{-# LANGUAGE TemplateHaskell #-}
{-# OPTIONS_GHC -Wno-orphans #-}
{-# LANGUAGE PartialTypeSignatures #-}
{-# OPTIONS_GHC -Wno-partial-type-signatures #-}
{-|
    Module      :  Numeric.MixedType.MinMaxAbs
    Description :  Bottom-up typed min, max and abs
    Copyright   :  (c) Michal Konecny
    License     :  BSD3

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

-}

module Numeric.MixedTypes.MinMaxAbs
(
  -- * Minimum and maximum
  CanMinMax, CanMinMaxAsymmetric(..), CanMinMaxThis, CanMinMaxSameType
  , minimum, maximum
  -- ** Tests
  , specCanMinMax, specCanMinMaxNotMixed
  -- * Absolute value
  , CanAbs(..), CanAbsSameType
  -- ** Tests
  , specCanNegNum, specCanAbs
)
where

import Utils.TH.DeclForTypes

import Numeric.MixedTypes.PreludeHiding
import qualified Prelude as P
import Text.Printf

import qualified Data.List as List

import Test.Hspec
import Test.QuickCheck

import Control.CollectErrors ( CollectErrors, CanBeErrors )
import qualified Control.CollectErrors as CE

import Numeric.MixedTypes.Literals
import Numeric.MixedTypes.Bool
import Numeric.MixedTypes.Eq
import Numeric.MixedTypes.Ord

{---- Min and max -----}

type CanMinMax t1 t2 =
  (CanMinMaxAsymmetric t1 t2, CanMinMaxAsymmetric t2 t1,
   MinMaxType t1 t2 ~ MinMaxType t2 t1)

{-|
  A replacement for Prelude's `P.min` and `P.max`.  If @t1 = t2@ and @Ord t1@,
  then one can use the default implementation to mirror Prelude's @min@ and @max@.
-}
class CanMinMaxAsymmetric t1 t2 where
  type MinMaxType t1 t2
  type MinMaxType t1 t2 = t1 -- default
  min :: t1 -> t2 -> MinMaxType t1 t2
  max :: t1 -> t2 -> MinMaxType t1 t2
  default min :: (MinMaxType t1 t2 ~ t1, t1~t2, P.Ord t1) => t1 -> t2 -> MinMaxType t1 t2
  min = P.min
  default max :: (MinMaxType t1 t2 ~ t1, t1~t2, P.Ord t1) => t1 -> t2 -> MinMaxType t1 t2
  max = P.max

type CanMinMaxThis t1 t2 =
  (CanMinMax t1 t2, MinMaxType t1 t2 ~ t1)
type CanMinMaxSameType t =
  CanMinMaxThis t t

maximum :: (CanMinMaxSameType t) => [t] -> t
maximum (x:xs) = List.foldl' max x xs
maximum [] = error $ "maximum: empty list"

minimum :: (CanMinMaxSameType t) => [t] -> t
minimum (x:xs) = List.foldl' min x xs
minimum [] = error $ "minimum: empty list"

{-|
  HSpec properties that each implementation of CanMinMax should satisfy.
 -}
specCanMinMax ::
  _ => T t1 -> T t2 -> T t3 -> Spec
specCanMinMax (T typeName1 :: T t1) (T typeName2 :: T t2) (T typeName3 :: T t3) =
  describe (printf "CanMinMax %s %s, CanMinMax %s %s" typeName1 typeName2 typeName2 typeName3) $ do
    it "`min` is not larger than its arguments" $ do
      property $ \ (x :: t1) (y :: t2) ->
        -- (x ?==? x) && (y ?==? y) ==> -- avoid NaN
        (isFinite x) && (isFinite y) ==>
          let m = x `min` y in (m ?<=?$ y) .&&. (m ?<=?$ x)
    it "`max` is not smaller than its arguments" $ do
      property $ \ (x :: t1) (y :: t2) ->
        -- (x ?==? x) && (y ?==? y) ==> -- avoid NaN
        (isFinite x) && (isFinite y) ==>
          let m = x `max` y in (m ?>=?$ y) .&&. (m ?>=?$ x)
    it "has idempotent `min`" $ do
      property $ \ (x :: t1) ->
        -- (x ?==? x) ==> -- avoid NaN
        (isFinite x) ==>
          (x `min` x) ?==?$ x
    it "has idempotent `max`" $ do
      property $ \ (x :: t1) ->
        -- (x ?==? x) ==> -- avoid NaN
        (isFinite x) ==>
          (x `max` x) ?==?$ x
    it "has commutative `min`" $ do
      property $ \ (x :: t1) (y :: t2) ->
        -- (x ?==? x) && (y ?==? y) ==> -- avoid NaN
        (isFinite x) && (isFinite y) ==>
          (x `min` y) ?==?$ (y `min` x)
    it "has commutative `max`" $ do
      property $ \ (x :: t1) (y :: t2) ->
        -- (x ?==? x) && (y ?==? y) ==> -- avoid NaN
        (isFinite x) && (isFinite y) ==>
          (x `max` y) ?==?$ (y `max` x)
    it "has associative `min`" $ do
      property $ \ (x :: t1) (y :: t2) (z :: t3) ->
        -- (x ?==? x) && (y ?==? y) && (z ?==? z) ==> -- avoid NaN
        (isFinite x) && (isFinite y) && (isFinite z) ==>
            (x `min` (y `min` z)) ?==?$ ((x `min` y) `min` z)
    it "has associative `max`" $ do
      property $ \ (x :: t1) (y :: t2) (z :: t3) ->
        -- (x ?==? x) && (y ?==? y) && (z ?==? z) ==> -- avoid NaN
        (isFinite x) && (isFinite y) && (isFinite z) ==>
            (x `max` (y `max` z)) ?==?$ ((x `max` y) `max` z)
  where
  (?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  (?==?$) = printArgsIfFails2 "?==?" (?==?)
  (?>=?$) :: (HasOrderCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  (?>=?$) = printArgsIfFails2 "?>=?" (?>=?)
  (?<=?$) :: (HasOrderCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  (?<=?$) = printArgsIfFails2 "?<=?" (?<=?)
--
{-|
  HSpec properties that each implementation of CanMinMax should satisfy.
 -}
specCanMinMaxNotMixed ::
  _ => T t -> Spec
specCanMinMaxNotMixed t = specCanMinMax t t t

instance CanMinMaxAsymmetric Int Int
instance CanMinMaxAsymmetric Integer Integer
instance CanMinMaxAsymmetric Rational Rational
instance CanMinMaxAsymmetric Double Double

instance CanMinMaxAsymmetric Int Integer where
  type MinMaxType Int Integer = Integer
  min = convertFirst min
  max = convertFirst max
instance CanMinMaxAsymmetric Integer Int where
  type MinMaxType Integer Int = Integer
  min = convertSecond min
  max = convertSecond max

$(declForTypes
  [[t| Integer |], [t| Int |]]
  (\ t -> [d|

  instance CanMinMaxAsymmetric $t Rational where
    type MinMaxType $t Rational = Rational
    min = convertFirst min
    max = convertFirst max
  instance CanMinMaxAsymmetric Rational $t where
    type MinMaxType Rational $t = Rational
    min = convertSecond min
    max = convertSecond max
  |]))

$(declForTypes
  [[t| Integer |], [t| Int |], [t| Rational |]]
  (\ t -> [d|

  instance
    CanMinMaxAsymmetric $t Double
    where
    type MinMaxType $t Double = Double
    min a b = min (double a) b
    max a b = max (double a) b

  instance
    CanMinMaxAsymmetric Double $t
    where
    type MinMaxType Double $t = Double
    min a b = min a (double b)
    max a b = max a (double b)

  |]))

instance (CanMinMaxAsymmetric a b) => CanMinMaxAsymmetric [a] [b] where
  type MinMaxType [a] [b] = [MinMaxType a b]
  min (x:xs) (y:ys) = (min x y) : (min xs ys)
  min _ _ = []
  max (x:xs) (y:ys) = (max x y) : (max xs ys)
  max _ _ = []

instance (CanMinMaxAsymmetric a b) => CanMinMaxAsymmetric (Maybe a) (Maybe b) where
  type MinMaxType (Maybe a) (Maybe b) = Maybe (MinMaxType a b)
  min (Just x) (Just y) = Just (min x y)
  min _ _ = Nothing
  max (Just x) (Just y) = Just (max x y)
  max _ _ = Nothing

instance
  (CanMinMaxAsymmetric a b, CanBeErrors es)
  =>
  CanMinMaxAsymmetric (CollectErrors es a) (CollectErrors es  b)
  where
  type MinMaxType (CollectErrors es a) (CollectErrors es b) =
    CollectErrors es (MinMaxType a b)
  min = CE.lift2 min
  max = CE.lift2 max

$(declForTypes
  [[t| Integer |], [t| Int |], [t| Rational |], [t| Double |]]
  (\ t -> [d|

    instance
      (CanMinMaxAsymmetric $t b, CanBeErrors es)
      =>
      CanMinMaxAsymmetric $t (CollectErrors es  b)
      where
      type MinMaxType $t (CollectErrors es  b) =
        CollectErrors es (MinMaxType $t b)
      min = CE.liftT1 min
      max = CE.liftT1 max

    instance
      (CanMinMaxAsymmetric a $t, CanBeErrors es)
      =>
      CanMinMaxAsymmetric (CollectErrors es a) $t
      where
      type MinMaxType (CollectErrors es  a) $t =
        CollectErrors es (MinMaxType a $t)
      min = CE.lift1T min
      max = CE.lift1T max

  |]))


{----  numeric negation tests and instances -----}

{-|
  HSpec properties that each numeric implementation of CanNeg should satisfy.
 -}
specCanNegNum ::
  _ => T t -> Spec
specCanNegNum (T typeName :: T t) =
  describe (printf "CanNeg %s" typeName) $ do
    it "ignores double negation" $ do
      property $ \ (x :: t) ->
        (x ?==? x) ==> -- avoid NaN
          (negate (negate x)) ?==?$ x
    it "takes 0 to 0" $ do
      let z = convertExactly 0 :: t in negate z ?==? z
    it "takes positive to negative" $ do
      property $ \ (x :: t) ->
        (isFinite x) ==> -- avoid NaN
        (isCertainlyPositive x) ==> (isCertainlyNegative (negate x))
    it "takes negative to positive" $ do
      property $ \ (x :: t) ->
        (isFinite x) ==> -- avoid NaN
        (isCertainlyNegative x) ==> (isCertainlyPositive (negate x))
  where
  (?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  (?==?$) = printArgsIfFails2 "?==?" (?==?)

instance CanNeg Int where negate = P.negate
instance CanNeg Integer where negate = P.negate
instance CanNeg Rational where negate = P.negate
instance CanNeg Double where negate = P.negate

{----  abs -----}

{-|
  A replacement for Prelude's `P.abs`.  If @Num t@,
  then one can use the default implementation to mirror Prelude's @abs@.
-}
class CanAbs t where
  type AbsType t
  type AbsType t = t -- default
  abs :: t -> AbsType t
  default abs :: (AbsType t ~ t, P.Num t) => t -> AbsType t
  abs = P.abs

type CanAbsSameType t = (CanAbs t, AbsType t ~ t)

instance CanAbs Int
instance CanAbs Integer
instance CanAbs Rational
instance CanAbs Double

instance
  (CanAbs a, CanBeErrors es)
  =>
  CanAbs (CollectErrors es a)
  where
  type AbsType (CollectErrors es a) = CollectErrors es (AbsType a)
  abs = CE.lift abs

{-|
  HSpec properties that each implementation of CanAbs should satisfy.
 -}
specCanAbs ::
  _ => T t -> Spec
specCanAbs (T typeName :: T t) =
  describe (printf "CanAbs %s" typeName) $ do
    it "is idempotent" $ do
      property $ \ (x :: t) ->
        (x ?==? x) ==> -- avoid NaN
          (abs (abs x)) ?==?$ (abs x)
    it "is identity on non-negative arguments" $ do
      property $ \ (x :: t) ->
        (isFinite x) ==>
        isCertainlyNonNegative x  ==> x ?==?$ (abs x)
    it "is negation on non-positive arguments" $ do
      property $ \ (x :: t) ->
        (isFinite x) ==>
        isCertainlyNonPositive x  ==> (negate x) ?==?$ (abs x)
    it "does not give negative results" $ do
      property $ \ (x :: t) -> 
        (isFinite x) ==>
        not $ isCertainlyNegative (abs x)
  where
  (?==?$) :: (HasEqCertainlyAsymmetric a b, Show a, Show b) => a -> b -> Property
  (?==?$) = printArgsIfFails2 "?==?" (?==?)