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)  primes#(x) => nats#(2, x)
      (2)  primes#(x) => sieve#(nats(2, x))
      (3)  nats#(x, y) => nats#(x + 1, y) | y >= x
      (4)  sieve#(cons(x, ys)) => filter#(x, ys)
      (5)  sieve#(cons(x, ys)) => sieve#(filter(x, ys))
      (6)  filter#(x, cons(y, zs)) => isdiv#(x, y)
      (7)  filter#(x, cons(y, zs)) => if#(isdiv(x, y), x, y, zs)
      (8)  if#(true, x, y, zs) => filter#(x, zs)
      (9)  if#(false, x, y, zs) => filter#(x, zs)
      (10) isdiv#(x, y) => isdiv#(x, y - x) | y >= x /\ x > 0

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

Considering the 4 SCCs, this DP problem is split into the following new problems.

  P2. (1) nats#(x, y) => nats#(x + 1, y) | y >= x

  P3. (1) isdiv#(x, y) => isdiv#(x, y - x) | y >= x /\ x > 0

  P4. (1) filter#(x, cons(y, zs)) => if#(isdiv(x, y), x, y, zs)
      (2) if#(true, x, y, zs) => filter#(x, zs)
      (3) if#(false, x, y, zs) => filter#(x, zs)

  P5. (1) sieve#(cons(x, ys)) => sieve#(filter(x, ys))

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

We use the following integer mapping:

  J(nats#) = arg_2 - arg_1

We thus have:

  (1) y >= x |= y - x > y - (x + 1) (and y - 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(isdiv#) = arg_2

We thus have:

  (1) y >= x /\ x > 0 |= y > y - x (and y >= 0)

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

***** We apply the Subterm Criterion Processor on P4.

We use the following projection function:

  nu(filter#) = 2
  nu(if#)     = 4

We thus have:

  (1) cons(y, zs) |>|  zs
  (2) zs          |>=| zs
  (3) zs          |>=| zs

We may remove the strictly oriented DPs, which yields:

  P6. (1) if#(true, x, y, zs) => filter#(x, zs)
      (2) if#(false, x, y, zs) => filter#(x, zs)

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