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) random#(x) => rand#(x, 0)
      (2) rand#(x, y) => id_inc#(y) | x > 0
      (3) rand#(x, y) => rand#(x - 1, id_inc(y)) | x > 0
      (4) rand#(x, y) => id_dec#(y) | 0 > x
      (5) rand#(x, y) => rand#(x + 1, id_dec(y)) | 0 > x

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

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

  P2. (1) rand#(x, y) => rand#(x - 1, id_inc(y)) | x > 0

  P3. (1) rand#(x, y) => rand#(x + 1, id_dec(y)) | 0 > x

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

We use the following integer mapping:

  J(rand#) = arg_1

We thus have:

  (1) x > 0 |= x > x - 1 (and x >= 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(rand#) = 0 - arg_1 - 1

We thus have:

  (1) 0 > x |= 0 - x - 1 > 0 - (x + 1) - 1 (and 0 - x - 1 >= 0)

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