We consider termination of the LCTRS with only rule scheme Calc:

  Signature: eval :: Int -> Int -> Int -> o

  Rules: eval(x, y, z) -> eval(x + 1, y + x, z) | x >= y /\ z = z
         eval(x, y, z) -> eval(x - z, y + z * z, z - 1) | x >= y /\ z = z

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, i, c), where:

  P1. (1) eval#(x, y, z) => eval#(x + 1, y + x, z) | x >= y /\ z = z
      (2) eval#(x, y, z) => eval#(x - z, y + z * z, z - 1) | x >= y /\ z = z

***** We apply the Usable Rules Processor on D1 = (P1, 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).