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) diff#(x, y) => cond1#(x = y, x, y)
      (2) cond1#(false, x, y) => cond2#(x > y, x, y)
      (3) cond2#(true, x, y) => diff#(x, y + 1)
      (4) cond2#(false, x, y) => diff#(x + 1, y)

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

We use the following theory arguments function:

  cond1# : [1, 2, 3]
  cond2# : [1, 2, 3]
  diff#  : [1, 2]

This yields the following new DP problems:

  P2. (1) diff#(x, y) => cond1#(x = y, x, y)
      (2) cond1#(false, x, y) => cond2#(x > y, x, y) { x, y }
      (3) cond2#(true, x, y) => diff#(x, y + 1) { x, y }
      (4) cond2#(false, x, y) => diff#(x + 1, y) { x, y }

  P3. (1) cond1#(false, x, y) => cond2#(x > y, x, y)
      (2) cond2#(true, x, y) => diff#(x, y + 1)
      (3) cond2#(false, x, y) => diff#(x + 1, y)

***** We apply the Theory Arguments Processor on P2.

We use the following theory arguments function:

  cond1# : [1, 2, 3]
  cond2# : [1, 2, 3]
  diff#  : [1, 2]

This yields the following new DP problems:

  P4. (1) diff#(x, y) => cond1#(x = y, x, y) { x, y }
      (2) cond1#(false, x, y) => cond2#(x > y, x, y) { x, y }
      (3) cond2#(true, x, y) => diff#(x, y + 1) { x, y }
      (4) cond2#(false, x, y) => diff#(x + 1, y) { x, y }

  P5. (1) diff#(x, y) => cond1#(x = y, x, y)

***** We apply the Graph Processor on P3.

As there are no SCCs, this DP problem is removed.

***** No progress could be made on DP problem P4.