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, z) | x >= 0
      (2) eval#(x, y) => eval#(x, y - 1) | y >= 0

***** We apply the Theory Arguments Processor on P1.

We use the following theory arguments function:

  eval# : [1]

This yields the following new DP problems:

  P2. (1) eval#(x, y) => eval#(x - 1, z) | x >= 0
      (2) eval#(x, y) => eval#(x, y - 1) | y >= 0 { x, y }

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

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

We use the following integer mapping:

  J(eval#) = arg_1

We thus have:

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

We may remove the strictly oriented DPs, which yields:

  P4. (1) eval#(x, y) => eval#(x, y - 1) | y >= 0 { x, y }

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

We use the following integer mapping:

  J(eval#) = arg_2 + 1

We thus have:

  (1) 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.

***** We apply the Integer Function Processor on P4.

We use the following integer mapping:

  J(eval#) = arg_2 + 1

We thus have:

  (1) 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.