{-|
    Module      :  MixedTypesNumPrelude
    Description :  Bottom-up typed numeric expressions
    Copyright   :  (c) Michal Konecny, Pieter Collins
    License     :  BSD3

    Maintainer  :  mikkonecny@gmail.com
    Stability   :  experimental
    Portability :  portable

    = Main purpose

    This package provides a version of Prelude where
    unary and binary operations such as @not@, @+@, @==@
    have their result type derived from the parameter type(s),
    allowing, /e.g./:

      * Dividing an integer by an integer, giving a rational, wrapped in the CN (ie Collecting NumErrors) monad:

      >>> :t let n = 1 :: Integer in n/(n+1)
      ...CN Rational

      >>> :t 1/2
      ...CN Rational

      (Integer literals are always of type @Integer@, not @Num t => t@.)

      * Adding an integer and a rational, giving a rational:

      >>> :t (length [])+1/3
      ...CN Rational

      The @CN@ monad is required because integer division can, in general, fail as it is a partial operation:

      >>> 1/0
      {[(ERROR,division by 0)]}

      Note that when evaluating @1/0@, it evaluates to the error value printed above.
      This is not an exception, but a special value.

      When one is certain the division is well defined, one can remove @CN@ in several ways:

      >>> :t (1%2)
      ...Rational

      Above we use (re-exported) Data.Ratio.(%), which means this trick works only for Integers.

      >>> :t (1/!2)
      ...Rational

      This works also for non-integer division.

      >>> :t (~!) (1/2)
      ...Rational

      The (~!) operator removes CN from any type, throwing an exception if there are collected errors.

      The operator (/!) stands for division which throws an exception is the
      denominator is 0.  It "propagates" any potential errors
      from the sub-expressions.  For example:

      >>> :t 1/!(1 - 1/n)
      ...CN Rational

      The above expression will throw an error exception when evaluated with @n=1@
      but when @n=0@, it will not throw an excetion but return an error value.

      * taking natural, integer and fractional power using the same operator:

      >>> :t 2^2
      ...CN Integer

      >>> :t 2.0^(-2)
      ...CN Rational

      >>> :t (double 2)^(1/!2)
      ...Double

      The following examples require package <https://github.com/michalkonecny/aern2 aern2-real>:

      >>> :t 2^(1/2)
      ...CauchyRealCN

      >>> :t pi
      ...CauchyReal

      >>> :t sqrt 2
      ...CauchyRealCN

      * comparing an integer with an (exact) real number, giving a sequence of @Maybe Bool@:

      >>> let abs2 x = if x < 0 then -x else x in (abs2 (pi - pi)) ? (bitsS 100)
      [0 ± <2^(-102)]

      In the last example, @if@ is overloaded so that it works for conditions
      of other types than @Bool@.  Here the condition has the type @Sequence (Maybe Bool)@.
      The whole expression is the sequence of balls in which those balls for which the condition
      is inconclusive are the union of the balls computed by both branches.

    = Type classes

    Arithmetic operations are provided via multi-parameter type classes
    and the result type is given by associated
    type families. For example:

    > (+) :: (CanAddAsymmetric t1 t2) => t1 -> t2 -> AddType t1 t2

    The type constraint @CanAdd t1 t2@ implies both
    @CanAddAsymmetric t1 t2@ and @CanAddAsymmetric t2 t1@.

    For convenience there are other aggregate type constraints such as
    @CanAddThis t1 t2@, which implies that the result is of type @t1@,
    and @CanAddSameType t@, which is a shortcut for @CanAddThis t t@.

    == Testable specification

    The arithmetic type classes are accompanied by generic hspec test suites,
    which are specialised to concrete instance types for their testing.
    These test suites include the expected algebraic properties of operations,
    such as commutativity and associativity of addition.

    = Limitations

    * Not all numerical operations are supported yet.
      Eg @tan@, @atan@ are missing at the moment.

    * Inferred types can be very large. Eg for @f a b c = sqrt (a + b * c + 1)@ the inferred type is:

    >  f: (CanMulAsymmetric t1 t2, CanAddAsymmetric t4 (MulType t1 t2),
    >      CanAddAsymmetric (AddType t4 (MulType t1 t2)) Integer,
    >      CanSqrt (AddType (AddType t4 (MulType t1 t2)) Integer)) =>
    >     t4
    >     -> t1
    >     -> t2
    >     -> SqrtType (AddType (AddType t4 (MulType t1 t2)) Integer)

    * Due to limitations of some versions of ghc, type inferrence sometimes fails.
      Eg @add1 = (+ 1)@ fails (eg with ghc 8.0.2) unless we explicitly declare the type
      @add1 :: (CanAdd Integer t) => t -> AddType t Integer@
      or use an explicit parameter, eg @add1 x = x + 1@.

    = Origin

    The idea of having numeric expressions in Haskell with types
    derived bottom-up was initially suggested and implemented by Pieter Collins.
    This version is a fresh rewrite by Michal Konečný.

    = More details

    This module facilitates a single-line import for the package
    mixed-types-num.  See the re-exported modules for further details.
-}
module MixedTypesNumPrelude
(
  -- * Re-exporting Prelude, hiding the operators we are changing
  module Numeric.MixedTypes.PreludeHiding,
  -- * Modules with Prelude alternatives
  module Numeric.MixedTypes.Literals,
  module Numeric.MixedTypes.Bool,
  module Numeric.MixedTypes.Eq,
  module Numeric.MixedTypes.Ord,
  module Numeric.MixedTypes.MinMaxAbs,
  module Numeric.MixedTypes.AddSub,
  module Numeric.MixedTypes.Round,
  module Numeric.MixedTypes.Ring,
  module Numeric.MixedTypes.Field,
  module Numeric.MixedTypes.Elementary,
  module Numeric.MixedTypes.Complex,
  module Numeric.CollectErrors,
  module Utils.TH.DeclForTypes,
  -- * Re-export for convenient Rational literals
  (%)
)
where

import Data.Ratio ((%))
import Utils.TH.DeclForTypes
import Numeric.CollectErrors
import Numeric.MixedTypes.PreludeHiding
import Numeric.MixedTypes.Literals
import Numeric.MixedTypes.Bool
import Numeric.MixedTypes.Eq
import Numeric.MixedTypes.Ord
import Numeric.MixedTypes.MinMaxAbs
import Numeric.MixedTypes.AddSub
import Numeric.MixedTypes.Round
import Numeric.MixedTypes.Ring
import Numeric.MixedTypes.Field
import Numeric.MixedTypes.Elementary
import Numeric.MixedTypes.Complex