We consider universal computability of the LCTRS with only rule scheme Calc:

  Signature: div :: Int -> Int -> Int

  Rules: div(x, y) -> 0 | 0 >= y /\ x = x
         div(x, y) -> 0 | y >= x
         div(x, y) -> div(x - y, y) + 1 | x > y /\ y > 0

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R UNION R_?, i, c), where:

  P1. (1) div#(x, y) => div#(x - y, y) | x > y /\ y > 0

***** We apply the Integer Function Processor on D1 = (P1, R UNION R_?, i, c).

We use the following integer mapping:

  J(div#) = arg_1 - 1

We thus have:

  (1) x > y /\ y > 0 |= x - 1 > x - y - 1 (and x - 1 >= 0)

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.