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

  Signature: 0       :: b
             cons    :: c -> d -> d
             false   :: a
             filter  :: (c -> a) -> d -> d
             filter2 :: a -> (c -> a) -> c -> d -> d
             gcd     :: b -> b -> b
             if_gcd  :: a -> b -> b -> b
             le      :: b -> b -> a
             map     :: (c -> c) -> d -> d
             minus   :: b -> b -> b
             nil     :: d
             pred    :: b -> b
             s       :: b -> b
             true    :: a

  Rules: le(0, X) -> true
         le(s(Y), 0) -> false
         le(s(U), s(V)) -> le(U, V)
         pred(s(W)) -> W
         minus(P, 0) -> P
         minus(X1, s(Y1)) -> pred(minus(X1, Y1))
         gcd(0, U1) -> U1
         gcd(s(V1), 0) -> s(V1)
         gcd(s(W1), s(P1)) -> if_gcd(le(P1, W1), s(W1), s(P1))
         if_gcd(true, s(X2), s(Y2)) -> gcd(minus(X2, Y2), s(Y2))
         if_gcd(false, s(U2), s(V2)) -> gcd(minus(V2, U2), s(U2))
         map(I2, nil) -> nil
         map(J2, cons(X3, Y3)) -> cons(J2(X3), map(J2, Y3))
         filter(G3, nil) -> nil
         filter(H3, cons(W3, P3)) -> filter2(H3(W3), H3, W3, P3)
         filter2(true, F4, Y4, U4) -> cons(Y4, filter(F4, U4))
         filter2(false, H4, W4, P4) -> filter(H4, P4)

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)  le#(s(U), s(V)) => le#(U, V)
      (2)  minus#(X1, s(Y1)) => minus#(X1, Y1)
      (3)  minus#(X1, s(Y1)) => pred#(minus(X1, Y1))
      (4)  gcd#(s(W1), s(P1)) => le#(P1, W1)
      (5)  gcd#(s(W1), s(P1)) => if_gcd#(le(P1, W1), s(W1), s(P1))
      (6)  if_gcd#(true, s(X2), s(Y2)) => minus#(X2, Y2)
      (7)  if_gcd#(true, s(X2), s(Y2)) => gcd#(minus(X2, Y2), s(Y2))
      (8)  if_gcd#(false, s(U2), s(V2)) => minus#(V2, U2)
      (9)  if_gcd#(false, s(U2), s(V2)) => gcd#(minus(V2, U2), s(U2))
      (10) map#(J2, cons(X3, Y3)) => map#(J2, Y3)
      (11) filter#(H3, cons(W3, P3)) => filter2#(H3(W3), H3, W3, P3)
      (12) filter2#(true, F4, Y4, U4) => filter#(F4, U4)
      (13) filter2#(false, H4, W4, P4) => filter#(H4, P4)

***** 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 3 
  3:
  4:  1 
  5:  6 7 8 9 
  6:  2 3 
  7:  4 5 
  8:  2 3 
  9:  4 5 
  10: 10 
  11: 12 13 
  12: 11 
  13: 11 

There are 5 SCCs.

Processor output: { D2 = (P2, R UNION R_?, i, c) ; D3 = (P3, R UNION R_?, i, c) ; D4 = (P4, R UNION R_?, i, c) ; D5 = (P5, R UNION R_?, i, c) ; D6 = (P6, R UNION R_?, i, c) }, where:

  P2. (1) le#(s(U), s(V)) => le#(U, V)

  P3. (1) minus#(X1, s(Y1)) => minus#(X1, Y1)

  P4. (1) gcd#(s(W1), s(P1)) => if_gcd#(le(P1, W1), s(W1), s(P1))
      (2) if_gcd#(true, s(X2), s(Y2)) => gcd#(minus(X2, Y2), s(Y2))
      (3) if_gcd#(false, s(U2), s(V2)) => gcd#(minus(V2, U2), s(U2))

  P5. (1) map#(J2, cons(X3, Y3)) => map#(J2, Y3)

  P6. (1) filter#(H3, cons(W3, P3)) => filter2#(H3(W3), H3, W3, P3)
      (2) filter2#(true, F4, Y4, U4) => filter#(F4, U4)
      (3) filter2#(false, H4, W4, P4) => filter#(H4, P4)

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

We use the following projection function:

  nu(le#) = 1

We thus have:

  (1) s(U) |>| U

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_?, i, c).

We use the following projection function:

  nu(minus#) = 2

We thus have:

  (1) s(Y1) |>| Y1

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 D4 = (P4, R UNION R_?, i, c).