We consider termination of the STRS with no additional rule schemes:

  Signature: 0         :: a
             cons      :: a -> c -> c
             false     :: b
             filter    :: (a -> b) -> c -> c
             filtersub :: b -> (a -> b) -> c -> c
             neq       :: a -> a -> b
             nil       :: c
             nonzero   :: c -> c
             s         :: a -> a
             true      :: b

  Rules: neq(0, 0) -> false
         neq(0, s(X)) -> true
         neq(s(Y), 0) -> true
         neq(s(U), s(V)) -> neq(U, V)
         filter(I, nil) -> nil
         filter(J, cons(X1, Y1)) -> filtersub(J(X1), J, cons(X1, Y1))
         filtersub(true, G1, cons(V1, W1)) -> cons(V1, filter(G1, W1))
         filtersub(false, J1, cons(X2, Y2)) -> filter(J1, Y2)
         nonzero -> filter(neq(0))

The system is accessible function passing by a sort ordering that equates all sorts.

We start by computing the initial DP problem D1 = (P1, R, i, c), where:

  P1. (1) neq#(s(U), s(V)) => neq#(U, V)
      (2) filter#(J, cons(X1, Y1)) => filtersub#(J(X1), J, cons(X1, Y1))
      (3) filtersub#(true, G1, cons(V1, W1)) => filter#(G1, W1)
      (4) filtersub#(false, J1, cons(X2, Y2)) => filter#(J1, Y2)
      (5) nonzero#(arg1) => neq#(0, fresh1)
      (6) nonzero#(arg1) => filter#(neq(0), arg1)

***** We apply the Graph Processor on D1 = (P1, R, i, c).

We compute a graph approximation with the following edges:

  1: 1 
  2: 3 4 
  3: 2 
  4: 2 
  5:
  6: 2 

There are 2 SCCs.

Processor output: { D2 = (P2, R, i, c) ; D3 = (P3, R, i, c) }, where:

  P2. (1) neq#(s(U), s(V)) => neq#(U, V)

  P3. (1) filter#(J, cons(X1, Y1)) => filtersub#(J(X1), J, cons(X1, Y1))
      (2) filtersub#(true, G1, cons(V1, W1)) => filter#(G1, W1)
      (3) filtersub#(false, J1, cons(X2, Y2)) => filter#(J1, Y2)

***** We apply the Subterm Criterion Processor on D2 = (P2, R, i, c).

We use the following projection function:

  nu(neq#) = 1

We thus have:

  (1) s(U) |>| U

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

Processor output: { }.

***** We apply the Subterm Criterion Processor on D3 = (P3, R, i, c).

We use the following projection function:

  nu(filter#)    = 2
  nu(filtersub#) = 3

We thus have:

  (1) cons(X1, Y1) |>=| cons(X1, Y1)
  (2) cons(V1, W1) |>|  W1
  (3) cons(X2, Y2) |>|  Y2

We may remove the strictly oriented DPs.

Processor output: { D4 = (P4, R, i, c) }, where:

  P4. (1) filter#(J, cons(X1, Y1)) => filtersub#(J(X1), J, cons(X1, Y1))

***** We apply the Graph Processor on D4 = (P4, R, i, c).

We compute a graph approximation with the following edges:

  1:

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

Processor output: { }.