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

  Signature: div :: Int -> Int -> Int
             if  :: Bool -> Int -> Int -> Int

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

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) => if#(x >= y /\ y > 0, x, y) | x = x /\ y = y
      (2) if#(true, x, y) => div#(x - y, y) | x = x /\ y = y

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

We obtain 0 usable rules (out of 2 rules in the input problem).

Processor output: { D2 = (P1, {}, i, c) }.

***** No progress could be made on DP problem D2 = (P1, {}, i, c).