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

  Signature: 0       :: a
             cons    :: c -> d -> d
             false   :: b
             filter  :: (c -> b) -> d -> d
             filter2 :: b -> (c -> b) -> c -> d -> d
             map     :: (c -> c) -> d -> d
             minus   :: a -> a -> a
             nil     :: d
             plus    :: a -> a -> a
             quot    :: a -> a -> a
             s       :: a -> a
             true    :: b

  Rules: minus(X, 0) -> X
         minus(s(Y), s(U)) -> minus(Y, U)
         quot(0, s(V)) -> 0
         quot(s(W), s(P)) -> s(quot(minus(W, P), s(P)))
         plus(0, X1) -> X1
         plus(s(Y1), U1) -> s(plus(Y1, U1))
         plus(minus(V1, s(0)), minus(W1, s(s(P1)))) -> plus(minus(W1, s(s(P1))), minus(V1, s(0)))
         plus(plus(X2, s(0)), plus(Y2, s(s(U2)))) -> plus(plus(Y2, s(s(U2))), plus(X2, s(0)))
         map(H2, nil) -> nil
         map(I2, cons(P2, X3)) -> cons(I2(P2), map(I2, X3))
         filter(Z3, nil) -> nil
         filter(G3, cons(V3, W3)) -> filter2(G3(V3), G3, V3, W3)
         filter2(true, J3, X4, Y4) -> cons(X4, filter(J3, Y4))
         filter2(false, G4, V4, W4) -> filter(G4, W4)

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)  minus#(s(Y), s(U)) => minus#(Y, U)
      (2)  quot#(s(W), s(P)) => minus#(W, P)
      (3)  quot#(s(W), s(P)) => quot#(minus(W, P), s(P))
      (4)  plus#(s(Y1), U1) => plus#(Y1, U1)
      (5)  plus#(minus(V1, s(0)), minus(W1, s(s(P1)))) => minus#(W1, s(s(P1)))
      (6)  plus#(minus(V1, s(0)), minus(W1, s(s(P1)))) => minus#(V1, s(0))
      (7)  plus#(minus(V1, s(0)), minus(W1, s(s(P1)))) => plus#(minus(W1, s(s(P1))), minus(V1, s(0)))
      (8)  plus#(plus(X2, s(0)), plus(Y2, s(s(U2)))) => plus#(Y2, s(s(U2)))
      (9)  plus#(plus(X2, s(0)), plus(Y2, s(s(U2)))) => plus#(X2, s(0))
      (10) plus#(plus(X2, s(0)), plus(Y2, s(s(U2)))) => plus#(plus(Y2, s(s(U2))), plus(X2, s(0)))
      (11) map#(I2, cons(P2, X3)) => map#(I2, X3)
      (12) filter#(G3, cons(V3, W3)) => filter2#(G3(V3), G3, V3, W3)
      (13) filter2#(true, J3, X4, Y4) => filter#(J3, Y4)
      (14) filter2#(false, G4, V4, W4) => filter#(G4, W4)

***** 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:  1 
  3:  2 3 
  4:  4 5 6 7 8 9 10 
  5:  1 
  6:  1 
  7:  4 5 6 7 8 9 10 
  8:  4 
  9:  4 
  10: 4 5 6 7 8 9 10 
  11: 11 
  12: 13 14 
  13: 12 
  14: 12 

There are 5 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) ; D6 = (P6, R UNION R_?, f, c) }, where:

  P2. (1) minus#(s(Y), s(U)) => minus#(Y, U)

  P3. (1) quot#(s(W), s(P)) => quot#(minus(W, P), s(P))

  P4. (1) plus#(s(Y1), U1) => plus#(Y1, U1)
      (2) plus#(minus(V1, s(0)), minus(W1, s(s(P1)))) => plus#(minus(W1, s(s(P1))), minus(V1, s(0)))
      (3) plus#(plus(X2, s(0)), plus(Y2, s(s(U2)))) => plus#(Y2, s(s(U2)))
      (4) plus#(plus(X2, s(0)), plus(Y2, s(s(U2)))) => plus#(X2, s(0))
      (5) plus#(plus(X2, s(0)), plus(Y2, s(s(U2)))) => plus#(plus(Y2, s(s(U2))), plus(X2, s(0)))

  P5. (1) map#(I2, cons(P2, X3)) => map#(I2, X3)

  P6. (1) filter#(G3, cons(V3, W3)) => filter2#(G3(V3), G3, V3, W3)
      (2) filter2#(true, J3, X4, Y4) => filter#(J3, Y4)
      (3) filter2#(false, G4, V4, W4) => filter#(G4, W4)

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

We use the following projection function:

  nu(minus#) = 1

We thus have:

  (1) s(Y) |>| Y

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).