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

  Signature: f :: Int -> Int

  Rules: f(x) -> f(x - 1) | x = x

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) f#(x) => f#(x - 1) | x = x

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

We obtain 0 usable rules (out of 1 rules in the input problem).

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

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