pattern` – This is what you should use most of the time. It provides common pattern functors that aren’t found elsewhere as well as other operations that are useful when you’re defining your own algebras.+* `Fold` – This (and its submodules) provides algebra transformers, fixed-point operators, and other things you use when applying folds.+* `Retrofit` – Utilities for making your existing data types compatible with recursion schemes.+* `Applied` – A number of commonly-useful utilies defined as folds. Intended both as examples and code that you can actually use in your projects.+* `Zoo` – Names that you may have seen in the recursion scheme literature, but that we generally avoid using here. In general, prefer the right-hand side of these definitions, which shouldn’t require importing this module.++## Some (hopefully) helpful guidelines++Greek characters (and names) for things can often add confusion, however, there are some that we’ve kept here because I think (with the right mnemonic) they are actually clarifying.++- `φ` – an algebra – “phi” (pronounced “fye” or “fee”)+- `ψ` – a coalgebra – “psi” (pronounced “sai” or “see”)++These are the symbols used in “the literature”, but I think they also provide a good visual mnemonic – φ, like an algebra, is folded inward, while ψ, like a coalgebra, opens up. So, I find these symbols more evocative than `f` and `g` or `algebra` and `coalgebra`, or any other pair of names I’ve come across for these concepts.++There are two other names, `Mu` and `Nu` (for the inductive and coinductive fixed-point operators), that I _don’t_ think have earned their place, but I just haven’t come up with something more helpful yet.++### Naming Conventions++There is a set of conventions around the naming of the operations. There are many variants of each operation (and they are all ultimately variants of `cata` and `ana`), so understanding this convention should help make it easier to understand the myriad possibilities rather than learning them by rote. The general pattern is++> [`e`][`g`]`operation`[`T`][`M`]++#### `g`++“Generalized” variant – This parameterizes the fold over some `DistributiveLaw` that generalizes the (co)algebra over some `Monad` or `Comonad`. This is normally only applied to the fundamental operations – `cata`, `ana`, and `hylo`, but there is also a `gapo` (dual to `zygo`) that really only coincidentally follows this naming pattern.++Many of the other “well-known” named folds are specializations of this:++- when specialized to `((,) T)`, it’s `para`;+- when `((,) B)`, `zygo`;+- when `Free f`, `futu`;+- etc.++#### `e`++“Elgot” variant – Named after the form of coalgebra used in an “Elgot algebra”. If there is an operation that takes some `f (x a) -> a`, the Elgot variant takes `x (f a) -> a`, which often has similar but distinct properties from the original.++As a mnemonic, you can read the `e` as “exterior” as with a regular generalized fold, the `x` is on the _inside_ of the `f`, while with the Elgot variant, it is on the _outside_ of the `f`.++#### `T`++“Transformer” variant – For some fold that takes an algebra like `f (x a) -> a`, and where `t` is the (monad or comonad) transformer of `x`, the transformer variant takes an algebra like `f (t m a) -> a`.++#### `M`++Kleisli (“monadic”) variant – This convention is much more widespread than simply recursion schemes. A fold that returns its result in a `Monad`, by applying a Kleisli algebra (i.e., `f a -> m a` rather than `f a -> a`. The dual of this might be something like `anaW` (taking a seed value in a `Comonad`), but those are uninteresting. Having Kleisli variants of unfolds is unsafe, as it can force traversal of an infinite structure. If you’re looking for an operation like that, you are better off with an effectful streaming library.
+ src/Yaya/Applied.hs view
@@ -0,0 +1,102 @@+module Yaya.Applied where++import Control.Monad.Trans.Free+import Data.Functor.Identity++import Yaya.Fold+import Yaya.Fold.Common+import Yaya.Pattern++now :: Steppable t (Either a) => a -> t+now = embed . Left++-- | This will collapse all the intermediate steps to get to the value that must+-- exist at the end.+runToEnd :: Recursive t (Either a) => t -> a+runToEnd = cata fromEither++-- | Converts exceptional divergence to non-termination.+fromMaybe :: (Steppable t (Either a), Corecursive t (Either a)) => Maybe a -> t+fromMaybe = maybe (ana (toRight . never) ()) now++type Void = Mu Identity++absurd :: Recursive t Identity => t -> a+absurd = cata runIdentity++vacuous :: (Functor f, Recursive t Identity) => f t -> f a+vacuous = fmap absurd++zeroN :: Steppable t Maybe => t+zeroN = embed Nothing++succN :: Steppable t Maybe => t -> t+succN = embed . Just++height :: (Foldable f, Steppable n Maybe, Ord n) => f n -> n+height = foldr (max . succN) zeroN++naturals :: (Steppable n Maybe, Corecursive t ((,) n)) => t+naturals = ana (unarySequence succN) zeroN++-- | Extracts _no more than_ `n` elements from the possibly-infinite sequence+-- `s`.+takeUpTo+ :: (Recursive n Maybe, Projectable s (XNor a), Steppable l (XNor a))+ => n -> s -> l+takeUpTo = cata (lowerDay (embed . takeAvailable))++-- | Extracts _exactly_ `n` elements from the infinite stream `s`.+take+ :: (Recursive n Maybe, Projectable s ((,) a), Steppable l (XNor a))+ => n -> s -> l+take = cata (lowerDay (embed . takeAnother))++-- | Turns part of a structure inductive, so it can be analyzed, without forcing+-- the entire tree.+maybeReify+ :: (Projectable s f, Steppable l (FreeF f s), Functor f)+ => Algebra Maybe (s -> l)+maybeReify Nothing = embed . Pure+maybeReify (Just f) = embed . Free . fmap f . project++reifyUpTo+ :: (Recursive n Maybe, Projectable s f, Steppable l (FreeF f s), Functor f)+ => n -> s -> l+reifyUpTo = cata maybeReify++fibonacciPolynomials :: (Integral i, Corecursive t ((,) i)) => i -> t+fibonacciPolynomials x = lucasSequenceU x (-1)++fibonacci :: Corecursive t ((,) Int) => t+fibonacci = fibonacciPolynomials 1++lucasSequenceU :: (Integral i, Corecursive t ((,) i)) => i -> i -> t+lucasSequenceU p q = lucasSequence' p q `ana` (0, 1)++lucasSequenceV :: (Integral i, Corecursive t ((,) i)) => i -> i -> t+lucasSequenceV p q = lucasSequence' p q `ana` (2, p)++lucas :: Integral i => Corecursive t ((,) i) => t+lucas = lucasSequenceV 1 (-1)++pell :: (Integral i, Corecursive t ((,) i)) => t+pell = lucasSequenceU 2 (-1)++jacobsthal :: (Integral i, Corecursive t ((,) i)) => t+jacobsthal = lucasSequenceU 1 (-2)++mersenne :: (Integral i, Corecursive t ((,) i)) => t+mersenne = lucasSequenceU 3 2++-- | Creates an infinite stream of the provided value.+constantly :: Corecursive t ((,) a) => a -> t+constantly = ana split++-- | Lops off the branches of the tree below a certain depth, turning a+-- potentially-infinite structure into a finite one. Like a generalized+-- 'take'.+truncate+ :: (Recursive n Maybe, Projectable t f, Steppable u (FreeF f ()), Functor f)+ => n -> t -> u+truncate = cata (lowerDay (embed . truncate'))
+ src/Yaya/Experimental/Foldable.hs view
@@ -0,0 +1,45 @@+-- | This shows how 'Data.Foldable' is basically 'Recursive' specialized to+-- lists. The true operation of 'Data.Foldable' is 'toList'.+--+-- As these few operations have the usual signatures, the rest of the type+-- class can be implemented in the as in 'base'.+module Yaya.Experimental.Foldable where++import Control.Monad.Trans.Free++import Yaya.Fold+import Yaya.Fold.Common+import Yaya.Pattern++foldMap :: (Recursive t (XNor a), Monoid m) => (a -> m) -> t -> m+foldMap = cata . lowerMonoid++-- | This class represents the ability of a structure to be converted to a+-- list. It is equivalent to `Foldable`, but designed to illustrate the+-- representation of `Foldable` as `Recursive` specialized to lists.+class Listable f where+ naturalList :: f a b -> Free (XNor a) b+ -- toColist :: (Projectable t (f a), Corecursive u (XNor a)) => t -> u+ -- toColist = elgotAna seqFree (naturalList . project)+ -- toList :: (Recursive t (f a), Steppable u (XNor a)) => t -> u+ -- toList = cata (embed . unFree . naturalList)++-- FIXME: Use `cata . liftCoEnv` instead of `iter`.++-- | This is simply `cata` applied to a list – the function is the `Cons`+-- case, while the initial value is the `Nil` case.+foldr :: (Listable f, Recursive t (f a)) => (a -> b -> b) -> b -> t -> b+foldr f b =+ cata (iter (\case+ Neither -> b+ Both a r -> f a r)+ . naturalList)++-- | Simply 'cata' with a carrier of 'b -> b'.+foldl :: (Listable f, Recursive t (f a)) => (b -> a -> b) -> b -> t -> b+foldl f =+ flip+ (cata (iter (\case+ Neither -> id+ Both a g -> g . flip f a)+ . naturalList))
+ src/Yaya/Fold.hs view
@@ -0,0 +1,396 @@+{-# LANGUAGE GADTs #-}++module Yaya.Fold where++import Control.Applicative+import Control.Arrow+import Control.Comonad+import Control.Comonad.Cofree+import Control.Comonad.Hoist.Class+import Control.Comonad.Trans.Env+import Control.Lens hiding ((:<))+import Control.Monad+import Control.Monad.Trans.Free+import Data.Bifunctor+import Data.Bitraversable+import Data.Distributive+import Data.Either.Combinators+import Data.Foldable+import Data.Functor.Classes+import Data.Functor.Day+import Data.Functor.Identity+import Data.List.NonEmpty (NonEmpty(..))+import Data.Void+import Numeric.Natural++import Yaya.Fold.Common+import Yaya.Pattern++type Algebra f a = f a -> a+type GAlgebra w f a = f (w a) -> a+type ElgotAlgebra w f a = w (f a) -> a+type AlgebraM m f a = f a -> m a+type GAlgebraM m w f a = f (w a) -> m a+type ElgotAlgebraM m w f a = w (f a) -> m a++type Coalgebra f a = a -> f a+type GCoalgebra m f a = a -> f (m a)+type ElgotCoalgebra m f a = a -> m (f a)+-- | Note that using a `CoalgebraM` “directly” is partial (e.g., with `anaM`).+-- However, `ana . Compose` can accept a `CoalgebraM` and produce something+-- like an effectful stream.+type CoalgebraM m f a = a -> m (f a)+type GCoalgebraM m n f a = a -> m (f (n a))++-- | This type class is lawless on its own, but there exist types that can’t+-- implement the corresponding `embed` operation. Laws are induced by+-- implementing either `Steppable` (which extends this) or `Corecursive`+-- (which doesn’t).+class Projectable t f | t -> f where+ project :: Coalgebra f t++-- | Structures you can walk through step-by-step.+class Projectable t f => Steppable t f | t -> f where+ embed :: Algebra f t++-- | Inductive structures that can be reasoned about in the way we usually do –+-- with pattern matching.+class Recursive t f | t -> f where+ cata :: Algebra f a -> t -> a++-- | Coinductive (potentially-infinite) structures that guarantee _productivity_+-- rather than termination.+class Corecursive t f | t -> f where+ ana :: Coalgebra f a -> a -> t++-- | An implementation of `Eq` for any `Recursive` instance. Note that this is+-- actually more general than `Eq`, as it can compare between different+-- fixed-point representations of the same functor.+recursiveEq+ :: (Recursive t f, Steppable u f, Functor f, Foldable f, Eq1 f)+ => t -> u -> Bool+recursiveEq = cata2 equal++-- | An implementation of `Show` for any `Recursive` instance.+recursiveShowsPrec :: (Recursive t f, Show1 f) => Int -> t -> ShowS+recursiveShowsPrec prec =+ cata (showParen True . liftShowsPrec (const id) (foldMap id) prec)++-- | A fixed-point operator for inductive / finite data structures.+data Mu f = Mu (forall a. Algebra f a -> a)+++instance Functor f => Projectable (Mu f) f where+ project = lambek++instance Functor f => Steppable (Mu f) f where+ embed m = Mu (\f -> f (fmap (cata f) m))++instance Recursive (Mu f) f where+ cata φ (Mu f) = f φ++instance Show1 f => Show (Mu f) where+ showsPrec = recursiveShowsPrec++instance (Functor f, Foldable f, Eq1 f) => Eq (Mu f) where+ (==) = recursiveEq++-- | A fixed-point operator for coinductive / potentially-infinite data+-- structures.+data Nu f where Nu :: Coalgebra f a -> a -> Nu f++instance Functor f => Projectable (Nu f) f where+ project (Nu f a) = Nu f <$> f a++instance Functor f => Steppable (Nu f) f where+ embed = colambek++instance Corecursive (Nu f) f where+ ana = Nu++instance Projectable [a] (XNor a) where+ project [] = Neither+ project (h : t) = Both h t++instance Steppable [a] (XNor a) where+ embed Neither = []+ embed (Both h t) = h : t++instance Projectable (NonEmpty a) (AndMaybe a) where+ project (a :| []) = Only a+ project (a :| b : bs) = Indeed a (b :| bs)++instance Steppable (NonEmpty a) (AndMaybe a) where+ embed (Only a) = a :| []+ embed (Indeed a b) = a :| toList b++instance Projectable Natural Maybe where+ project 0 = Nothing+ project n = Just (pred n)++instance Steppable Natural Maybe where+ embed = maybe 0 succ++instance Projectable Void Identity where+ project = Identity++instance Steppable Void Identity where+ embed = runIdentity++instance Recursive Void Identity where+ cata _ = absurd++instance Projectable (Cofree f a) (EnvT a f) where+ project (a :< ft) = EnvT a ft++instance Steppable (Cofree f a) (EnvT a f) where+ embed (EnvT a ft) = a :< ft++instance Projectable (Free f a) (FreeF f a) where+ project = runFree++instance Steppable (Free f a) (FreeF f a) where+ embed = free++-- | Combines two `Algebra`s with different carriers into a single tupled+-- `Algebra`.+zipAlgebras :: Functor f => Algebra f a -> Algebra f b -> Algebra f (a, b)+zipAlgebras f g = (f . fmap fst &&& g . fmap snd)++-- | Algebras over Day convolution are convenient for binary operations, but+-- aren’t directly handleable by `cata`.+lowerDay :: Projectable t g => Algebra (Day f g) a -> Algebra f (t -> a)+lowerDay φ fta t = φ (Day fta (project t) ($))++-- | By analogy with `liftA2` (which also relies on `Day`, at least+-- conceptually).+cata2 :: (Recursive t f, Projectable u g) => Algebra (Day f g) a -> t -> u -> a+cata2 = cata . lowerDay++-- | Makes it possible to provide a 'GAlgebra' to 'cata'.+lowerAlgebra+ :: (Functor f, Comonad w)+ => DistributiveLaw f w+ -> GAlgebra w f a+ -> Algebra f (w a)+lowerAlgebra k φ = fmap φ . k . fmap duplicate++-- | Makes it possible to provide a 'GAlgebraM' to 'cataM'.+lowerAlgebraM+ :: (Applicative m, Traversable f, Comonad w, Traversable w)+ => DistributiveLaw f w+ -> GAlgebraM m w f a+ -> AlgebraM m f (w a)+lowerAlgebraM k φ = traverse φ . k . fmap duplicate++-- | Makes it possible to provide a 'GCoalgebra' to 'ana'.+lowerCoalgebra+ :: (Functor f, Monad m)+ => DistributiveLaw m f+ -> GCoalgebra m f a+ -> Coalgebra f (m a)+lowerCoalgebra k ψ = fmap join . k . fmap ψ++-- | Makes it possible to provide a 'GCoalgebraM' to 'anaM'.+lowerCoalgebraM+ :: (Applicative m, Traversable f, Monad n, Traversable n)+ => DistributiveLaw n f+ -> GCoalgebraM m n f a+ -> CoalgebraM m f (n a)+lowerCoalgebraM k ψ = fmap (fmap join . k) . traverse ψ++gcata+ :: (Recursive t f, Functor f, Comonad w)+ => DistributiveLaw f w+ -> GAlgebra w f a+ -> t+ -> a+gcata k φ = extract . cata (lowerAlgebra k φ)++elgotCata+ :: (Recursive t f, Functor f, Comonad w)+ => DistributiveLaw f w+ -> ElgotAlgebra w f a+ -> t+ -> a+elgotCata k φ = φ . cata (k . fmap (extend φ))++gcataM+ :: (Monad m, Recursive t f, Traversable f, Comonad w, Traversable w)+ => DistributiveLaw f w+ -> GAlgebraM m w f a+ -> t+ -> m a+gcataM w φ = fmap extract . cata (lowerAlgebraM w φ <=< sequenceA)++elgotCataM+ :: (Monad m, Recursive t f, Traversable f, Comonad w, Traversable w)+ => DistributiveLaw f w+ -> ElgotAlgebraM m w f a+ -> t+ -> m a+elgotCataM w φ = φ <=< cata (fmap w . traverse (sequence . extend φ) <=< sequenceA)++ezygoM+ :: (Monad m, Recursive t f, Traversable f)+ => AlgebraM m f b+ -> ElgotAlgebraM m ((,) b) f a+ -> t+ -> m a+ezygoM φ' φ =+ fmap snd+ . cata ((\x@(b, _) -> (b,) <$> φ x)+ <=< bisequence . (φ' . fmap fst &&& pure . fmap snd)+ <=< sequenceA)++gana+ :: (Corecursive t f, Functor f, Monad m)+ => DistributiveLaw m f+ -> GCoalgebra m f a+ -> a+ -> t+gana k ψ = ana (lowerCoalgebra k ψ) . pure++elgotAna+ :: (Corecursive t f, Functor f, Monad m)+ => DistributiveLaw m f+ -> ElgotCoalgebra m f a+ -> a+ -> t+elgotAna k ψ = ana (fmap (>>= ψ) . k) . ψ++lambek :: (Steppable t f, Recursive t f, Functor f) => Coalgebra f t+lambek = cata (fmap embed)++colambek :: (Projectable t f, Corecursive t f, Functor f) => Algebra f t+colambek = ana (fmap project)++-- | There are a number of distributive laws, including+-- `Data.Traversable.sequenceA`, `Data.Distributive.distribute`, and+-- `Data.Align.sequenceL`. Yaya also provides others for specific recursion+-- schemes.+type DistributiveLaw f g = forall a. f (g a) -> g (f a)++-- | A less-constrained `distribute` for `Identity`.+distIdentity :: Functor f => DistributiveLaw f Identity+distIdentity = Identity . fmap runIdentity++-- | A less-constrained `sequenceA` for `Identity`.+seqIdentity :: Functor f => DistributiveLaw Identity f+seqIdentity = fmap Identity . runIdentity++distTuple :: Functor f => Algebra f a -> DistributiveLaw f ((,) a)+distTuple φ = φ . fmap fst &&& fmap snd++distEnvT+ :: Functor f+ => Algebra f a+ -> DistributiveLaw f w+ -> DistributiveLaw f (EnvT a w)+distEnvT φ k = uncurry EnvT . (φ . fmap ask &&& k . fmap lowerEnvT)++seqEither :: Functor f => Coalgebra f a -> DistributiveLaw (Either a) f+seqEither ψ = fmap Left . ψ ||| fmap Right++-- | Converts an `Algebra` to one that annotates the tree with the result for+-- each node.+attributeAlgebra+ :: (Steppable t (EnvT a f), Functor f)+ => Algebra f a -> Algebra f t+attributeAlgebra φ ft = embed $ EnvT (φ (fmap (fst . runEnvT . project) ft)) ft++-- | Converts a `Coalgebra` to one that annotates the tree with the seed that+-- generated each node.+attributeCoalgebra :: Coalgebra f a -> Coalgebra (EnvT a f) a+attributeCoalgebra ψ = uncurry EnvT . (id &&& ψ)++-- | This is just a more obvious name for composing `lowerEnvT` with your+-- algebra directly.+ignoringAttribute :: Algebra f a -> Algebra (EnvT b f) a+ignoringAttribute φ = φ . lowerEnvT++unFree :: Steppable t f => Algebra (FreeF f t) t+unFree = \case+ Pure t -> t+ Free ft -> embed ft++-- preservingAttribute :: (forall a. f a -> g a) -> EnvT a f b -> EnvT a g b+-- preservingAttribute = cohoist++-- instances for non-recursive types++constEmbed :: Algebra (Const a) a+constEmbed = getConst++constProject :: Coalgebra (Const a) a+constProject = Const++constCata :: Algebra (Const b) a -> b -> a+constCata φ = φ . Const++constAna :: Coalgebra (Const b) a -> a -> b+constAna ψ = getConst . ψ++instance Projectable (Either a b) (Const (Either a b)) where+ project = constProject++instance Steppable (Either a b) (Const (Either a b)) where+ embed = constEmbed++instance Recursive (Either a b) (Const (Either a b)) where+ cata = constCata++instance Corecursive (Either a b) (Const (Either a b)) where+ ana = constAna++instance Projectable (Maybe a) (Const (Maybe a)) where+ project = constProject++instance Steppable (Maybe a) (Const (Maybe a)) where+ embed = constEmbed++instance Recursive (Maybe a) (Const (Maybe a)) where+ cata = constCata++instance Corecursive (Maybe a) (Const (Maybe a)) where+ ana = constAna++-- | An endofunctor in the category of endofunctors.+class HFunctor h where+ hmap :: (forall a. f a -> g a) -> h f a -> h g a+++-- | A functor from the category of endofunctors to *Hask*.+class DFunctor d where+ dmap :: (forall a. f a -> g a) -> d f -> d g++-- instance DFunctor Mu where+-- dmap f = cata (embed . f)++-- instance DFunctor Fix where+-- dmap f = ana (f . project)++-- instance DFunctor Nu where+-- dmap f = ana (f . project)++type BialgebraIso f a = Iso' (f a) a+type AlgebraPrism f a = Prism' (f a) a+type CoalgebraPrism f a = Prism' a (f a)++cursiveIso :: Steppable t f => BialgebraIso f t+cursiveIso = iso embed project++birecursiveIso+ :: (Recursive t f, Corecursive t f)+ => BialgebraIso f a+ -> Iso' t a+birecursiveIso alg = iso (cata (view alg)) (ana (review alg))++recursivePrism+ :: (Recursive t f, Corecursive t f, Traversable f)+ => AlgebraPrism f a+ -> Prism' t a+recursivePrism alg =+ prism+ (ana (review alg))+ (\t -> mapLeft (const t) $ cata (matching alg <=< sequenceA) t)
+ src/Yaya/Fold/Common.hs view
@@ -0,0 +1,108 @@+-- | Common algebras that are useful when folding.+module Yaya.Fold.Common where++import Control.Arrow+import Control.Monad+import Control.Monad.Trans.Free+import Data.Foldable+import Data.Functor+import Data.Functor.Classes+import Data.Functor.Day+import Data.Functor.Identity+import Data.Semigroup+import Numeric.Natural++import Yaya.Pattern++-- | Converts the free monoid (a list) into some other monoid.+lowerMonoid :: Monoid m => (a -> m) -> XNor a m -> m+lowerMonoid f = \case+ Neither -> mempty+ Both a b -> mappend (f a) b++-- | Converts the free semigroup (a non-empty list) into some other semigroup.+lowerSemigroup :: Semigroup m => (a -> m) -> AndMaybe a m -> m+lowerSemigroup f = \case+ Only a -> f a+ Indeed a b -> f a <> b++lowerMonad :: Monad m => (forall a. f a -> m a) -> FreeF f a (m a) -> m a+lowerMonad f = \case+ Pure a -> pure a+ Free fm -> join (f fm)++equal :: (Functor f, Foldable f, Eq1 f) => Day f f Bool -> Bool+equal (Day f1 f2 fn) =+ liftEq (==) (void f1) (void f2)+ && and (zipWith fn (toList f1) (toList f2))++-- TODO: Redefine this using `Natural`+-- | When folded, returns the height of the data structure.+height :: Foldable f => f Integer -> Integer+height = (+ 1) . foldr max (-1)++-- NB: It seems like this could be some more general notion of this, like+-- size :: (Foldable f, Semiring a) => f a -> a+-- size = foldr (+) one+-- | When folded, returns the number ef nodes in the data structure.+size :: Foldable f => f Natural -> Natural+size = foldr (+) 1++toRight :: Identity b -> Either a b+toRight = Right . runIdentity++-- | Returns the last 'Just' result.+while :: (a -> Maybe a) -> a -> Either a a+while f a = maybe (Left a) Right $ f a++fromEither :: Either a a -> a+fromEither = \case+ Left a -> a+ Right a -> a++never :: a -> Identity a+never = Identity++le :: Day Maybe Maybe Bool -> Bool+le = \case+ Day Nothing _ _ -> True+ Day (Just a) (Just b) f -> f a b+ Day (Just _) Nothing _ -> False++takeAnother :: Day Maybe ((,) a) b -> XNor a b+takeAnother = \case+ Day Nothing _ _ -> Neither+ Day (Just x) (h, t) f -> Both h (f x t)++takeAvailable :: Day Maybe (XNor a) b -> XNor a b+takeAvailable = \case+ Day Nothing _ _ -> Neither+ Day (Just x) t f -> fmap (f x) t++truncate' :: Functor f => Day Maybe f a -> FreeF f () a+truncate' = \case+ Day Nothing fa _ -> Pure ()+ Day (Just n) fa f -> Free (fmap (f n) fa)++-- | Converts a single value into a tuple with the same value on both sides.+-- > x &&& y = (x *** y) . split+split :: a -> (a, a)+split x = (x, x)++-- * sequence generators+--+-- These functions are defined with different type parameters in order to+-- constrain the implementation, but to be used as coalgebras, all of the+-- parameters need to be specialized to the same type.++unarySequence :: (a -> b) -> a -> (a, b)+unarySequence f a = (a, f a)++binarySequence :: (a -> b -> c) -> (a, b) -> (a, (b, c))+binarySequence f (a, b) = (a, (b, f a b))++ternarySequence :: (a -> b -> c -> d) -> (a, b, c) -> (a, (b, c, d))+ternarySequence f (a, b, c) = (a, (b, c, f a b c))++lucasSequence' :: Integral i => i -> i -> (i, i) -> (i, (i, i))+lucasSequence' p q = binarySequence (\n2 n1 -> p * n1 - q * n2)
+ src/Yaya/Fold/Native.hs view
@@ -0,0 +1,60 @@+-- | Uses of recursion schemes that use Haskell’s built-in recursion in a total+-- manner.+module Yaya.Fold.Native where++import Control.Arrow+import Control.Comonad+import Control.Comonad.Cofree+import Control.Comonad.Trans.Env+import Control.Monad.Trans.Free+import Data.List.NonEmpty+import Numeric.Natural++import Yaya.Fold+import Yaya.Pattern++newtype Fix f = Fix { unFix :: f (Fix f) }++instance Projectable (Fix f) f where+ project = unFix++instance Steppable (Fix f) f where+ embed = Fix++instance Functor f => Corecursive (Fix f) f where+ ana φ = embed . fmap (ana φ) . φ++instance Recursive Natural Maybe where+ cata ɸ = ɸ . fmap (cata ɸ) . project++instance Corecursive [a] (XNor a) where+ ana ψ =+ (\case+ Neither -> []+ Both h t -> h : ana ψ t)+ . ψ++instance Corecursive (NonEmpty a) (AndMaybe a) where+ ana ψ =+ (\case+ Only h -> h :| []+ Indeed h t -> h :| toList (ana ψ t))+ . ψ++instance Functor f => Corecursive (Free f a) (FreeF f a) where+ ana ψ =+ free+ . (\case+ Pure a -> Pure a+ Free fb -> Free . fmap (ana ψ) $ fb)+ . ψ++instance Functor f => Corecursive (Cofree f a) (EnvT a f) where+ ana ψ = uncurry (:<) . fmap (fmap (ana ψ)) . runEnvT . ψ++distCofreeT+ :: (Functor f, Functor h)+ => DistributiveLaw f h+ -> DistributiveLaw f (Cofree h)+distCofreeT k = ana $ uncurry EnvT . (fmap extract &&& k . fmap unwrap)+
+ src/Yaya/Pattern.hs view
@@ -0,0 +1,36 @@+module Yaya.Pattern where++import Control.Applicative+import Control.Arrow+import Control.Comonad+import Control.Comonad.Cofree+import Control.Comonad.Env+import Control.Monad+import Control.Monad.Trans.Free+import Data.Bifunctor+import Data.Bitraversable+import Data.Distributive+import Data.Foldable+import Data.Functor.Classes+import Data.Functor.Day+import Data.Functor.Identity+import Data.Void+import Numeric.Natural++-- | Isomorphic to 'Maybe (a, b)', it’s also the pattern functor for lists.+data XNor a b = Neither | Both a b deriving (Functor, Foldable, Traversable)++instance Bifunctor XNor where+ bimap f g = \case+ Neither -> Neither+ Both a b -> Both (f a) (g b)++-- | Isomorphic to `(a, Maybe b)`, it’s also the pattern functor for non-empty+-- lists.+data AndMaybe a b = Only a | Indeed a b deriving (Functor, Foldable, Traversable)++instance Bifunctor AndMaybe where+ bimap f g = \case+ Only a -> Only (f a)+ Indeed a b -> Indeed (f a) (g b)+
+ src/Yaya/Retrofit.hs view
@@ -0,0 +1,14 @@+-- | This module re-exports a subset of `Yaya.Fold`, intended for when you want+-- to define recursion scheme instances for your existing recursive types.+module Yaya.Retrofit+ ( module Yaya.Fold+ ) where++import Yaya.Fold+ ( Corecursive+ , Projectable+ , Recursive+ , recursiveEq+ , recursiveShowsPrec+ , Steppable+ )
+ src/Yaya/Zoo.hs view
@@ -0,0 +1,220 @@+-- | Contains all the commonly-named folds that aren’t core to the library. In+-- general, this can be seen as a mapping from names you may have heard or+-- read in a paper to how Yaya expects you to achieve the same end. Of course,+-- you can always import this module and use the “common” name as well.+module Yaya.Zoo where++import Control.Arrow hiding (first)+import Control.Comonad.Cofree+import Control.Comonad.Env+import Control.Monad+import Control.Monad.Trans.Free+import Data.Bifunctor+import Data.Bitraversable+import Data.Either.Combinators+import Data.Function+import Data.Profunctor+import Data.Tuple++import Yaya.Fold+import Yaya.Fold.Native (distCofreeT)+import Yaya.Pattern++-- | A recursion scheme that allows you to return a complete branch when+-- unfolding.+apo+ :: (Projectable t f, Corecursive t f, Functor f)+ => GCoalgebra (Either t) f a+ -> a+ -> t+apo = gana (seqEither project)++-- | If you have a monadic algebra, you can fold it by distributing the monad+-- over the algebra.+cataM :: (Monad m, Recursive t f, Traversable f) => AlgebraM m f a -> t -> m a+cataM φ = cata (φ <=< sequenceA)++-- | A recursion scheme that allows to algebras to see each others’ results. (A+-- generalization of 'zygo'.) This is an example that falls outside the scope+-- of “comonadic folds”, but _would_ be covered by “adjoint folds”.+mutu+ :: (Recursive t f, Functor f)+ => GAlgebra ((,) a) f b+ -> GAlgebra ((,) b) f a+ -> t+ -> a+mutu φ' φ = extract . cata (φ' . fmap swap &&& φ)++gmutu+ :: (Comonad w, Comonad v, Recursive t f, Functor f)+ => DistributiveLaw f w+ -> DistributiveLaw f v+ -> GAlgebra (EnvT a w) f b+ -> GAlgebra (EnvT b v) f a+ -> t+ -> a+gmutu w v φ' φ = extract . mutu (lowerEnv w φ') (lowerEnv v φ)+ where+ lowerEnv x φ'' =+ fmap φ''+ . x+ . fmap (fmap (uncurry EnvT) . distProd . (extract *** duplicate))+ distProd p =+ let a = fst p+ in fmap (\b -> (a , b)) (snd p)++-- | This could use a better name.+comutu+ :: (Corecursive t f, Functor f)+ => GCoalgebra (Either a) f b+ -> GCoalgebra (Either b) f a+ -> a+ -> t+comutu ψ' ψ = ana (fmap swapEither . ψ' ||| ψ) . pure++-- gcomutu+-- :: (Monad m, Monad n, Corecursive t f, Functor f)+-- => DistributiveLaw m f+-- -> DistributiveLaw n f+-- -> GCoalgebra (FreeF m a) f b+-- -> GCoalgebra (FreeF n b) f a+-- -> a+-- -> t+-- gcomutu m n ψ' ψ = comutu (lowerFree m ψ') (lowerFree n ψ) . pure+-- where+-- lowerFree x ψ'' =+-- fmap ((pure +++ join) . distProd . fmap (uncurry EnvT))+-- . x+-- . fmap ψ''+-- distProd :: DistributiveLaw f (Either a)+-- distProd p =+-- let a = fst p+-- in fmap (\b -> (a , b)) (snd p)++mutuM+ :: (Monad m, Recursive t f, Traversable f)+ => GAlgebraM m ((,) a) f b+ -> GAlgebraM m ((,) b) f a+ -> t+ -> m a+mutuM φ' φ = fmap snd . cataM (bisequence . (φ' . fmap swap &&& φ))++histo :: (Recursive t f, Functor f) => GAlgebra (Cofree f) f a -> t -> a+histo = gcata (distCofreeT id)++-- | A recursion scheme that gives you access to the original structure as you+-- fold. (A specialization of 'zygo'.)+para+ :: (Steppable t f, Recursive t f, Functor f)+ => GAlgebra ((,) t) f a+ -> t+ -> a+para = gcata (distTuple embed)++-- | A recursion scheme that uses a “helper algebra” to provide additional+-- information when folding. (A generalization of 'para', and specialization+-- of 'mutu'.)+zygo+ :: (Recursive t f, Functor f)+ => Algebra f b+ -> GAlgebra ((,) b) f a+ -> t+ -> a+zygo φ = gcata (distTuple φ)++-- | This definition is different from the one given by `gcataM (distTuple φ')`+-- because it has a monadic “helper” algebra. But at least it gives us the+-- opportunity to show how 'zygo' is a specialization of 'mutu'.+zygoM+ :: (Monad m, Recursive t f, Traversable f)+ => AlgebraM m f b+ -> GAlgebraM m ((,) b) f a+ -> t+ -> m a+zygoM φ' φ = mutuM (φ' . fmap snd) φ++-- | Potentially-infinite lists, like 'Data.List'.+type Colist a = Nu (XNor a)++-- | Finite lists.+type List a = Mu (XNor a)++-- | Finite non-empty lists.+type NonEmptyList a = Mu (AndMaybe a)++-- | Finite natural numbers.+type Nat = Mu Maybe++-- | Represents partial functions that may eventually return a value ('Left').+-- NB: This is a newtype so we can create the usual instances.+newtype Partial a = Partial { fromPartial :: Nu (Either a) }++-- TODO: There may be some way to do this over an arbitrary 'newtype', or at+-- least a way to do it over an arbitrary 'Iso'.+insidePartial :: (Nu (Either a) -> Nu (Either b)) -> Partial a -> Partial b+insidePartial f = Partial . f . fromPartial++instance Functor Partial where+ fmap f = insidePartial (comap f)++instance Applicative Partial where+ pure = Partial . embed . Left+ ff <*> fa =+ flip insidePartial ff+ $ elgotAna (seqEither project)+ ((fromPartial . flip fmap fa +++ Right) . project)++instance Monad Partial where+ pa >>= f = join (fmap f pa)+ where+ join =+ insidePartial+ $ elgotAna (seqEither project) ((fromPartial +++ Right) . project)++-- | Always-infinite streams (as opposed to 'Colist', which _may_ terminate).+type Stream a = Nu ((,) a)++-- | A more general implementation of 'fmap', because it can also work to, from,+-- or within monomorphic structures, obviating the need for classes like+-- 'MonoFunctor'.+map :: (Recursive t (f a), Steppable u (f b), Bifunctor f) => (a -> b) -> t -> u+map f = cata (embed . first f)++-- | A version of `map` that applies to Corecursive structures.+comap+ :: (Projectable t (f a), Corecursive u (f b), Bifunctor f)+ => (a -> b)+ -> t+ -> u+comap f = ana (first f . project)++-- | A more general implementation of 'traverse', because it can also work to,+-- from, or within monomorphic structures, obviating the need for classes like+-- 'MonoTraversable'.+-- TODO: Weaken the 'Monad' constraint to 'Applicative'.+traverse+ :: ( Recursive t (f a)+ , Steppable u (f b)+ , Bitraversable f+ , Traversable (f a)+ , Monad m)+ => (a -> m b)+ -> t+ -> m u+traverse f = cata (fmap embed . bitraverse f pure <=< sequenceA)++-- | A more general implementation of 'contramap', because it can also work to,+-- from, or within monomorphic structures.+contramap+ :: (Recursive t (f b), Steppable u (f a), Profunctor f)+ => (a -> b)+ -> t+ -> u+contramap f = cata (embed . lmap f)++cocontramap+ :: (Projectable t (f b), Corecursive u (f a), Profunctor f)+ => (a -> b)+ -> t+ -> u+cocontramap f = ana (lmap f . project)
+ test/Test/Fold.hs view
@@ -0,0 +1,25 @@+{-# LANGUAGE TemplateHaskell #-}++module Test.Fold where++import Hedgehog+import qualified Hedgehog.Gen as Gen++import Yaya.Fold.Common+import Yaya.Hedgehog.Expr+import Yaya.Hedgehog.Fold++prop_muCataCancel :: Property+prop_muCataCancel =+ property $ law_cataCancel size =<< forAll (genExpr (Gen.sized genMuExpr))++prop_muCataRefl :: Property+prop_muCataRefl =+ property $ law_cataRefl =<< forAll (Gen.sized genMuExpr)++-- prop_muCataCompose :: Property+-- prop_muCataCompose =+-- property $ law_cataCompose size id =<< forAll genMuExpr++tests :: IO Bool+tests = checkParallel $$(discover)
+ test/Test/Fold/Common.hs view
@@ -0,0 +1,19 @@+{-# LANGUAGE TemplateHaskell #-}++module Test.Fold.Common where++import Hedgehog+import qualified Hedgehog.Gen as Gen++import Yaya.Fold+import Yaya.Fold.Common+import Yaya.Hedgehog.Expr++prop_heightLtSize :: Property+prop_heightLtSize =+ property+ (assert . uncurry (<) . fmap toInteger . cata (zipAlgebras height size)+ =<< forAll (Gen.sized genMuExpr))++tests :: IO Bool+tests = checkParallel $$(discover)
+ test/test.hs view
@@ -0,0 +1,17 @@+import Control.Monad+import System.Exit (exitFailure)+import System.IO (BufferMode(..), hSetBuffering, stdout, stderr)++import qualified Test.Fold as Fold+import qualified Test.Fold.Common as Fold.Common++main :: IO ()+main = do+ hSetBuffering stdout LineBuffering+ hSetBuffering stderr LineBuffering++ results <- sequence [ Fold.tests+ , Fold.Common.tests+ ]++ unless (and results) exitFailure
+ yaya.cabal view
@@ -0,0 +1,74 @@+name: yaya+version: 0.1.0.0+synopsis: Total recursion schemes.+description: Recursion schemes allow you to separate recursion from your+ business logic – making your own operations simpler, more+ modular, and less error-prone. This library also provides+ tools for combining your operations in ways that reduce the+ number of passes over your data and is designed to+ encourage total (i.e., successfully terminating) functions.+homepage: https://github.com/sellout/yaya#readme+author: Greg Pfeil+maintainer: greg@technomadic.org+copyright: 2017 Greg Pfeil+license: AGPL-3+license-file: LICENSE+category: Recursion+build-type: Simple+extra-source-files: README.md+cabal-version: >=1.10++library+ hs-source-dirs: src+ exposed-modules: Yaya.Pattern+ , Yaya.Fold+ , Yaya.Fold.Common+ , Yaya.Fold.Native+ , Yaya.Retrofit+ , Yaya.Applied+ , Yaya.Zoo+ , Yaya.Experimental.Foldable+ build-depends: base >= 4.7 && < 5+ , bifunctors+ , comonad+ , constraints+ , containers+ , distributive+ , either+ , errors+ , free+ , kan-extensions+ , lens+ , profunctors+ , template-haskell+ , transformers+ default-extensions: ConstraintKinds+ , DeriveTraversable+ , FlexibleContexts+ , FlexibleInstances+ , FunctionalDependencies+ , LambdaCase+ , MultiParamTypeClasses+ , PolyKinds+ , RankNTypes+ , ScopedTypeVariables+ , TupleSections+ default-language: Haskell2010++test-suite yaya-test+ type: exitcode-stdio-1.0+ hs-source-dirs: test+ main-is: test.hs+ other-modules: Test.Fold+ , Test.Fold.Common+ build-depends: base+ , deriving-compat+ , hedgehog+ , yaya+ , yaya-hedgehog+ ghc-options: -threaded -rtsopts -with-rtsopts=-N -Wall+ default-language: Haskell2010++source-repository head+ type: git+ location: https://github.com/sellout/yaya