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

  Signature: cons      :: b -> c -> c
             false     :: a
             filter    :: (b -> a) -> c -> c
             filtersub :: a -> (b -> a) -> c -> c
             nil       :: c
             true      :: a

  Rules: filter(F, nil) -> nil
         filter(Z, cons(U, V)) -> filtersub(Z(U), Z, cons(U, V))
         filtersub(true, I, cons(P, X1)) -> cons(P, filter(I, X1))
         filtersub(false, Z1, cons(U1, V1)) -> filter(Z1, V1)

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) filter#(Z, cons(U, V)) => filtersub#(Z(U), Z, cons(U, V))
      (2) filtersub#(true, I, cons(P, X1)) => filter#(I, X1)
      (3) filtersub#(false, Z1, cons(U1, V1)) => filter#(Z1, V1)

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

We use the following projection function:

  nu(filter#)    = 2
  nu(filtersub#) = 3

We thus have:

  (1) cons(U, V)   |>=| cons(U, V)
  (2) cons(P, X1)  |>|  X1
  (3) cons(U1, V1) |>|  V1

We may remove the strictly oriented DPs.

Processor output: { D2 = (P2, R UNION R_?, i, c) }, where:

  P2. (1) filter#(Z, cons(U, V)) => filtersub#(Z(U), Z, cons(U, V))

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

We compute a graph approximation with the following edges:

  1:

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

Processor output: { }.