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, y) => fNat#(x >= 0 /\ y >= 0, x, y)
      (2) fNat#(true, x, y) => trunc#(x)
      (3) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1)
      (4) trunc#(x) => if#(x % 2 = 0, x, x - 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) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)
      (2) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1)

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

We use the following theory arguments function:

  f#    : [3]
  fNat# : [1, 2, 3]

This yields the following new DP problems:

  P3. (1) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)
      (2) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1) { x, y }

  P4. (1) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1)

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

We use the following theory arguments function:

  f#    : [3]
  fNat# : [1, 2, 3]

This yields the following new DP problems:

  P5. (1) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y) { y }
      (2) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1) { x, y }

  P6. (1) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)

***** 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.