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GHC 9.10.3 · lts/ghc-9.10.x · 248f8f0 · 2026-10-05

Modulemonoid-subclasses-1.2.5.1Haskell2010

Data.Monoid.Factorial

This module defines the FactorialMonoid class and some of its instances.

  • 1 type
  • 1 class

module Data.Semigroup.Factorial

classclass (Factorial m, MonoidNull m) => FactorialMonoid m where
#

Class of monoids that can be split into irreducible (i.e., atomic or prime) factors in a unique way. Note that mempty is not considered a factor. Factors of a Product are literally its prime factors:

Property
factors (Product 12) == [Product 2, Product 2, Product 3]

Factors of a list are not its elements but all its single-item sublists:

Property
factors "abc" == ["a", "b", "c"]

The methods of this class satisfy the following laws in addition to those of Factorial:

null == List.null . factors
factors == unfoldr splitPrimePrefix == List.reverse . unfoldr (fmap swap . splitPrimeSuffix)
reverse == mconcat . List.reverse . factors
primePrefix == maybe mempty fst . splitPrimePrefix
primeSuffix == maybe mempty snd . splitPrimeSuffix
inits == List.map mconcat . List.inits . factors
tails == List.map mconcat . List.tails . factors
span p m == (mconcat l, mconcat r) where (l, r) = List.span p (factors m)
List.all (List.all (not . pred) . factors) . split pred
mconcat . intersperse prime . split (== prime) == id
splitAt i m == (mconcat l, mconcat r) where (l, r) = List.splitAt i (factors m)
spanMaybe () (const $ bool Nothing (Maybe ()) . p) m == (takeWhile p m, dropWhile p m, ())
spanMaybe s0 (\s m-> Just $ f s m) m0 == (m0, mempty, foldl f s0 m0)
let (prefix, suffix, s') = spanMaybe s f m
    foldMaybe = foldl g (Just s)
    g s m = s >>= flip f m
in all ((Nothing ==) . foldMaybe) (inits prefix)
   && prefix == last (filter (isJust . foldMaybe) $ inits m)
   && Just s' == foldMaybe prefix
   && m == prefix <> suffix

A minimal instance definition should implement splitPrimePrefix for performance reasons, and other methods where beneficial.

Methods

Instances26FactorialMonoid, …