The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the following initial DP problem:

  P1. (1) eval_0#(x, y, z) => eval_1#(x, y, z) | y > 0
      (2) eval_1#(x, y, z) => eval_1#(x + y, y, z) | y > x /\ z > y /\ y > 0
      (3) eval_1#(x, y, z) => eval_1#(x, y, x - y) | y > x /\ z > y /\ y > 0

***** We apply the Graph Processor on P1.

There is only one SCC, so all DPs not inside the SCC can be removed:

  P2. (1) eval_1#(x, y, z) => eval_1#(x + y, y, z) | y > x /\ z > y /\ y > 0

***** We apply the Integer Function Processor on P2.

We use the following integer mapping:

  J(eval_1#) = arg_2 - arg_1 - 1

We thus have:

  (1) y > x /\ z > y /\ y > 0 |= y - x - 1 > y - (x + y) - 1 (and y - x - 1 >= 0)

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