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

  Signature: 0       :: a
             cons    :: c -> c -> c
             copy    :: a -> c -> c -> c
             f       :: c -> c
             false   :: b
             filter  :: (c -> b) -> c -> c
             filter2 :: b -> (c -> b) -> c -> c -> c
             map     :: (c -> c) -> c -> c
             n       :: a
             nil     :: c
             s       :: a -> a
             true    :: b

  Rules: f(cons(nil, X)) -> X
         f(cons(f(cons(nil, Y)), U)) -> copy(n, Y, U)
         copy(0, V, W) -> f(W)
         copy(s(P), X1, Y1) -> copy(P, X1, cons(f(X1), Y1))
         map(G1, nil) -> nil
         map(H1, cons(W1, P1)) -> cons(H1(W1), map(H1, P1))
         filter(F2, nil) -> nil
         filter(Z2, cons(U2, V2)) -> filter2(Z2(U2), Z2, U2, V2)
         filter2(true, I2, P2, X3) -> cons(P2, filter(I2, X3))
         filter2(false, Z3, U3, V3) -> filter(Z3, V3)

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) f#(cons(f(cons(nil, Y)), U)) => copy#(n, Y, U)
      (2) copy#(0, V, W) => f#(W)
      (3) copy#(s(P), X1, Y1) => f#(X1)
      (4) copy#(s(P), X1, Y1) => copy#(P, X1, cons(f(X1), Y1))
      (5) map#(H1, cons(W1, P1)) => map#(H1, P1)
      (6) filter#(Z2, cons(U2, V2)) => filter2#(Z2(U2), Z2, U2, V2)
      (7) filter2#(true, I2, P2, X3) => filter#(I2, X3)
      (8) filter2#(false, Z3, U3, V3) => filter#(Z3, V3)

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

We compute a graph approximation with the following edges:

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

There are 3 SCCs.

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

  P2. (1) copy#(s(P), X1, Y1) => copy#(P, X1, cons(f(X1), Y1))

  P3. (1) map#(H1, cons(W1, P1)) => map#(H1, P1)

  P4. (1) filter#(Z2, cons(U2, V2)) => filter2#(Z2(U2), Z2, U2, V2)
      (2) filter2#(true, I2, P2, X3) => filter#(I2, X3)
      (3) filter2#(false, Z3, U3, V3) => filter#(Z3, V3)

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

We use the following projection function:

  nu(copy#) = 1

We thus have:

  (1) s(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 D3 = (P3, R UNION R_?, i, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(W1, P1) |>| P1

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

We thus have:

  (1) cons(U2, V2) |>|  V2
  (2) X3           |>=| X3
  (3) V3           |>=| V3

We may remove the strictly oriented DPs.

Processor output: { D5 = (P5, R UNION R_?, i, c) }, where:

  P5. (1) filter2#(true, I2, P2, X3) => filter#(I2, X3)
      (2) filter2#(false, Z3, U3, V3) => filter#(Z3, V3)

***** We apply the Graph Processor on D5 = (P5, 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: { }.