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|
{-# LANGUAGE EmptyDataDecls #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
-- |
-- Module : TypeLevel.Number.Classes
-- Copyright : Alexey Khudyakov
-- License : BSD3-style (see LICENSE)
--
-- Maintainer : Alexey Khudyakov <alexey.skladnoy@gmail.com>
-- Stability : unstable
-- Portability : unportable (GHC only)
--
-- This module contain interface type classes for operations with type
-- level numbers.
module TypeLevel.Number.Classes ( -- * Comparison of numbers
Compare
, compareN
-- ** Data labels for types comparison
, IsLesser
, IsEqual
, IsGreater
-- ** Specialized type classes
-- $comparing
, Lesser
, LesserEq
, Greater
, GreaterEq
-- ** Special traits
, Positive
, NonZero
-- * Arithmetic operations on numbers
, Next
, nextN
, Prev
, prevN
, Negate
, negateN
, Add
, addN
, Sub
, subN
, Mul
, mulN
, Div
, divN
-- * Special classes
, Normalized
) where
----------------------------------------------------------------
-- Comparison
----------------------------------------------------------------
-- | Type family for comparing two numbers. It's expected that for any
-- two valid 'n' and 'm' 'Compare n m' is equal to IsLess when 'n<m', IsEqual
-- when 'n=m' and IsGreater when 'n>m'.
type family Compare n m :: *
compareN :: n -> m -> Compare n m
compareN _ _ = undefined
data IsLesser
data IsEqual
data IsGreater
instance Show IsLesser where show _ = "IsLesser"
instance Show IsEqual where show _ = "IsEqual"
instance Show IsGreater where show _ = "IsGreater"
----------------------------------------------------------------
-- $comparing
-- These type classes are meant to be used in contexts to ensure
-- relations between numbers. For example:
--
-- > someFunction :: Lesser n m => Data n -> Data m -> Data n
-- > someFunction = ...
--
-- They have generic instances and every number which is instance of
-- Compare type family is instance of these type classes.
--
-- These instance could have problems. They weren't exensively tested.
-- Also error messages are really unhelpful.
-- | Numbers n and m are instances of this class if and only is n < m.
class Lesser n m
-- | Numbers n and m are instances of this class if and only is n > m.
class Greater n m
-- | Numbers n and m are instances of this class if and only is n <= m.
class LesserEq n m
-- | Numbers n and m are instances of this class if and only is n >= m.
class GreaterEq n m
-- a b c are instance of class only when a ~ b or a ~ c. Require ovelapping.
class OneOfTwo a b c
instance OneOfTwo a a b
instance OneOfTwo a b a
instance OneOfTwo a a a
instance (Compare n m ~ IsLesser ) => Lesser n m
instance (Compare n m ~ IsGreater) => Greater n m
-- Instances for LessEq and GreaterEq are trickier.
instance (OneOfTwo (Compare n m) IsLesser IsEqual) => LesserEq n m
instance (OneOfTwo (Compare n m) IsGreater IsEqual) => GreaterEq n m
-- | Non-zero number. For naturals it's same as positive
class NonZero n
-- | Positive number.
class Positive n
----------------------------------------------------------------
-- | Next number.
type family Next n :: *
nextN :: n -> Next n
nextN _ = undefined
-- | Previous number
type family Prev n :: *
prevN :: n -> Prev n
prevN _ = undefined
-- | Negate number.
type family Negate n :: *
negateN :: n -> Negate n
negateN _ = undefined
----------------------------------------------------------------
-- | Sum of two numbers.
type family Add n m :: *
addN :: n -> m -> Add n m
addN _ _ = undefined
-- | Difference of two numbers.
type family Sub n m :: *
subN :: n -> m -> Sub n m
subN _ _ = undefined
-- | Product of two numbers.
type family Mul n m :: *
mulN :: n -> m -> Mul n m
mulN _ _ = undefined
-- | Division of two numbers. 'n' and 'm' should be instances of this
-- class only if remainder of 'n/m' is zero.
type family Div n m :: *
divN :: n -> m -> Div n m
divN _ _ = undefined
----------------------------------------------------------------
-- | Usually numbers have non-unique representation. This type family
-- is canonical representation of number.
type family Normalized n :: *
|
