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
             log     :: a -> a
             map     :: (c -> c) -> d -> d
             minus   :: a -> a -> a
             nil     :: d
             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)))
         log(s(0)) -> 0
         log(s(s(X1))) -> s(log(s(quot(X1, s(s(0))))))
         map(Z1, nil) -> nil
         map(G1, cons(V1, W1)) -> cons(G1(V1), map(G1, W1))
         filter(J1, nil) -> nil
         filter(F2, cons(Y2, U2)) -> filter2(F2(Y2), F2, Y2, U2)
         filter2(true, H2, W2, P2) -> cons(W2, filter(H2, P2))
         filter2(false, F3, Y3, U3) -> filter(F3, U3)

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) 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) log#(s(s(X1))) => quot#(X1, s(s(0)))
      (5) log#(s(s(X1))) => log#(s(quot(X1, s(s(0)))))
      (6) map#(G1, cons(V1, W1)) => map#(G1, W1)
      (7) filter#(F2, cons(Y2, U2)) => filter2#(F2(Y2), F2, Y2, U2)
      (8) filter2#(true, H2, W2, P2) => filter#(H2, P2)
      (9) filter2#(false, F3, Y3, U3) => filter#(F3, U3)

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

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) minus#(s(Y), s(U)) => minus#(Y, U)

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

  P4. (1) log#(s(s(X1))) => log#(s(quot(X1, s(s(0)))))

  P5. (1) map#(G1, cons(V1, W1)) => map#(G1, W1)

  P6. (1) filter#(F2, cons(Y2, U2)) => filter2#(F2(Y2), F2, Y2, U2)
      (2) filter2#(true, H2, W2, P2) => filter#(H2, P2)
      (3) filter2#(false, F3, Y3, U3) => filter#(F3, U3)

***** We apply the Subterm Criterion Processor on D2 = (P2, R UNION R_?, i, 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: { }.

***** We apply the Usable Rules Processor on D3 = (P3, R UNION R_?, i, c).

We obtain 2 usable rules (out of 12 rules in the input problem).

Processor output: { D7 = (P3, R2, i, c) }, where:

  R2. (1) minus(X, 0) -> X
      (2) minus(s(Y), s(U)) -> minus(Y, U)

***** We apply the Usable Rules Processor on D4 = (P4, R UNION R_?, i, c).

We obtain 4 usable rules (out of 12 rules in the input problem).

Processor output: { D8 = (P4, R3, i, c) }, where:

  R3. (1) minus(X, 0) -> X
      (2) minus(s(Y), s(U)) -> minus(Y, U)
      (3) quot(0, s(V)) -> 0
      (4) quot(s(W), s(P)) -> s(quot(minus(W, P), s(P)))

***** We apply the Subterm Criterion Processor on D5 = (P5, R UNION R_?, i, c).

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(V1, 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 D6 = (P6, R UNION R_?, i, c).

We use the following projection function:

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

We thus have:

  (1) cons(Y2, U2) |>|  U2
  (2) P2           |>=| P2
  (3) U3           |>=| U3

We may remove the strictly oriented DPs.

Processor output: { D9 = (P7, R UNION R_?, i, c) }, where:

  P7. (1) filter2#(true, H2, W2, P2) => filter#(H2, P2)
      (2) filter2#(false, F3, Y3, U3) => filter#(F3, U3)

***** We apply the Reduction Pair [with HORPO] Processor on D7 = (P3, R2, i, c).

Constrained HORPO yields:

  quot#(s(W), s(P)) (>) quot#(minus(W, P), s(P))
  minus(X, 0) (>=) X
  minus(s(Y), s(U)) (>=) minus(Y, U)

We do this using the following settings:

* Disregarded arguments:

  minus 2 
  quot# 2 

* Precedence and permutation:

    s      { 1 }
  > 0      { }
  = minus  { }   1
  = quot#  { }   1

* Well-founded theory orderings:

  [>]_{Bool} = {(true,false)}
  [>]_{Int}  = {(x,y) | x < 1000 /\ x < y }

All dependency pairs were removed.

Processor output: { }.

***** No progress could be made on DP problem D8 = (P4, R3, i, c).