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

  Signature: sif :: Bool -> Int -> Int -> Int
             sum :: Int -> Int -> Int

  Rules: sum(x, y) -> sif(x >= y, x, y) | x = x /\ y = y
         sif(true, x, y) -> y + sum(x, y + 1) | x = x /\ y = y
         sif(false, x, y) -> 0 | 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, i, c), where:

  P1. (1) sum#(x, y) => sif#(x >= y, x, y) | x = x /\ y = y
      (2) sif#(true, x, y) => sum#(x, y + 1) | x = x /\ y = y

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