```-- |Support for natural numbers.
module Data.Natural (

Natural,
monus,
View,
view,
fold

) where

import Data.Monoid

{-FIXME:
Maybe, we should have more rewrite rules. See

Should we also include an alternative of plus/minus with swapped order of (+)-arguments?
-}
{-# RULES
"pred/succ"             forall nat       . pred (succ nat)             = nat
"plus/minus"            forall nat1 nat2 . (nat1 + nat2) - nat2        = nat1
"plus/monus"            forall nat1 nat2 . (nat1 + nat2) `monus` nat2  = nat1
"signum/succ"           forall nat       . signum (succ nat)           = Natural 1
"fromInteger/toInteger" forall nat       . fromInteger (toInteger nat) = nat
"view/succ"             forall nat       . view (succ nat)             = Succ (uncheckedPred nat)
#-}

-- An unchecked version of pred, which is only used internally for the rewrite rule view/succ.
uncheckedPred :: Natural -> Natural
uncheckedPred (Natural integer) = Natural (pred integer)

{-|
The type of natural numbers.

Note that matching a natural number against a negative pattern might not work as you expect.
For example, evaluating the following expression results in a run-time error, instead of the
result @\"plus five\"@:

@
case 5 :: Natural of
-5 -> \"minus five\"
5 -> \"plus five\"
@

The reason is that the @==@ operator of @Natural@ is used for checking if the patterns
match, making it necessary to convert @-5@ to Natural.
-}
newtype Natural = Natural Integer deriving (Eq, Ord)

{-
We do not just provide a minimal complete definition. The reason is that all default
implementations use toEnum and fromEnum internally. Unfortunately, these use Int instead of
Integer, which can lead to overflows. Furthermore, we can avoid the negativity check in some
methods.
-}
instance Enum Natural where

succ (Natural integer) = Natural (succ integer)

pred = fromInteger . pred . toInteger

toEnum = fromInteger . toEnum

enumFrom (Natural start) = map Natural (enumFrom start)

enumFromThen (Natural start) (Natural next) = map Natural \$
if start > next
then enumFromThenTo start next 0
else enumFromThen start next

enumFromTo (Natural start) (Natural end) = map Natural (enumFromTo start end)

enumFromThenTo (Natural start) (Natural next) (Natural end) = map Natural \$
enumFromThenTo start next end

instance Show Natural where

showsPrec prec (Natural integer) = showsPrec prec integer

-- This is Control.Arrow.first, specialized to (->).
first :: (val -> val') -> (val,other) -> (val',other)
first fun (val,other) = (fun val,other)

instance Num Natural where

Natural integer1 + Natural integer2 = Natural (integer1 + integer2)

Natural integer1 - Natural integer2 = fromInteger (integer1 - integer2)

Natural integer1 * Natural integer2 = Natural (integer1 * integer2)

abs = id

signum (Natural integer) = Natural (signum integer)

fromInteger integer | integer >= 0 = Natural integer
| otherwise    = error "Data.Natural: natural cannot be negative"

instance Real Natural where

toRational = toRational . toInteger

instance Integral Natural where

quotRem (Natural integer1) (Natural integer2) = let

(quot,rem) = quotRem integer1 integer2

in (Natural quot,Natural rem)

{-
Although an implementation of divMod is generally not needed for a minimal complete
definition, we have to include one, since the default implementation relies on negative
numbers being available.
-}
divMod = quotRem

toInteger (Natural integer) = integer

{-|
Yields the monus of two natural numbers, which is their difference if the first number is
greater than the second, and zero otherwise.
-}
monus :: Natural -> Natural -> Natural
monus (Natural integer1) (Natural integer2) | integer1 > integer2 = Natural (integer1 - integer2)
| otherwise           = Natural 0

{-|
A data type for views of natural numbers.
-}
data View = Zero | Succ Natural

{-|
Yields the view of a natural number.
-}
view :: Natural -> View
view (Natural 0)       = Zero
view (Natural integer) = Succ (Natural (pred integer))

{-|
Folding of natural numbers.
-}
fold :: accu -> (accu -> accu) -> Natural -> accu
fold zeroAccu _        (Natural 0)       = zeroAccu
fold zeroAccu succAccu (Natural integer) = succAccu \$
fold zeroAccu succAccu (Natural (pred integer))
```