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

  Signature: if  :: Bool -> Int -> Int -> Int
             pow :: Int -> Int -> Int

  Rules: pow(x, y) -> if(y > 0, x, y) | x = x /\ y = y
         if(true, x, y) -> x * pow(x, y - 1) | x = x /\ y = y
         if(false, x, 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) pow#(x, y) => if#(y > 0, x, y) | x = x /\ y = y
      (2) if#(true, x, y) => pow#(x, y - 1) | 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 3 rules in the input problem).

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

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