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) div#(x, y) => divNat#(x >= 0 /\ y >= 1, x, y)
      (2) divNat#(true, x, y) => d#(x, y, 0)
      (3) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
      (4) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z)
      (5) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z)

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

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

  P2. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
      (2) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z)
      (3) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z)

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

We use the following theory arguments function:

  cond# : [1, 2, 3, 4]
  d#    : [1, 2, 3]
  dNat# : [1, 2, 3, 4]

This yields the following new DP problems:

  P3. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
      (2) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z) { x, y, z }
      (3) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z) { x, y, z }

  P4. (1) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z)
      (2) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z)

***** We apply the Theory Arguments Processor on P3.

We use the following theory arguments function:

  cond# : [1, 2, 3, 4]
  d#    : [1, 2, 3]
  dNat# : [1, 2, 3, 4]

This yields the following new DP problems:

  P5. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z) { x, y, z }
      (2) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z) { x, y, z }
      (3) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z) { x, y, z }

  P6. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)

***** We apply the Graph Processor on P4.

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

***** No progress could be made on DP problem P5.