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

  Signature: eval_1 :: Int -> Int -> Int -> o
             eval_2 :: Int -> Int -> Int -> o

  Rules: eval_1(x, y, z) -> eval_2(x, y, z) | x > y /\ z = z
         eval_2(x, y, z) -> eval_1(x, y + 1, z) | x > z /\ y = y
         eval_2(x, y, z) -> eval_1(x, y, z + 1) | x > z /\ y = y
         eval_2(x, y, z) -> eval_1(x - 1, y, z) | z >= 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) eval_1#(x, y, z) => eval_2#(x, y, z) | x > y /\ z = z
      (2) eval_2#(x, y, z) => eval_1#(x, y + 1, z) | x > z /\ y = y
      (3) eval_2#(x, y, z) => eval_1#(x, y, z + 1) | x > z /\ y = y
      (4) eval_2#(x, y, z) => eval_1#(x - 1, y, z) | z >= x /\ y = y

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

We obtain 0 usable rules (out of 4 rules in the input problem).

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

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