The Uniqueness of the Natural Numbers

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{-# LANGUAGE NoImplicitPrelude, BangPatterns #-}
module NaturalNumbers
  ( Natural(..), Unary()
  , isZero, prev, naturalRec

  , _test_nats, main_nats
  ) where

import Testing
import Functions
import Flip
import Clone
import Composition
import Booleans
import Tuples
import DisjointUnions
import Unary

class Natural n where
  zero :: n

  next :: n -> n

  unnext :: n -> Union () n
  
  natural :: Integer -> n

isZero :: (Natural n) => n -> Bool
isZero = (either (const true) (const false)) . unnext

prev :: (Natural n) => n -> n
prev = (either (const zero) id) . unnext

naturalRec :: (Natural n)
  => a -> (a -> a) -> n -> a
naturalRec e phi n =
  let
    tau !x k = case unnext k of
      Left () -> x
      Right m -> tau (phi x) m
  in tau e n

instance Natural Unary where
  zero = Z

  next = N

  unnext k = case k of
    Z   -> Left ()
    N m -> Right m

  natural = mkUnary

_test_unnext_zero :: (Natural n, Equal n)
  => n -> Test Bool
_test_unnext_zero m =
  testName "unnext(0) == lft(*)" $
  eq
    (unnext (zero `withTypeOf` m))
    (lft ())


_test_unnext_next :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_unnext_next _ =
  testName "unnext(next(m)) == rgt(m)" $
  \m -> eq (unnext (next m)) (rgt m)


_test_prev_zero :: (Natural n, Equal n)
  => n -> Test Bool
_test_prev_zero m =
  testName "prev(0) == 0" $
  eq
    (prev (zero `withTypeOf` m))
    zero


_test_prev_next :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_prev_next _ =
  testName "prev(next(m)) == m" $
  \m -> eq (prev (next m)) m


_test_isZero_zero :: (Natural n, Equal n)
  => n -> Test Bool
_test_isZero_zero m =
  testName "isZero(0) == true" $
  eq
    (isZero (zero `withTypeOf` m))
    true


_test_isZero_next :: (Natural n, Equal n)
  => n -> Test (n -> Bool)
_test_isZero_next _ =
  testName "isZero(next(m)) == false" $
  \m -> eq (isZero (next m)) false

_test_nats :: (TypeName n, Natural n, Show n, Arbitrary n, Equal n)
  => Int -> Int -> n -> IO ()
_test_nats size cases n = do
  testLabel1 "nats" n
  let args = testArgs size cases

  runTest args (_test_unnext_zero n)
  runTest args (_test_unnext_next n)
  runTest args (_test_prev_zero n)
  runTest args (_test_prev_next n)
  runTest args (_test_isZero_zero n)
  runTest args (_test_isZero_next n)

main_nats :: IO ()
main_nats = do
  _test_nats 100 100 (zero :: Unary)

  let a = zero :: Unary
  let b = true :: Bool
  let c = tup zero zero :: Pair Unary Unary

  _test_functions      20 200 a a a a
  _test_functions      20 200 a b a b
  _test_flip           20 200 a a a a a a a
  _test_flip           20 200 a b c a b c a
  _test_clone          20 200 a a
  _test_clone          20 200 a c
  _test_compose        20 200 a a a a a a a
  _test_compose        20 200 a b c a b c a
  _test_tuple          20 200 a a a
  _test_tuple          20 200 a b c
  _test_disjoint_union 20 200 a a a
  _test_disjoint_union 20 200 a b c