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

  Signature: sum :: Int -> Int

  Rules: sum(x) -> 0 | x >= 0
         sum(x) -> x + sum(x + 1) | x < 0

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) => sum#(x + 1) | x < 0

***** We apply the Integer Function Processor on D1 = (P1, R, i, c).

We use the following integer mapping:

  J(sum#) = 0 - arg_1 - 1

We thus have:

  (1) x < 0 |= 0 - x - 1 > 0 - (x + 1) - 1 (and 0 - x - 1 >= 0)

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.