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

  Signature: 0       :: b
             1       :: b
             c       :: b -> b
             cons    :: c -> d -> d
             f       :: b -> a
             false   :: a
             filter  :: (c -> a) -> d -> d
             filter2 :: a -> (c -> a) -> c -> d -> d
             g       :: b -> b -> b
             if      :: a -> b -> b -> b
             map     :: (c -> c) -> d -> d
             nil     :: d
             s       :: b -> b
             true    :: a

  Rules: f(0) -> true
         f(1) -> false
         f(s(X)) -> f(X)
         if(true, s(Y), s(U)) -> s(Y)
         if(false, s(V), s(W)) -> s(W)
         g(P, c(X1)) -> c(g(P, X1))
         g(Y1, c(U1)) -> g(Y1, if(f(Y1), c(g(s(Y1), U1)), c(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_?, f, c), where:

  P1. (1)  f#(s(X)) => f#(X)
      (2)  g#(P, c(X1)) => g#(P, X1)
      (3)  g#(Y1, c(U1)) => f#(Y1)
      (4)  g#(Y1, c(U1)) => g#(s(Y1), U1)
      (5)  g#(Y1, c(U1)) => if#(f(Y1), c(g(s(Y1), U1)), c(U1))
      (6)  g#(Y1, c(U1)) => g#(Y1, if(f(Y1), c(g(s(Y1), U1)), c(U1)))
      (7)  map#(I1, cons(P1, X2)) => map#(I1, X2)
      (8)  filter#(G2, cons(V2, W2)) => filter2#(G2(V2), G2, V2, W2)
      (9)  filter2#(true, J2, X3, Y3) => filter#(J2, Y3)
      (10) filter2#(false, G3, V3, W3) => filter#(G3, W3)

***** 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:  2 3 4 5 6 
  3:  1 
  4:  2 3 4 5 6 
  5:
  6:  2 3 4 5 6 
  7:  7 
  8:  9 10 
  9:  8 
  10: 8 

There are 4 SCCs.

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

  P2. (1) f#(s(X)) => f#(X)

  P3. (1) g#(P, c(X1)) => g#(P, X1)
      (2) g#(Y1, c(U1)) => g#(s(Y1), U1)
      (3) g#(Y1, c(U1)) => g#(Y1, if(f(Y1), c(g(s(Y1), U1)), c(U1)))

  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_?, f, c).

We use the following projection function:

  nu(f#) = 1

We thus have:

  (1) s(X) |>| X

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

Processor output: { }.

***** No progress could be made on DP problem D3 = (P3, R UNION R_?, f, c).