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#(i, j) => eval_2#(i, 0) | i >= 0
      (2) eval_2#(i, j) => eval_2#(i, j + 1) | j <= i - 1
      (3) eval_2#(i, j) => eval_1#(i - 1, j) | j > i - 1

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

We use the following integer mapping:

  J(eval_1#) = arg_1 + 1
  J(eval_2#) = arg_1

We thus have:

  (1) i >= 0     |= i + 1 >  i         (and i + 1 >= 0)
  (2) j <= i - 1 |= i     >= i
  (3) j > i - 1  |= i     >= i - 1 + 1

We may remove the strictly oriented DPs, which yields:

  P2. (1) eval_2#(i, j) => eval_2#(i, j + 1) | j <= i - 1
      (2) eval_2#(i, j) => eval_1#(i - 1, j) | j > i - 1

***** 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#(i, j) => eval_2#(i, j + 1) | j <= i - 1

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

We use the following integer mapping:

  J(eval_2#) = arg_1 - 1 - arg_2

We thus have:

  (1) j <= i - 1 |= i - 1 - j > i - 1 - (j + 1) (and i - 1 - j >= 0)

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