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

  Signature: *       :: a -> a -> a
             1       :: a
             cons    :: c -> d -> d
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             i       :: a -> a
             map     :: (c -> c) -> d -> d
             nil     :: d
             true    :: b

  Rules: *(1, X) -> X
         *(Y, 1) -> Y
         *(i(U), U) -> 1
         *(V, i(V)) -> 1
         *(i(W), *(W, P)) -> P
         *(X1, *(i(X1), Y1)) -> Y1
         i(1) -> 1
         i(i(U1)) -> U1
         map(H1, nil) -> nil
         map(I1, cons(P1, X2)) -> cons(I1(P1), map(I1, X2))
         filter(Z2, nil) -> nil
         filter(G2, cons(V2, W2)) -> filter2(G2(V2), G2, V2, W2)
         filter2(true, J2, X3, Y3) -> cons(X3, filter(J2, Y3))
         filter2(false, G3, V3, W3) -> filter(G3, W3)

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) map#(I1, cons(P1, X2)) => map#(I1, X2)
      (2) filter#(G2, cons(V2, W2)) => filter2#(G2(V2), G2, V2, W2)
      (3) filter2#(true, J2, X3, Y3) => filter#(J2, Y3)
      (4) filter2#(false, G3, V3, W3) => filter#(G3, W3)

***** 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 

There are 2 SCCs.

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

  P2. (1) map#(I1, cons(P1, X2)) => map#(I1, X2)

  P3. (1) filter#(G2, cons(V2, W2)) => filter2#(G2(V2), G2, V2, W2)
      (2) filter2#(true, J2, X3, Y3) => filter#(J2, Y3)
      (3) filter2#(false, G3, V3, W3) => filter#(G3, W3)

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

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(P1, X2) |>| X2

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(filter2#) = 4

We thus have:

  (1) cons(V2, W2) |>|  W2
  (2) Y3           |>=| Y3
  (3) W3           |>=| W3

We may remove the strictly oriented DPs.

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

  P4. (1) filter2#(true, J2, X3, Y3) => filter#(J2, Y3)
      (2) filter2#(false, G3, V3, W3) => filter#(G3, W3)

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

We compute a graph approximation with the following edges:

  1:
  2:

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

Processor output: { }.