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# Solution to Assignment 1 (by Hongwei Xi)
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# please note that only tail-recursion is used in the solution.

plus =
  fix f. lam x. lam y.
    case x of Z => y | S x => f (x) (S y)

mult =
  let val aux =
     fix f. lam x. lam y. lam res.
       case x of Z => res | S x => f (x) (y) (plus y res)
  in
     lam x. lam y. aux x y Z

exp =
  let val aux =
     fix f. lam x. lam y. lam res.
       case x of Z => res | S x => f (x) (y) (mult y res)
  in
     lam x. lam y. aux x y (S Z)

# 'subgeq x y' returns the pair '(x - y, x >= y)'
subgeq =
  fix f. lam x. lam y.
    case y of Z => (x, S Z) | S y => (case x of Z => (Z, Z) | S x => f x y)

quorem =
  let val aux
    fix f. lam x. lam y. lam res
      let val p = subgeq x y
      in
         case (snd p) of Z => (res, x) | S u => f (fst p) y (S res)
  in
     lam x. lam y. aux x y Z