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_1#(x, y) => eval_2#(x, y) | x > 0
      (2) eval_2#(x, y) => eval_2#(x, y - 1) | x > 0 /\ y > 0
      (3) eval_2#(x, y) => eval_1#(x - 1, y) | x > 0 /\ 0 >= y

***** We apply the Integer Function Processor on P1.

We use the following integer mapping:

  J(eval_1#) = arg_1
  J(eval_2#) = arg_1

We thus have:

  (1) x > 0           |= x >= x
  (2) x > 0 /\ y > 0  |= x >= x
  (3) x > 0 /\ 0 >= y |= x >  x - 1 (and x >= 0)

We may remove the strictly oriented DPs, which yields:

  P2. (1) eval_1#(x, y) => eval_2#(x, y) | x > 0
      (2) eval_2#(x, y) => eval_2#(x, y - 1) | x > 0 /\ y > 0

***** We apply the Graph Processor on P2.

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

  P3. (1) eval_2#(x, y) => eval_2#(x, y - 1) | x > 0 /\ y > 0

***** We apply the Integer Function Processor on P3.

We use the following integer mapping:

  J(eval_2#) = arg_2 - 1

We thus have:

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

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