-- necessary funcs
chAdd :: ChurchNumber -> ChurchNumber -> ChurchNumber
chAdd a (Succ b) = chAdd (Succ a) b
chAdd a churchZero = a

chSucc :: ChurchNumber -> ChurchNumber
chSucc = Succ

-- for testing purposes
churchZero :: ChurchNumber
churchZero = Zero

churchOne :: ChurchNumber
churchOne = chSucc churchZero

churchTwo :: ChurchNumber
churchTwo = chSucc churchOne

churchThree :: ChurchNumber
churchThree = chSucc churchTwo

churchIntZero :: (ChurchNumber, ChurchNumber)
churchIntZero = (churchZero, churchZero)

churchIntOne :: (ChurchNumber, ChurchNumber)
churchIntOne = (churchOne, Zero)

churchIntOneNeg :: (ChurchNumber, ChurchNumber)
churchIntOneNeg = churchIntNegation churchIntOne

churchIntTwo :: (ChurchNumber, ChurchNumber)
churchIntTwo = churchIntCalcSum churchIntOne churchIntOne

churchIntTwoNeg :: (ChurchNumber, ChurchNumber)
churchIntTwoNeg = churchIntCalcSum churchIntOneNeg churchIntOneNeg

churchIntThree :: (ChurchNumber, ChurchNumber)
churchIntThree = churchIntCalcSum churchIntTwo churchIntOne

churchIntThreeNeg :: (ChurchNumber, ChurchNumber)
churchIntThreeNeg = churchIntCalcSum churchIntTwoNeg churchIntOneNeg

(+&) :: ChurchNumber -> ChurchNumber -> ChurchNumber
(+&) a (Succ b) = (+&) (Succ a) b
(+&) a Zero = a

(*&) :: ChurchNumber -> ChurchNumber -> ChurchNumber
(*&) a (Succ b) = (+&) a (a *& b)
(*&) a Zero = Zero

chMult :: ChurchNumber -> ChurchNumber -> ChurchNumber
chMult a (Succ b) = chAdd a (a `chMult` b)
chMult a Zero = Zero

churchNullify :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchNullify (Zero, Zero) = (Zero, Zero)
churchNullify (pos, Zero) = (pos, Zero)
churchNullify (Zero, neg) = (Zero, neg)
churchNullify (Succ pos, Succ neg) = churchNullify (pos, neg)

-- (pos1 + neg2), (pos2 + neg1)
churchIntCalcDifference :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchIntCalcDifference (pos1, neg1) (pos2, neg2) = churchNullify (pos1 +& neg2, neg1 +& pos2)

-- (pos1 + neg2), (pos2 + neg1)
churchIntCalcSum :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchIntCalcSum (pos1, neg1) (pos2, neg2) = churchNullify (pos1 +& pos2, neg1 +& neg2)

churchIntCalcProd :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchIntCalcProd chN1 chN2 = helper (churchNullify chN1) (churchNullify chN2)
  where
    -- helper chN1 chN2 = chN2
    helper (_, _) (Zero, Zero) = (Zero, Zero)
    helper (Zero, Zero) (_, _)  = (Zero, Zero)
    helper (Zero, b) (c, Zero) = (Zero, b *& c)
    helper (a, Zero) (c, Zero) = (a *& c, Zero)
    helper (a, Zero) (Zero, d) = (Zero, a *& d)
    helper (Zero, b) (Zero, d) = (b *& d, Zero)

churchIntAbs :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchIntAbs chN = helper (churchNullify chN)
  where
    helper (Zero, Zero) = (Zero, Zero)
    helper (a, Zero) = (a, Zero)
    helper (Zero, a) = (a, Zero)

churchIntSignum :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchIntSignum chN = helper (churchNullify chN)
  where
    helper (Zero, Zero) = (Zero, Zero)
    helper (a, Zero) = (Succ Zero, Zero)
    helper (Zero, a) = (Zero, Succ Zero)

fromInteger' :: Integer -> (ChurchNumber, ChurchNumber)
fromInteger' int = helper int (Zero, Zero)
  where
    helper 0 (a, b) = (a, b)
    helper int (a, b) = helper (int - 1) (Succ a, Zero)

churchIntNegation :: (ChurchNumber, ChurchNumber) -> (ChurchNumber, ChurchNumber)
churchIntNegation (pos1, neg1) = churchNullify (neg1, pos1)

churchIntIsPositive :: (ChurchNumber, ChurchNumber) -> Bool
churchIntIsPositive chN = helper (churchNullify chN)
  where
    helper (Zero, Zero) = False
    helper (_, Zero) = True
    helper (Zero, _) = False

churchIntIsZero :: (ChurchNumber, ChurchNumber) -> Bool
churchIntIsZero (Zero, Zero) = True
churchIntIsZero (_, _) = False