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

  Signature: $       :: a -> a -> a
             *       :: a -> a -> a
             /       :: a -> a -> a
             cons    :: c -> d -> d
             e       :: a
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             map     :: (c -> c) -> d -> d
             nil     :: d
             true    :: b

  Rules: $(X, X) -> e
         $(e, Y) -> Y
         $(U, *(U, V)) -> V
         $(/(W, P), W) -> P
         /(X1, X1) -> e
         /(Y1, e) -> Y1
         /(*(V1, U1), U1) -> V1
         /(W1, $(P1, W1)) -> P1
         *(e, X2) -> X2
         *(Y2, e) -> Y2
         *(U2, $(U2, V2)) -> V2
         *(/(P2, W2), W2) -> P2
         map(F3, nil) -> nil
         map(Z3, cons(U3, V3)) -> cons(Z3(U3), map(Z3, V3))
         filter(I3, nil) -> nil
         filter(J3, cons(X4, Y4)) -> filter2(J3(X4), J3, X4, Y4)
         filter2(true, G4, V4, W4) -> cons(V4, filter(G4, W4))
         filter2(false, J4, X5, Y5) -> filter(J4, Y5)

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_?, f, c), where:

  P1. (1) map#(Z3, cons(U3, V3)) => map#(Z3, V3)
      (2) filter#(J3, cons(X4, Y4)) => filter2#(J3(X4), J3, X4, Y4)
      (3) filter2#(true, G4, V4, W4) => filter#(G4, W4)
      (4) filter2#(false, J4, X5, Y5) => filter#(J4, Y5)

***** We apply the Graph Processor on D1 = (P1, R UNION R_?, f, 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 UNION R_?, f, c) ; D3 = (P3, R UNION R_?, f, c) }, where:

  P2. (1) map#(Z3, cons(U3, V3)) => map#(Z3, V3)

  P3. (1) filter#(J3, cons(X4, Y4)) => filter2#(J3(X4), J3, X4, Y4)
      (2) filter2#(true, G4, V4, W4) => filter#(G4, W4)
      (3) filter2#(false, J4, X5, Y5) => filter#(J4, Y5)

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

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(U3, V3) |>| V3

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_?, f, c).

We use the following projection function:

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

We thus have:

  (1) cons(X4, Y4) |>|  Y4
  (2) W4           |>=| W4
  (3) Y5           |>=| Y5

We may remove the strictly oriented DPs.

Processor output: { D4 = (P4, R UNION R_?, f, c) }, where:

  P4. (1) filter2#(true, G4, V4, W4) => filter#(G4, W4)
      (2) filter2#(false, J4, X5, Y5) => filter#(J4, Y5)

***** We apply the Graph Processor on D4 = (P4, R UNION 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: { }.