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

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

  Rules: consif(true, X, Y) -> cons(X, Y)
         consif(false, U, V) -> V
         filter(I, nil) -> nil
         filter(J, cons(X1, Y1)) -> consif(J(X1), X1, filter(J, Y1))

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#(J, cons(X1, Y1)) => filter#(J, Y1)
      (2) filter#(J, cons(X1, Y1)) => consif#(J(X1), X1, filter(J, Y1))

***** 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:

There is only one SCC, so all DPs not inside the SCC can be removed.

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

  P2. (1) filter#(J, cons(X1, Y1)) => filter#(J, Y1)

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

We use the following projection function:

  nu(filter#) = 2

We thus have:

  (1) cons(X1, Y1) |>| Y1

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

Processor output: { }.