catamorphism: Exposes a Template Haskell function for generating catamorphisms.

This module exposes a makeCata function which can create catamorphisms for arbitrary Haskell types. Catamorphisms are functions which deconstruct some value by replacing each data constructor with a custom function yielding a new value. See http://www.haskell.org/haskellwiki/Catamorphisms for a more in-depth discussion of catamorphisms in Haskell.

The Haskell base package already comes with a couple of standard catamorphisms, such as maybe (for Maybe values). The maybe function could have been generated using makeCata as follows:

-- Defines 'maybe :: b -> (a -> b) -> Maybe a -> b'
$(makeCata defaultOptions ''Maybe)

However, catamorphisms are especially useful for recursive data structures. Consider the following simple example which defines a basic data type for modelling sums of numbers, supporting variables:

import Data.Morphism.Cata
import Data.Maybe (fromJust)

data Expr a = Number a
            | Variable Char
            | Sum (Expr a) (Expr a)

-- Defines 'expr :: (a -> b) -> (Char -> b) -> (b -> b -> b) -> Expr a -> b'
$(makeCata defaultOptions ''Expr)

The makeCata invocation defines a expr function which works like a fold on Expr values; it can be used to implement various useful other functions:

-- Evaluate an Expr, given some variable bindings
eval :: Num a => [(Char, a)] -> Expr a -> a
eval vars = expr id (fromJust . (`lookup` vars)) (+)

-- Pretty-prints an Expr
pprint :: Show a => Expr a -> String
pprint = expr show show (\a b -> a ++ " + " ++ b)

-- Counts the number of variables used in an expr
numVars :: Expr a -> Int
numVars = expr (const 1) (const 0) (+)

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