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

  Signature: cons    :: c -> d -> d
             f       :: a -> a
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             g       :: a -> a
             map     :: (c -> c) -> d -> d
             nil     :: d
             true    :: b

  Rules: f(g(X)) -> g(g(f(X)))
         f(g(Y)) -> g(g(g(Y)))
         map(G, nil) -> nil
         map(H, cons(W, P)) -> cons(H(W), map(H, P))
         filter(F1, nil) -> nil
         filter(Z1, cons(U1, V1)) -> filter2(Z1(U1), Z1, U1, V1)
         filter2(true, I1, P1, X2) -> cons(P1, filter(I1, X2))
         filter2(false, Z2, U2, V2) -> filter(Z2, V2)

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, f, c), where:

  P1. (1) f#(g(X)) => f#(X)
      (2) map#(H, cons(W, P)) => map#(H, P)
      (3) filter#(Z1, cons(U1, V1)) => filter2#(Z1(U1), Z1, U1, V1)
      (4) filter2#(true, I1, P1, X2) => filter#(I1, X2)
      (5) filter2#(false, Z2, U2, V2) => filter#(Z2, V2)

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

We compute a graph approximation with the following edges:

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

There are 3 SCCs.

Processor output: { D2 = (P2, R, f, c) ; D3 = (P3, R, f, c) ; D4 = (P4, R, f, c) }, where:

  P2. (1) f#(g(X)) => f#(X)

  P3. (1) map#(H, cons(W, P)) => map#(H, P)

  P4. (1) filter#(Z1, cons(U1, V1)) => filter2#(Z1(U1), Z1, U1, V1)
      (2) filter2#(true, I1, P1, X2) => filter#(I1, X2)
      (3) filter2#(false, Z2, U2, V2) => filter#(Z2, V2)

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

We use the following projection function:

  nu(f#) = 1

We thus have:

  (1) g(X) |>| X

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, f, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(W, P) |>| P

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

Processor output: { }.

***** We apply the Subterm Criterion Processor on D4 = (P4, R, f, c).

We use the following projection function:

  nu(filter#)  = 2
  nu(filter2#) = 4

We thus have:

  (1) cons(U1, V1) |>|  V1
  (2) X2           |>=| X2
  (3) V2           |>=| V2

We may remove the strictly oriented DPs.

Processor output: { D5 = (P5, R, f, c) }, where:

  P5. (1) filter2#(true, I1, P1, X2) => filter#(I1, X2)
      (2) filter2#(false, Z2, U2, V2) => filter#(Z2, V2)

***** We apply the Graph Processor on D5 = (P5, R, f, c).

We compute a graph approximation with the following edges:

  1:
  2:

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

Processor output: { }.