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) quot#(s(U), s(V), W) => quot#(U, V, W)
      (2) quot#(P, 0, s(X1)) => quot#(P, s(X1), s(X1))
      (3) map#(G1, cons(V1, W1)) => map#(G1, W1)
      (4) filter#(F2, cons(Y2, U2)) => filter2#(F2(Y2), F2, Y2, U2)
      (5) filter2#(true, H2, W2, P2) => filter#(H2, P2)
      (6) filter2#(false, F3, Y3, U3) => filter#(F3, U3)

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

Considering the 3 SCCs, this DP problem is split into the following new problems.

  P2. (1) quot#(s(U), s(V), W) => quot#(U, V, W)
      (2) quot#(P, 0, s(X1)) => quot#(P, s(X1), s(X1))

  P3. (1) map#(G1, cons(V1, W1)) => map#(G1, W1)

  P4. (1) filter#(F2, cons(Y2, U2)) => filter2#(F2(Y2), F2, Y2, U2)
      (2) filter2#(true, H2, W2, P2) => filter#(H2, P2)
      (3) filter2#(false, F3, Y3, U3) => filter#(F3, U3)

***** We apply the Subterm Criterion Processor on P2.

We use the following projection function:

  nu(quot#) = 1

We thus have:

  (1) s(U) |>|  U
  (2) P    |>=| P

We may remove the strictly oriented DPs, which yields:

  P5. (1) quot#(P, 0, s(X1)) => quot#(P, s(X1), s(X1))

***** We apply the Subterm Criterion Processor on P3.

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(V1, W1) |>| W1

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(filter2#) = 4

We thus have:

  (1) cons(Y2, U2) |>|  U2
  (2) P2           |>=| P2
  (3) U3           |>=| U3

We may remove the strictly oriented DPs, which yields:

  P6. (1) filter2#(true, H2, W2, P2) => filter#(H2, P2)
      (2) filter2#(false, F3, Y3, U3) => filter#(F3, U3)

***** We apply the Graph Processor on P5.

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

***** We apply the Graph Processor on P6.

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