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#(x, y) => eval#(x - 1, y) | x > 0
      (2) eval#(x, y) => eval#(x - 1, y) | y > 0 /\ x > 0
      (3) eval#(x, y) => eval#(x, y - 1) | x > 0 /\ 0 >= x /\ y > 0
      (4) eval#(x, y) => eval#(x, y - 1) | y > 0 /\ 0 >= x
      (5) eval#(x, y) => eval#(x, y) | x > 0 /\ 0 >= x /\ 0 >= y
      (6) eval#(x, y) => eval#(x, y) | y > 0 /\ 0 >= x /\ 0 >= y

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

Considering the 2 SCCs, this DP problem is split into the following new problems.

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

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

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

We use the following integer mapping:

  J(eval#) = arg_2

We thus have:

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

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

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

We use the following integer mapping:

  J(eval#) = arg_1 - 1

We thus have:

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

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