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) f#(true, x) => ack#(10, 10)
      (2) f#(true, x) => f#(ack(10, 10) >= x, x + 1)
      (3) ack#(x, y) => ackNat#(x >= 0 /\ y >= 0, x, y)
      (4) ackNat#(true, x, y) => cond1#(x = 0, x, y)
      (5) cond1#(false, x, y) => cond2#(y = 0, x, y)
      (6) cond2#(true, x, y) => ack#(x - 1, y + 1)
      (7) cond2#(false, x, y) => ack#(x, y - 1)
      (8) cond2#(false, x, y) => ack#(x - 1, ack(x, y - 1))

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

There is only one SCC, so all DPs not inside the SCC can be removed:

  P2. (1) ack#(x, y) => ackNat#(x >= 0 /\ y >= 0, x, y)
      (2) ackNat#(true, x, y) => cond1#(x = 0, x, y)
      (3) cond1#(false, x, y) => cond2#(y = 0, x, y)
      (4) cond2#(true, x, y) => ack#(x - 1, y + 1)
      (5) cond2#(false, x, y) => ack#(x, y - 1)
      (6) cond2#(false, x, y) => ack#(x - 1, ack(x, y - 1))

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

We use the following theory arguments function:

  ack#    : []
  ackNat# : [1, 2, 3]
  cond1#  : [1, 2, 3]
  cond2#  : [1, 2, 3]

This yields the following new DP problems:

  P3. (1) ack#(x, y) => ackNat#(x >= 0 /\ y >= 0, x, y)
      (2) ackNat#(true, x, y) => cond1#(x = 0, x, y) { x, y }
      (3) cond1#(false, x, y) => cond2#(y = 0, x, y) { x, y }
      (4) cond2#(true, x, y) => ack#(x - 1, y + 1) { x, y }
      (5) cond2#(false, x, y) => ack#(x, y - 1) { x, y }
      (6) cond2#(false, x, y) => ack#(x - 1, ack(x, y - 1)) { x, y }

  P4. (1) ackNat#(true, x, y) => cond1#(x = 0, x, y)
      (2) cond1#(false, x, y) => cond2#(y = 0, x, y)
      (3) cond2#(true, x, y) => ack#(x - 1, y + 1)
      (4) cond2#(false, x, y) => ack#(x, y - 1)
      (5) cond2#(false, x, y) => ack#(x - 1, ack(x, y - 1))

***** No progress could be made on DP problem P3.