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

  Signature: 0        :: b
             cons     :: d -> e -> e
             f        :: b -> b -> b -> b -> c
             false    :: c
             filter   :: (d -> c) -> e -> e
             filter2  :: c -> (d -> c) -> d -> e -> e
             if       :: a -> c -> c -> c
             le       :: b -> b -> a
             map      :: (d -> d) -> e -> e
             minus    :: b -> b -> b
             nil      :: e
             perfectp :: b -> c
             s        :: b -> b
             true     :: c

  Rules: perfectp(0) -> false
         perfectp(s(X)) -> f(X, s(0), s(X), s(X))
         f(0, U, 0, Y) -> true
         f(0, W, s(P), V) -> false
         f(s(Y1), 0, U1, X1) -> f(Y1, X1, minus(U1, s(Y1)), X1)
         f(s(W1), s(P1), X2, V1) -> if(le(W1, P1), f(s(W1), minus(P1, W1), X2, V1), f(W1, V1, X2, V1))
         map(Z2, nil) -> nil
         map(G2, cons(V2, W2)) -> cons(G2(V2), map(G2, W2))
         filter(J2, nil) -> nil
         filter(F3, cons(Y3, U3)) -> filter2(F3(Y3), F3, Y3, U3)
         filter2(true, H3, W3, P3) -> cons(W3, filter(H3, P3))
         filter2(false, F4, Y4, U4) -> filter(F4, U4)

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) perfectp#(s(X)) => f#(X, s(0), s(X), s(X))
      (2) f#(s(Y1), 0, U1, X1) => f#(Y1, X1, minus(U1, s(Y1)), X1)
      (3) f#(s(W1), s(P1), X2, V1) => f#(s(W1), minus(P1, W1), X2, V1)
      (4) f#(s(W1), s(P1), X2, V1) => f#(W1, V1, X2, V1)
      (5) map#(G2, cons(V2, W2)) => map#(G2, W2)
      (6) filter#(F3, cons(Y3, U3)) => filter2#(F3(Y3), F3, Y3, U3)
      (7) filter2#(true, H3, W3, P3) => filter#(H3, P3)
      (8) filter2#(false, F4, Y4, U4) => filter#(F4, U4)

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

We compute a graph approximation with the following edges:

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

There are 3 SCCs.

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

  P2. (1) f#(s(Y1), 0, U1, X1) => f#(Y1, X1, minus(U1, s(Y1)), X1)
      (2) f#(s(W1), s(P1), X2, V1) => f#(W1, V1, X2, V1)

  P3. (1) map#(G2, cons(V2, W2)) => map#(G2, W2)

  P4. (1) filter#(F3, cons(Y3, U3)) => filter2#(F3(Y3), F3, Y3, U3)
      (2) filter2#(true, H3, W3, P3) => filter#(H3, P3)
      (3) filter2#(false, F4, Y4, U4) => filter#(F4, U4)

***** 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(Y1) |>| Y1
  (2) s(W1) |>| W1

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(map#) = 2

We thus have:

  (1) cons(V2, W2) |>| W2

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_?, f, c).

We use the following projection function:

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

We thus have:

  (1) cons(Y3, U3) |>|  U3
  (2) P3           |>=| P3
  (3) U4           |>=| U4

We may remove the strictly oriented DPs.

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

  P5. (1) filter2#(true, H3, W3, P3) => filter#(H3, P3)
      (2) filter2#(false, F4, Y4, U4) => filter#(F4, U4)

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