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/src/Constrained/NumOrd.hs

{-# LANGUAGE AllowAmbiguousTypes #-}
{-# LANGUAGE CPP #-}
{-# LANGUAGE ConstraintKinds #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE DefaultSignatures #-}
{-# LANGUAGE DerivingVia #-}
{-# LANGUAGE FlexibleContexts #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE LambdaCase #-}
{-# LANGUAGE PatternSynonyms #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE TypeApplications #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeOperators #-}
{-# LANGUAGE UndecidableInstances #-}
{-# LANGUAGE UndecidableSuperClasses #-}
{-# LANGUAGE ViewPatterns #-}
-- Random Natural, Arbitrary Natural, Uniform Natural
{-# OPTIONS_GHC -Wno-orphans #-}

-- | Everything we need to deal with numbers and comparisons between them
module Constrained.NumOrd (
  NumSpec (..),
  (>.),
  (<.),
  (-.),
  (>=.),
  (<=.),
  (+.),
  (*.),
  negate_,
  cardinality,
  caseBoolSpec,
  addSpecInt,
  emptyNumSpec,
  cardinalNumSpec,
  combineNumSpec,
  genFromNumSpec,
  shrinkWithNumSpec,
  fixupWithNumSpec,
  fixupWithTypeSpec,
  conformsToNumSpec,
  toPredsNumSpec,
  OrdLike (..),
  MaybeBounded (..),
  NumLike (..),
  HasDivision (..),
  Numeric,
  Number,
  nubOrd,
  IntW (..),
  OrdW (..),
) where

import Constrained.AbstractSyntax
import Constrained.Base
import Constrained.Conformance
import Constrained.Core (Value (..), unionWithMaybe)
import Constrained.FunctionSymbol
import Constrained.GenT
import Constrained.Generic
import Constrained.List
import Constrained.PrettyUtils
import Control.Applicative ((<|>))
import Control.Arrow (first)
import Data.Containers.ListUtils
import Data.Foldable
import Data.Kind
import Data.List (nub)
import Data.List.NonEmpty (NonEmpty ((:|)))
import qualified Data.List.NonEmpty as NE
import Data.Maybe
import Data.Typeable (typeOf)
import Data.Word
import GHC.Int
import GHC.Natural
import GHC.Real
import System.Random.Stateful (Random (..), Uniform (..))
import Test.QuickCheck (Arbitrary (arbitrary, shrink), choose, frequency)

-- | Witnesses for comparison operations (<=. and <. and <=. and >=.) on numbers
-- The other operations are defined in terms of these.
data OrdW (dom :: [Type]) (rng :: Type) where
  LessOrEqualW :: OrdLike a => OrdW '[a, a] Bool
  LessW :: OrdLike a => OrdW '[a, a] Bool
  GreaterOrEqualW :: OrdLike a => OrdW '[a, a] Bool
  GreaterW :: OrdLike a => OrdW '[a, a] Bool

deriving instance Eq (OrdW ds r)

instance Show (OrdW ds r) where
  show LessOrEqualW = "<=."
  show LessW = "<."
  show GreaterOrEqualW = ">=."
  show GreaterW = ">."

instance Semantics OrdW where
  semantics LessOrEqualW = (<=)
  semantics LessW = (<)
  semantics GreaterW = (>)
  semantics GreaterOrEqualW = (>=)

instance Syntax OrdW where
  isInfix _ = True

-- =============================================
-- OrdLike. Ord for Numbers in the Logic
-- =============================================

-- | Ancillary things we need to be able to implement `Logic` instances for
-- `OrdW` that make sense for a given type we are comparing things on.
class (Ord a, HasSpec a) => OrdLike a where
  leqSpec :: a -> Specification a
  default leqSpec ::
    ( GenericRequires a
    , OrdLike (SimpleRep a)
    ) =>
    a ->
    Specification a
  leqSpec = fromSimpleRepSpec . leqSpec . toSimpleRep

  ltSpec :: a -> Specification a
  default ltSpec ::
    ( OrdLike (SimpleRep a)
    , GenericRequires a
    ) =>
    a ->
    Specification a
  ltSpec = fromSimpleRepSpec . ltSpec . toSimpleRep

  geqSpec :: a -> Specification a
  default geqSpec ::
    ( OrdLike (SimpleRep a)
    , GenericRequires a
    ) =>
    a ->
    Specification a
  geqSpec = fromSimpleRepSpec . geqSpec . toSimpleRep

  gtSpec :: a -> Specification a
  default gtSpec ::
    ( OrdLike (SimpleRep a)
    , GenericRequires a
    ) =>
    a ->
    Specification a
  gtSpec = fromSimpleRepSpec . gtSpec . toSimpleRep

-- | This instance should be general enough for every type of Number that has a NumSpec as its TypeSpec
instance {-# OVERLAPPABLE #-} (Ord a, HasSpec a, MaybeBounded a, Num a, TypeSpec a ~ NumSpec a) => OrdLike a where
  leqSpec l = typeSpec $ NumSpecInterval Nothing (Just l)
  ltSpec l
    | Just b <- lowerBound
    , l == b =
        ErrorSpec (pure ("ltSpec @" ++ show (typeOf l) ++ " " ++ show l))
    | otherwise = typeSpec $ NumSpecInterval Nothing (Just (l - 1))
  geqSpec l = typeSpec $ NumSpecInterval (Just l) Nothing
  gtSpec l
    | Just b <- upperBound
    , l == b =
        ErrorSpec (pure ("gtSpec @" ++ show (typeOf l) ++ " " ++ show l))
    | otherwise = typeSpec $ NumSpecInterval (Just (l + 1)) Nothing

-- ========================================================================
-- helper functions for the TypeSpec for Numbers
-- ========================================================================

-- | Helper class for talking about things that _might_ be `Bounded`
class MaybeBounded a where
  lowerBound :: Maybe a
  upperBound :: Maybe a

  default lowerBound :: Bounded a => Maybe a
  lowerBound = Just minBound

  default upperBound :: Bounded a => Maybe a
  upperBound = Just maxBound

newtype Unbounded a = Unbounded a

instance MaybeBounded (Unbounded a) where
  lowerBound = Nothing
  upperBound = Nothing

instance MaybeBounded Int

instance MaybeBounded Int64

instance MaybeBounded Int32

instance MaybeBounded Int16

instance MaybeBounded Int8

instance MaybeBounded Word64

instance MaybeBounded Word32

instance MaybeBounded Word16

instance MaybeBounded Word8

deriving via Unbounded Integer instance MaybeBounded Integer

deriving via Unbounded (Ratio Integer) instance MaybeBounded (Ratio Integer)

deriving via Unbounded Float instance MaybeBounded Float

deriving via Unbounded Double instance MaybeBounded Double

instance MaybeBounded Natural where
  lowerBound = Just 0
  upperBound = Nothing

-- ===================================================================
-- The TypeSpec for numbers
-- ===================================================================

-- | t`TypeSpec` for numbers - represented as a single interval
data NumSpec n = NumSpecInterval (Maybe n) (Maybe n)

instance Ord n => Eq (NumSpec n) where
  NumSpecInterval ml mh == NumSpecInterval ml' mh'
    | isEmpty ml mh = isEmpty ml' mh'
    | isEmpty ml' mh' = isEmpty ml mh
    | otherwise = ml == ml' && mh == mh'
    where
      isEmpty (Just a) (Just b) = a > b
      isEmpty _ _ = False

instance Show n => Show (NumSpec n) where
  show (NumSpecInterval ml mu) = lb ++ ".." ++ ub
    where
      lb = "[" ++ maybe "" show ml
      ub = maybe "" show mu ++ "]"

instance Ord n => Semigroup (NumSpec n) where
  NumSpecInterval ml mu <> NumSpecInterval ml' mu' =
    NumSpecInterval
      (unionWithMaybe max ml ml')
      (unionWithMaybe min mu mu')

instance Ord n => Monoid (NumSpec n) where
  mempty = NumSpecInterval Nothing Nothing

-- ===========================================
-- Arbitrary for Num like things
-- ===========================================

instance (Arbitrary a, Ord a) => Arbitrary (NumSpec a) where
  arbitrary = do
    m <- arbitrary
    m' <- arbitrary
    frequency [(10, pure $ mkLoHiInterval m m'), (1, pure $ NumSpecInterval m m')]
    where
      mkLoHiInterval (Just a) (Just b) = NumSpecInterval (Just $ min a b) (Just $ max a b)
      mkLoHiInterval m m' = NumSpecInterval m m'
  shrink (NumSpecInterval m m') =
    uncurry NumSpecInterval <$> shrink (m, m')

#if !MIN_VERSION_QuickCheck(2, 17, 0)
instance Arbitrary Natural where
  arbitrary = wordToNatural . abs <$> arbitrary
  shrink n = [wordToNatural w | w <- shrink (naturalToWord n)]
#endif

instance Uniform Natural where
  uniformM g = wordToNatural . abs <$> uniformM g

instance Random Natural where
  randomR (lo, hi) g = first fromIntegral $ randomR (toInteger lo, toInteger hi) g

instance Random (Ratio Integer) where
  randomR (lo, hi) g =
    let (r, g') = random g
     in (lo + (hi - lo) * r, g')
  random g =
    let (d, g') = first ((+ 1) . abs) $ random g
        (n, g'') = randomR (0, d) g'
     in (n % d, g'')

-- ==============================================================================
-- Operations on NumSpec, that give it the required properties of a TypeSpec
-- ==============================================================================

-- | Admits anything
emptyNumSpec :: Ord a => NumSpec a
emptyNumSpec = mempty

guardNumSpec ::
  (Ord n, HasSpec n, TypeSpec n ~ NumSpec n) =>
  [String] ->
  NumSpec n ->
  Specification n
guardNumSpec msg s@(NumSpecInterval (Just a) (Just b))
  | a > b = ErrorSpec ("NumSpec has low bound greater than hi bound" :| (("   " ++ show s) : msg))
  | a == b = equalSpec a
guardNumSpec _ s = typeSpec s

-- | Conjunction
combineNumSpec ::
  (HasSpec n, Ord n, TypeSpec n ~ NumSpec n) =>
  NumSpec n ->
  NumSpec n ->
  Specification n
combineNumSpec s s' = guardNumSpec ["when combining two NumSpecs", "   " ++ show s, "   " ++ show s'] (s <> s')

-- | Generate a value that satisfies the spec
genFromNumSpec ::
  (MonadGenError m, Show n, Random n, Ord n, Num n, MaybeBounded n) =>
  NumSpec n ->
  GenT m n
genFromNumSpec (NumSpecInterval ml mu) = do
  n <- sizeT
  pureGen . choose =<< constrainInterval (ml <|> lowerBound) (mu <|> upperBound) (fromIntegral n)

-- TODO: fixme

-- | Try to shrink using a `NumSpec`
shrinkWithNumSpec :: Arbitrary n => NumSpec n -> n -> [n]
shrinkWithNumSpec _ = shrink

-- TODO: fixme

fixupWithNumSpec :: Arbitrary n => NumSpec n -> n -> Maybe n
fixupWithNumSpec _ = listToMaybe . shrink

constrainInterval ::
  (MonadGenError m, Ord a, Num a, Show a) => Maybe a -> Maybe a -> Integer -> m (a, a)
constrainInterval ml mu r =
  case (ml, mu) of
    (Nothing, Nothing) -> pure (-r', r')
    (Just l, Nothing)
      | l < 0 -> pure (max l (negate r'), r')
      | otherwise -> pure (l, l + 2 * r')
    (Nothing, Just u)
      | u > 0 -> pure (negate r', min u r')
      | otherwise -> pure (u - r' - r', u)
    (Just l, Just u)
      | l > u -> genError ("bad interval: " ++ show l ++ " " ++ show u)
      | u < 0 -> pure (safeSub l (safeSub l u r') r', u)
      | l >= 0 -> pure (l, safeAdd u (safeAdd u l r') r')
      -- TODO: this is a bit suspect if the bounds are lopsided
      | otherwise -> pure (max l (-r'), min u r')
  where
    r' = abs $ fromInteger r
    safeSub l a b
      | a - b > a = l
      | otherwise = max l (a - b)
    safeAdd u a b
      | a + b < a = u
      | otherwise = min u (a + b)

-- | Check that a value is in the spec
conformsToNumSpec :: Ord n => n -> NumSpec n -> Bool
conformsToNumSpec i (NumSpecInterval ml mu) = maybe True (<= i) ml && maybe True (i <=) mu

-- =======================================================================
-- Several of the methods of HasSpec that have default implementations
-- could benefit from type specific implementations for numbers. Those
-- implementations are found here
-- =====================================================================

-- | Builds a MemberSpec, but returns an Error spec if the list is empty
nubOrdMemberSpec :: Ord a => String -> [a] -> Specification a
nubOrdMemberSpec message xs =
  memberSpec
    (nubOrd xs)
    ( NE.fromList
        [ "In call to nubOrdMemberSpec"
        , "Called from context"
        , message
        , "The input is the empty list."
        ]
    )

lowBound :: Bounded n => Maybe n -> n
lowBound Nothing = minBound
lowBound (Just n) = n

highBound :: Bounded n => Maybe n -> n
highBound Nothing = maxBound
highBound (Just n) = n

-- | The exact count of the number elements in a Bounded NumSpec
countSpec :: forall n. (Bounded n, Integral n) => NumSpec n -> Integer
countSpec (NumSpecInterval lo hi) = if lo > hi then 0 else toInteger high - toInteger low + 1
  where
    high = highBound hi
    low = lowBound lo

-- | The exact number of elements in a Bounded Integral type.
finiteSize :: forall n. (Integral n, Bounded n) => Integer
finiteSize = toInteger (maxBound @n) - toInteger (minBound @n) + 1

-- | This is an optimizing version of  TypeSpec :: TypeSpec n -> [n] -> Specification n
--   for Bounded NumSpecs.
--                    notInNumSpec :: Bounded n => TypeSpec n -> [n] -> Specification n
--   We use this function to specialize the (HasSpec t) method 'typeSpecOpt' for Bounded n.
--   So given (TypeSpec interval badlist) we might want to transform it to (MemberSpec goodlist)
--   There are 2 opportunities where this can payoff big time.
--   1) Suppose the total count of the elements in the interval is < length badlist
--      we can then return (MemberSpec (filter elements (`notElem` badlist)))
--      this must be smaller than (TypeSpec interval badlist) because the filtered list must be smaller than badlist
--   2) Suppose the type 't' is finite with size N. If the length of the badlist > (N/2), then the number of possible
--      good things must be smaller than (length badlist), because (possible good + bad == N), so regardless of the
--      count of the interval (MemberSpec (filter elements (`notElem` badlist))) is better. Sometimes much better.
--      Example, let 'n' be the finite set {0,1,2,3,4,5,6,7,8,9} and the bad list be [0,1,3,4,5,6,8,9]
--      (TypeSpec [0..9]  [0,1,3,4,5,6,8,9]) = filter  {0,1,2,3,4,5,6,7,8,9} (`notElem` [0,1,3,4,5,6,8,9]) = [2,7]
--      So (MemberSpec [2,7]) is better than  (TypeSpec [0..9]  [0,1,3,4,5,6,8,9]). This works no matter what
--      the count of interval is. We only need the (length badlist > (N/2)).
notInNumSpec ::
  forall n.
  ( HasSpec n
  , TypeSpec n ~ NumSpec n
  , Bounded n
  , Integral n
  ) =>
  NumSpec n ->
  [n] ->
  Specification n
notInNumSpec ns@(NumSpecInterval a b) bad
  | toInteger (length bad) > (finiteSize @n `div` 2) || countSpec ns < toInteger (length bad) =
      nubOrdMemberSpec
        ("call to: (notInNumSpec " ++ show ns ++ " " ++ show bad ++ ")")
        [x | x <- [lowBound a .. highBound b], notElem x bad]
  | otherwise = TypeSpec @n ns bad

-- ==========================================================================
-- Num n => (NumSpec n) can support operation of Num as interval arithmetic.
-- So we will make a (Num (NumSpec Integer)) instance. We won't make other
-- instances, because  they would be subject to overflow.
-- Given operator ☉, then (a,b) ☉ (c,d) = (minimum s, maximum s) where s = [a ☉ c, a ☉ d, b ☉ c, b ☉ d]
-- There are simpler rules for (+) and (-), but for (*) we need to use the general rule.
-- ==========================================================================

guardEmpty :: (Ord n, Num n) => Maybe n -> Maybe n -> NumSpec n -> NumSpec n
guardEmpty (Just a) (Just b) s
  | a <= b = s
  | otherwise = NumSpecInterval (Just 1) (Just 0)
guardEmpty _ _ s = s

addNumSpec :: (Ord n, Num n) => NumSpec n -> NumSpec n -> NumSpec n
addNumSpec (NumSpecInterval x y) (NumSpecInterval a b) =
  guardEmpty x y $
    guardEmpty a b $
      NumSpecInterval ((+) <$> x <*> a) ((+) <$> y <*> b)

subNumSpec :: (Ord n, Num n) => NumSpec n -> NumSpec n -> NumSpec n
subNumSpec (NumSpecInterval x y) (NumSpecInterval a b) =
  guardEmpty x y $
    guardEmpty a b $
      NumSpecInterval ((-) <$> x <*> b) ((-) <$> y <*> a)

multNumSpec :: (Ord n, Num n) => NumSpec n -> NumSpec n -> NumSpec n
multNumSpec (NumSpecInterval a b) (NumSpecInterval c d) =
  guardEmpty a b $
    guardEmpty c d $
      NumSpecInterval (unT (minimum s)) (unT (maximum s))
  where
    s = [multT (neg a) (neg c), multT (neg a) (pos d), multT (pos b) (neg c), multT (pos b) (pos d)]

negNumSpec :: Num n => NumSpec n -> NumSpec n
negNumSpec (NumSpecInterval lo hi) = NumSpecInterval (negate <$> hi) (negate <$> lo)

instance Num (NumSpec Integer) where
  (+) = addNumSpec
  (-) = subNumSpec
  (*) = multNumSpec
  negate = negNumSpec
  fromInteger n = NumSpecInterval (Just (fromInteger n)) (Just (fromInteger n))
  abs = error "No abs in the Num (NumSpec  Integer) instance"
  signum = error "No signum in the Num (NumSpec  Integer) instance"

-- ========================================================================
-- Helper functions for interval multiplication
--  (a,b) * (c,d) = (minimum s, maximum s) where s = [a * c, a * d, b * c, b * d]

-- | T is a sort of special version of Maybe, with two Nothings.
--   Given:: NumSpecInterval (Maybe n) (Maybe n) -> Numspec
--   We can't distinguish between the two Nothings in (NumSpecInterval Nothing Nothing)
--   But using (NumSpecInterval NegInf PosInf) we can, In fact we can make a total ordering on 'T'
--   So an ascending Sorted [T x] would all the NegInf on the left and all the PosInf on the right, with
--   the Ok's sorted in between. I.e. [NegInf, NegInf, Ok 3, Ok 6, Ok 12, Pos Inf]
data T x = NegInf | Ok x | PosInf
  deriving (Show, Eq, Ord)

-- \| Conversion between (T x) and (Maybe x)
unT :: T x -> Maybe x
unT (Ok x) = Just x
unT _ = Nothing

-- | Use this on the lower bound. I.e. lo from pair (lo,hi)
neg :: Maybe x -> T x
neg Nothing = NegInf
neg (Just x) = Ok x

-- | Use this on the upper bound. I.e. hi from pair (lo,hi)
pos :: Maybe x -> T x
pos Nothing = PosInf
pos (Just x) = Ok x

-- | multiply two (T x), correctly handling the infinities NegInf and PosInf
multT :: Num x => T x -> T x -> T x
multT NegInf NegInf = PosInf
multT NegInf PosInf = NegInf
multT NegInf (Ok _) = NegInf
multT (Ok _) NegInf = NegInf
multT (Ok x) (Ok y) = Ok (x * y)
multT (Ok _) PosInf = PosInf
multT PosInf PosInf = PosInf
multT PosInf NegInf = NegInf
multT PosInf (Ok _) = PosInf

-- ========================================================================
-- We have
-- (1) Num Integer
-- (2) Num (NumSpec Integer)   And we need
-- (3) Num (Specification Integer)
-- We need this to implement the method cardinalTypeSpec of (HasSpec t).
-- cardinalTypeSpec :: HasSpec a => TypeSpec a -> Specification Integer
-- Basically for defining these two cases
-- cardinalTypeSpec (Cartesian x y) = (cardinality x) * (cardinality y)
-- cardinalTypeSpec (SumSpec leftspec rightspec) = (cardinality leftspec) + (cardinality rightspec)
-- So we define addSpecInt for (+)   and  multSpecInt for (*)

-- | What constraints we need to make HasSpec instance for a Haskell numeric type.
--   By abstracting over this, we can avoid making actual HasSpec instances until
--   all the requirements (HasSpec Bool, HasSpec(Sum a b)) have been met in
--   Constrained.TheKnot.
type Number n = (Num n, Enum n, TypeSpec n ~ NumSpec n, Num (NumSpec n), HasSpec n, Ord n)

-- | Addition on `Specification` for `Number`
addSpecInt ::
  Number n =>
  Specification n ->
  Specification n ->
  Specification n
addSpecInt x y = operateSpec " + " (+) (+) x y

subSpecInt ::
  Number n =>
  Specification n ->
  Specification n ->
  Specification n
subSpecInt x y = operateSpec " - " (-) (-) x y

multSpecInt ::
  Number n =>
  Specification n ->
  Specification n ->
  Specification n
multSpecInt x y = operateSpec " * " (*) (*) x y

-- | let 'n' be some numeric type, and 'f' and 'ft' be operations on 'n' and (TypeSpec n)
--   Then lift these operations from (TypeSpec n) to (Specification n)
--   Normally 'f' will be a (Num n) instance method (+,-,*) on n,
--   and 'ft' will be a a (Num (TypeSpec n)) instance method (+,-,*) on (TypeSpec n)
--   But this will work for any operations 'f' and 'ft' with the right types
operateSpec ::
  Number n =>
  String ->
  (n -> n -> n) ->
  (TypeSpec n -> TypeSpec n -> TypeSpec n) ->
  Specification n ->
  Specification n ->
  Specification n
operateSpec operator f ft (ExplainSpec es x) y = explainSpec es $ operateSpec operator f ft x y
operateSpec operator f ft x (ExplainSpec es y) = explainSpec es $ operateSpec operator f ft x y
operateSpec operator f ft x y = case (x, y) of
  (ErrorSpec xs, ErrorSpec ys) -> ErrorSpec (xs <> ys)
  (ErrorSpec xs, _) -> ErrorSpec xs
  (_, ErrorSpec ys) -> ErrorSpec ys
  (TrueSpec, _) -> TrueSpec
  (_, TrueSpec) -> TrueSpec
  (_, SuspendedSpec _ _) -> TrueSpec
  (SuspendedSpec _ _, _) -> TrueSpec
  (TypeSpec a bad1, TypeSpec b bad2) -> TypeSpec (ft a b) [f b1 b2 | b1 <- bad1, b2 <- bad2]
  (MemberSpec xs, MemberSpec ys) ->
    nubOrdMemberSpec
      (show x ++ operator ++ show y)
      [f x1 y1 | x1 <- NE.toList xs, y1 <- NE.toList ys]
  -- This block is all (MemberSpec{}, TypeSpec{}) with MemberSpec on the left
  (MemberSpec ys, TypeSpec (NumSpecInterval (Just i) (Just j)) bad) ->
    let xs = NE.toList ys
     in nubOrdMemberSpec
          (show x ++ operator ++ show y)
          [f x1 y1 | x1 <- xs, y1 <- [i .. j], not (elem y1 bad)]
  -- Somewhat loose spec here, but more accurate then TrueSpec, it is exact if 'xs' has one element (i.e. 'xs' = [i])
  (MemberSpec ys, TypeSpec (NumSpecInterval lo hi) bads) ->
    -- We use the specialized version of 'TypeSpec' 'typeSpecOpt'
    let xs = NE.toList ys
     in typeSpecOpt
          (NumSpecInterval (f (minimum xs) <$> lo) (f (maximum xs) <$> hi))
          [f x1 b | x1 <- xs, b <- bads]
  -- we flip the arguments, so we need to flip the functions as well
  (sleft, sright) -> operateSpec operator (\a b -> f b a) (\u v -> ft v u) sright sleft

-- | This is very liberal, since in lots of cases it returns TrueSpec.
--  for example all operations on SuspendedSpec, and certain
--  operations between TypeSpec and MemberSpec. Perhaps we should
--  remove it. Only the addSpec (+) and multSpec (*) methods are used.
--  But, it is kind of cool ...
--  In Fact we can use this to make Num(Specification n) instance for any 'n'.
--  But, only Integer is safe, because in all other types (+) and especially
--  (-) can lead to overflow or underflow failures.
instance Number Integer => Num (Specification Integer) where
  (+) = addSpecInt
  (-) = subSpecInt
  (*) = multSpecInt
  fromInteger n = TypeSpec (NumSpecInterval (Just n) (Just n)) []
  abs _ = TrueSpec
  signum _ = TrueSpec

-- ===========================================================================

-- | Put some (admittedly loose bounds) on the number of solutions that
--   'genFromTypeSpec' might return. For lots of types, there is no way to be very accurate.
--   Here we lift the HasSpec methods 'cardinalTrueSpec' and 'cardinalTypeSpec'
--   from (TypeSpec Integer) to (Specification Integer)
cardinality ::
  forall a. (Number Integer, HasSpec a) => Specification a -> Specification Integer
cardinality (ExplainSpec es s) = explainSpec es (cardinality s)
cardinality TrueSpec = cardinalTrueSpec @a
cardinality (MemberSpec es) = equalSpec (toInteger $ length (nub (NE.toList es)))
cardinality ErrorSpec {} = equalSpec 0
cardinality (TypeSpec s cant) =
  subSpecInt
    (cardinalTypeSpec @a s)
    (equalSpec (toInteger $ length (nub $ filter (\c -> conformsTo @a c s) cant)))
cardinality SuspendedSpec {} = cardinalTrueSpec @a

-- | A generic function to use as an instance for the HasSpec method
--   cardinalTypeSpec :: HasSpec a => TypeSpec a -> Specification Integer
--   for types 'n' such that (TypeSpec n ~ NumSpec n)
cardinalNumSpec ::
  forall n. (Integral n, MaybeBounded n, HasSpec n) => NumSpec n -> Specification Integer
cardinalNumSpec (NumSpecInterval (Just lo) (Just hi)) =
  if hi >= lo
    then equalSpec (toInteger hi - toInteger lo + 1)
    else equalSpec 0
cardinalNumSpec (NumSpecInterval Nothing (Just hi)) =
  case lowerBound @n of
    Just lo -> equalSpec (toInteger hi - toInteger lo)
    Nothing -> TrueSpec
cardinalNumSpec (NumSpecInterval (Just lo) Nothing) =
  case upperBound @n of
    Just hi -> equalSpec (toInteger hi - toInteger lo)
    Nothing -> TrueSpec
cardinalNumSpec (NumSpecInterval Nothing Nothing) = cardinalTrueSpec @n

-- ====================================================================
-- Now the operations on Numbers

-- | Everything we need to make the number operations make sense on a given type
class (Num a, HasSpec a, HasDivision a, OrdLike a) => NumLike a where
  subtractSpec :: a -> TypeSpec a -> Specification a
  default subtractSpec ::
    ( NumLike (SimpleRep a)
    , GenericRequires a
    ) =>
    a ->
    TypeSpec a ->
    Specification a
  subtractSpec a ts = fromSimpleRepSpec $ subtractSpec (toSimpleRep a) ts

  negateSpec :: TypeSpec a -> Specification a
  default negateSpec ::
    ( NumLike (SimpleRep a)
    , GenericRequires a
    ) =>
    TypeSpec a ->
    Specification a
  negateSpec = fromSimpleRepSpec . negateSpec @(SimpleRep a)

  safeSubtract :: a -> a -> Maybe a
  default safeSubtract ::
    (HasSimpleRep a, NumLike (SimpleRep a)) =>
    a ->
    a ->
    Maybe a
  safeSubtract a b = fromSimpleRep <$> safeSubtract @(SimpleRep a) (toSimpleRep a) (toSimpleRep b)

-- | Operations on numbers.
-- The reason there is no implementation of abs here is that you can't easily deal with abs
-- without specifications becoming very large. Consider the following example:
-- > constrained $ \ x -> [1000 <. abs_ x, abs_ x <. 1050]
-- The natural `Specification` here would be something like `(-1050, -1000) || (1000, 1050)`
-- - the disjoint union of two open, non-overlapping, intervals. However, this doesn't work
-- because number type-specs only support a single interval. You could fudge it in all sorts of ways
-- by using `chooseSpec` or by using the can't set (which would blow up to be 2000 elements large in this
-- case). In short, there is no _satisfactory_ solution here.
data IntW (as :: [Type]) b where
  AddW :: NumLike a => IntW '[a, a] a
  MultW :: NumLike a => IntW '[a, a] a
  NegateW :: NumLike a => IntW '[a] a
  SignumW :: NumLike a => IntW '[a] a

deriving instance Eq (IntW dom rng)

instance Show (IntW d r) where
  show AddW = "+"
  show NegateW = "negate_"
  show MultW = "*"
  show SignumW = "signum_"

instance Semantics IntW where
  semantics AddW = (+)
  semantics NegateW = negate
  semantics MultW = (*)
  semantics SignumW = signum

instance Syntax IntW where
  isInfix AddW = True
  isInfix NegateW = False
  isInfix MultW = True
  isInfix SignumW = False

class HasDivision a where
  doDivide :: a -> a -> a
  default doDivide ::
    ( HasDivision (SimpleRep a)
    , GenericRequires a
    ) =>
    a ->
    a ->
    a
  doDivide a b = fromSimpleRep $ doDivide (toSimpleRep a) (toSimpleRep b)

  divideSpec :: a -> TypeSpec a -> Specification a
  default divideSpec ::
    ( HasDivision (SimpleRep a)
    , GenericRequires a
    ) =>
    a ->
    TypeSpec a ->
    Specification a
  divideSpec a ts = fromSimpleRepSpec $ divideSpec (toSimpleRep a) ts

instance {-# OVERLAPPABLE #-} (HasSpec a, MaybeBounded a, Integral a, TypeSpec a ~ NumSpec a) => HasDivision a where
  doDivide = div

  divideSpec 0 _ = TrueSpec
  divideSpec a (NumSpecInterval (unionWithMaybe max lowerBound -> ml) (unionWithMaybe min upperBound -> mu)) = typeSpec ts
    where
      ts
        | a > 0 = NumSpecInterval ml' mu'
        | otherwise = NumSpecInterval mu' ml'
      ml' = adjustLowerBound <$> ml
      mu' = adjustUpperBound <$> mu

      -- NOTE: negate has different overflow semantics than div, so that's why we use negate below...

      adjustLowerBound l
        | a == 1 = l
        | a == -1 = negate l
        | otherwise =
            let r = l `div` a
             in if toInteger r * toInteger a < toInteger l
                  then r + signum a
                  else r

      adjustUpperBound u
        | a == 1 = u
        | a == -1 = negate u
        | otherwise =
            let r = u `div` a
             in if toInteger r * toInteger a > toInteger u
                  then r - signum a
                  else r

instance HasDivision (Ratio Integer) where
  doDivide = (/)

  divideSpec 0 _ = TrueSpec
  divideSpec a (NumSpecInterval ml mu) = typeSpec ts
    where
      ts
        | a > 0 = NumSpecInterval ml' mu'
        | otherwise = NumSpecInterval mu' ml'
      ml' = adjustLowerBound <$> ml
      mu' = adjustUpperBound <$> mu
      adjustLowerBound l =
        let r = l / a
            l' = r * a
         in if l' < l
              then r + (l - l') * 2 / a
              else r

      adjustUpperBound u =
        let r = u / a
            u' = r * a
         in if u < u'
              then r - (u' - u) * 2 / a
              else r

instance HasDivision Float where
  doDivide = (/)

  divideSpec 0 _ = TrueSpec
  divideSpec a (NumSpecInterval ml mu) = typeSpec ts
    where
      ts
        | a > 0 = NumSpecInterval ml' mu'
        | otherwise = NumSpecInterval mu' ml'
      ml' = adjustLowerBound <$> ml
      mu' = adjustUpperBound <$> mu
      adjustLowerBound l =
        let r = l / a
            l' = r * a
         in if l' < l
              then r + (l - l') * 2 / a
              else r

      adjustUpperBound u =
        let r = u / a
            u' = r * a
         in if u < u'
              then r - (u' - u) * 2 / a
              else r

instance HasDivision Double where
  doDivide = (/)

  divideSpec 0 _ = TrueSpec
  divideSpec a (NumSpecInterval ml mu) = typeSpec ts
    where
      ts
        | a > 0 = NumSpecInterval ml' mu'
        | otherwise = NumSpecInterval mu' ml'
      ml' = adjustLowerBound <$> ml
      mu' = adjustUpperBound <$> mu
      adjustLowerBound l =
        let r = l / a
            l' = r * a
         in if l' < l
              then r + (l - l') * 2 / a
              else r

      adjustUpperBound u =
        let r = u / a
            u' = r * a
         in if u < u'
              then r - (u' - u) * 2 / a
              else r

-- | A type that we can reason numerically about in constraints
type Numeric a = (HasSpec a, Ord a, Num a, TypeSpec a ~ NumSpec a, MaybeBounded a, HasDivision a)

instance {-# OVERLAPPABLE #-} Numeric a => NumLike a where
  subtractSpec a ts@(NumSpecInterval ml mu)
    | Just u <- mu
    , a > 0
    , Nothing <- safeSubtract a u =
        ErrorSpec $
          NE.fromList
            [ "Underflow in subtractSpec (" ++ showType @a ++ "):"
            , "  a = " ++ show a
            , "  ts = " ++ show ts
            ]
    | Just l <- ml
    , a < 0
    , Nothing <- safeSubtract a l =
        ErrorSpec $
          NE.fromList
            [ "Overflow in subtractSpec (" ++ showType @a ++ "):"
            , "  a = " ++ show a
            , "  ts = " ++ show ts
            ]
    | otherwise = typeSpec $ NumSpecInterval (safeSub a <$> ml) (safeSub a <$> mu)
    where
      safeSub :: a -> a -> a
      safeSub a1 x
        | Just r <- safeSubtract a1 x = r
        | a1 < 0 = fromJust upperBound
        | otherwise = fromJust lowerBound

  negateSpec (NumSpecInterval ml mu) = typeSpec $ NumSpecInterval (negate <$> mu) (negate <$> ml)

  safeSubtract a x
    | a > 0
    , Just lb <- lowerBound
    , lb + a > x =
        Nothing
    | a < 0
    , Just ub <- upperBound
    , ub + a < x =
        Nothing
    | otherwise = Just $ x - a

instance NumLike a => Num (Term a) where
  (+) = (+.)
  negate = negate_
  fromInteger = Lit . fromInteger
  (*) = (*.)
  signum = signum_
  abs = error "No implementation for abs @(Term a)"

invertMult :: (HasSpec a, Num a, HasDivision a) => a -> a -> Maybe a
invertMult a b =
  let r = a `doDivide` b in if r * b == a then Just r else Nothing

-- | Just a note that these instances won't work until we are in a context where
--   there is a HasSpec instance of 'a', which (NumLike a) demands.
--   This happens in Constrained.Experiment.TheKnot
instance Logic IntW where
  propagateTypeSpec AddW (HOLE :<: i) ts cant = subtractSpec i ts <> notMemberSpec (mapMaybe (safeSubtract i) cant)
  propagateTypeSpec AddW ctx ts cant = propagateTypeSpec AddW (flipCtx ctx) ts cant
  propagateTypeSpec NegateW (Unary HOLE) ts cant = negateSpec ts <> notMemberSpec (map negate cant)
  propagateTypeSpec MultW (HOLE :<: 0) ts cant
    | 0 `conformsToSpec` TypeSpec ts cant = TrueSpec
    | otherwise = ErrorSpec $ NE.fromList ["zero"]
  propagateTypeSpec MultW (HOLE :<: i) ts cant = divideSpec i ts <> notMemberSpec (mapMaybe (flip invertMult i) cant)
  propagateTypeSpec MultW ctx ts cant = propagateTypeSpec MultW (flipCtx ctx) ts cant
  propagateTypeSpec SignumW (Unary HOLE) ts cant =
    constrained $ \x ->
      [x `satisfies` notMemberSpec [0] | not $ ok 0]
        ++ [Assert $ 0 <=. x | not $ ok (-1)]
        ++ [Assert $ x <=. 0 | not $ ok 1]
    where
      ok = flip conformsToSpec (TypeSpec ts cant)

  propagateMemberSpec AddW (HOLE :<: i) es =
    memberSpec
      (nubOrd $ mapMaybe (safeSubtract i) (NE.toList es))
      ( NE.fromList
          [ "propagateSpecFn on (" ++ show i ++ " +. HOLE)"
          , "The Spec is a MemberSpec = " ++ show es -- show (MemberSpec @HasSpec @TS es)
          , "We can't safely subtract " ++ show i ++ " from any choice in the MemberSpec."
          ]
      )
  propagateMemberSpec AddW ctx es = propagateMemberSpec AddW (flipCtx ctx) es
  propagateMemberSpec NegateW (Unary HOLE) es = MemberSpec $ NE.nub $ fmap negate es
  propagateMemberSpec MultW (HOLE :<: 0) es
    | 0 `elem` es = TrueSpec
    | otherwise = ErrorSpec $ NE.fromList ["zero"]
  propagateMemberSpec MultW (HOLE :<: i) es = memberSpec (mapMaybe (flip invertMult i) (NE.toList es)) (NE.fromList ["propagateSpec"])
  propagateMemberSpec MultW ctx es = propagateMemberSpec MultW (flipCtx ctx) es
  propagateMemberSpec SignumW (Unary HOLE) es
    | all ((`notElem` [-1, 0, 1]) . signum) es =
        ErrorSpec $ NE.fromList ["signum for invalid member spec", show es]
    | otherwise = constrained $ \x ->
        [x `satisfies` notMemberSpec [0] | 0 `notElem` es]
          ++ [Assert $ 0 <=. x | -1 `notElem` es]
          ++ [Assert $ x <=. 0 | 1 `notElem` es]

  rewriteRules AddW (x :> y :> Nil) _ | x == y = Just $ 2 * x
  rewriteRules _ _ _ = Nothing

infix 4 +.

-- | `Term`-level `(+)`
(+.) :: NumLike a => Term a -> Term a -> Term a
(+.) = appTerm AddW

infixl 7 *.

-- | `Term`-level `(+)`
(*.) :: NumLike a => Term a -> Term a -> Term a
(*.) = appTerm MultW

-- | `Term`-level `negate`
negate_ :: NumLike a => Term a -> Term a
negate_ = appTerm NegateW

-- | `Term`-level `signum`
signum_ :: NumLike a => Term a -> Term a
signum_ = appTerm SignumW

infix 4 -.

-- | `Term`-level `(-)`
(-.) :: Numeric n => Term n -> Term n -> Term n
(-.) x y = x +. negate_ y

infixr 4 <=.

-- | `Term`-level `(<=)`
(<=.) :: forall a. OrdLike a => Term a -> Term a -> Term Bool
(<=.) = appTerm LessOrEqualW

infixr 4 <.

-- | `Term`-level `(<)`
(<.) :: forall a. OrdLike a => Term a -> Term a -> Term Bool
(<.) = appTerm LessW

infixr 4 >=.

-- | `Term`-level `(>=)`
(>=.) :: forall a. OrdLike a => Term a -> Term a -> Term Bool
(>=.) = appTerm GreaterOrEqualW

infixr 4 >.

-- | `Term`-level `(>)`
(>.) :: forall a. OrdLike a => Term a -> Term a -> Term Bool
(>.) = appTerm GreaterW

-- | t`TypeSpec`-level `satisfies` to implement `toPreds` in
-- `HasSpec` instance
toPredsNumSpec ::
  OrdLike n =>
  Term n ->
  NumSpec n ->
  Pred
toPredsNumSpec v (NumSpecInterval ml mu) =
  fold $
    [Assert $ Lit l <=. v | l <- maybeToList ml]
      ++ [Assert $ v <=. Lit u | u <- maybeToList mu]

instance Logic OrdW where
  propagate f ctxt (ExplainSpec [] s) = propagate f ctxt s
  propagate f ctxt (ExplainSpec es s) = ExplainSpec es $ propagate f ctxt s
  propagate _ _ TrueSpec = TrueSpec
  propagate _ _ (ErrorSpec msgs) = ErrorSpec msgs
  propagate GreaterW (HOLE :? x :> Nil) spec =
    propagate LessW (x :! Unary HOLE) spec
  propagate GreaterW (x :! Unary HOLE) spec =
    propagate LessW (HOLE :? x :> Nil) spec
  propagate LessOrEqualW (HOLE :? Value x :> Nil) (SuspendedSpec v ps) =
    constrained $ \v' -> Let (App LessOrEqualW (v' :> Lit x :> Nil)) (v :-> ps)
  propagate LessOrEqualW (Value x :! Unary HOLE) (SuspendedSpec v ps) =
    constrained $ \v' -> Let (App LessOrEqualW (Lit x :> v' :> Nil)) (v :-> ps)
  propagate LessOrEqualW (HOLE :? Value l :> Nil) spec =
    caseBoolSpec spec $ \case True -> leqSpec l; False -> gtSpec l
  propagate LessOrEqualW (Value l :! Unary HOLE) spec =
    caseBoolSpec spec $ \case True -> geqSpec l; False -> ltSpec l
  propagate GreaterOrEqualW (HOLE :? Value x :> Nil) spec =
    propagate LessOrEqualW (Value x :! Unary HOLE) spec
  propagate GreaterOrEqualW (x :! Unary HOLE) spec =
    propagate LessOrEqualW (HOLE :? x :> Nil) spec
  propagate LessW (HOLE :? Value x :> Nil) (SuspendedSpec v ps) =
    constrained $ \v' -> Let (App LessW (v' :> Lit x :> Nil)) (v :-> ps)
  propagate LessW (Value x :! Unary HOLE) (SuspendedSpec v ps) =
    constrained $ \v' -> Let (App LessW (Lit x :> v' :> Nil)) (v :-> ps)
  propagate LessW (HOLE :? Value l :> Nil) spec =
    caseBoolSpec spec $ \case True -> ltSpec l; False -> geqSpec l
  propagate LessW (Value l :! Unary HOLE) spec =
    caseBoolSpec spec $ \case True -> gtSpec l; False -> leqSpec l

-- | @if-then-else@ on a specification, useful for writing `propagate` implementations
-- of predicates, e.g.:
-- > propagate LessW (Value l :! Unary HOLE) spec =
-- >   caseBoolSpec spec $ \case True -> gtSpec l; False -> leqSpec l
caseBoolSpec ::
  HasSpec a => Specification Bool -> (Bool -> Specification a) -> Specification a
caseBoolSpec spec cont = case possibleValues spec of
  [] -> ErrorSpec (NE.fromList ["No possible values in caseBoolSpec"])
  [b] -> cont b
  _ -> mempty
  where
    -- This will always get the same result, and probably faster since running 2
    -- conformsToSpec on True and False takes less time than simplifying the spec.
    -- Since we are in TheKnot, we could keep the simplifySpec. Is there a good reason to?
    possibleValues s = filter (flip conformsToSpec s) [True, False]

------------------------------------------------------------------------
-- Instances of HasSpec for numeric types
------------------------------------------------------------------------

instance HasSpec Integer where
  type TypeSpec Integer = NumSpec Integer
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Int where
  type TypeSpec Int = NumSpec Int
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec (Ratio Integer) where
  type TypeSpec (Ratio Integer) = NumSpec (Ratio Integer)
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec _ = TrueSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Natural where
  type TypeSpec Natural = NumSpec Natural
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec (NumSpecInterval (fromMaybe 0 -> lo) (Just hi)) =
    if lo < hi
      then equalSpec (fromIntegral $ hi - lo + 1)
      else equalSpec 0
  cardinalTypeSpec _ = TrueSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Word8 where
  type TypeSpec Word8 = NumSpec Word8
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  cardinalTrueSpec = equalSpec 256
  typeSpecOpt = notInNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Word16 where
  type TypeSpec Word16 = NumSpec Word16
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  cardinalTrueSpec = equalSpec 65536
  guardTypeSpec = guardNumSpec

instance HasSpec Word32 where
  type TypeSpec Word32 = NumSpec Word32
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Word64 where
  type TypeSpec Word64 = NumSpec Word64
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Int8 where
  type TypeSpec Int8 = NumSpec Int8
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTrueSpec = equalSpec 256
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Int16 where
  type TypeSpec Int16 = NumSpec Int16
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  cardinalTrueSpec = equalSpec 65536
  guardTypeSpec = guardNumSpec

instance HasSpec Int32 where
  type TypeSpec Int32 = NumSpec Int32
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Int64 where
  type TypeSpec Int64 = NumSpec Int64
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec = cardinalNumSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Float where
  type TypeSpec Float = NumSpec Float
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec _ = TrueSpec
  guardTypeSpec = guardNumSpec

instance HasSpec Double where
  type TypeSpec Double = NumSpec Double
  emptySpec = emptyNumSpec
  combineSpec = combineNumSpec
  genFromTypeSpec = genFromNumSpec
  shrinkWithTypeSpec = shrinkWithNumSpec
  fixupWithTypeSpec = fixupWithNumSpec
  conformsTo = conformsToNumSpec
  toPreds = toPredsNumSpec
  cardinalTypeSpec _ = TrueSpec
  guardTypeSpec = guardNumSpec

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