We consider universal computability of the STRS with no additional rule schemes:

  Signature: 0       :: a
             cons    :: c -> d -> d
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             map     :: (c -> c) -> d -> d
             nil     :: d
             plus    :: a -> a -> a
             s       :: a -> a
             times   :: a -> a -> a
             true    :: b

  Rules: times(X, 0) -> 0
         times(Y, s(U)) -> plus(times(Y, U), Y)
         plus(V, 0) -> V
         plus(0, W) -> W
         plus(P, s(X1)) -> s(plus(P, X1))
         plus(s(Y1), U1) -> s(plus(Y1, 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 UNION R_?, i, c), where:

  P1. (1) times#(Y, s(U)) => times#(Y, U)
      (2) times#(Y, s(U)) => plus#(times(Y, U), Y)
      (3) plus#(P, s(X1)) => plus#(P, X1)
      (4) plus#(s(Y1), U1) => plus#(Y1, U1)
      (5) map#(I1, cons(P1, X2)) => map#(I1, X2)
      (6) filter#(G2, cons(V2, W2)) => filter2#(G2(V2), G2, V2, W2)
      (7) filter2#(true, J2, X3, Y3) => filter#(J2, Y3)
      (8) filter2#(false, G3, V3, W3) => filter#(G3, W3)

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

We compute a graph approximation with the following edges:

  1: 1 2 
  2: 3 4 
  3: 3 4 
  4: 3 4 
  5: 5 
  6: 7 8 
  7: 6 
  8: 6 

There are 4 SCCs.

Processor output: { D2 = (P2, R UNION R_?, i, c) ; D3 = (P3, R UNION R_?, i, c) ; D4 = (P4, R UNION R_?, i, c) ; D5 = (P5, R UNION R_?, i, c) }, where:

  P2. (1) plus#(P, s(X1)) => plus#(P, X1)
      (2) plus#(s(Y1), U1) => plus#(Y1, U1)

  P3. (1) times#(Y, s(U)) => times#(Y, U)

  P4. (1) map#(I1, cons(P1, X2)) => map#(I1, X2)

  P5. (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 UNION R_?, i, c).

We use the following projection function:

  nu(plus#) = 1

We thus have:

  (1) P     |>=| P
  (2) s(Y1) |>|  Y1

We may remove the strictly oriented DPs.

Processor output: { D6 = (P6, R UNION R_?, i, c) }, where:

  P6. (1) plus#(P, s(X1)) => plus#(P, X1)

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

We use the following projection function:

  nu(times#) = 2

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 D4 = (P4, R UNION 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 D5 = (P5, R UNION 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: { D7 = (P7, R UNION R_?, i, c) }, where:

  P7. (1) filter2#(true, J2, X3, Y3) => filter#(J2, Y3)
      (2) filter2#(false, G3, V3, W3) => filter#(G3, W3)

***** We apply the Subterm Criterion Processor on D6 = (P6, R UNION R_?, i, c).

We use the following projection function:

  nu(plus#) = 2

We thus have:

  (1) s(X1) |>| X1

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

Processor output: { }.

***** We apply the Graph Processor on D7 = (P7, R UNION 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: { }.