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

  Signature: append  :: a -> a -> a
             concat  :: a -> a
             cons    :: a -> a -> a
             flatten :: a -> a
             map     :: (a -> a) -> a -> a
             nil     :: a
             node    :: a -> a -> a

  Rules: map(F, nil) -> nil
         map(Z, cons(U, V)) -> cons(Z(U), map(Z, V))
         flatten(node(W, P)) -> cons(W, concat(map(flatten, P)))
         concat(nil) -> nil
         concat(cons(X1, Y1)) -> append(X1, concat(Y1))
         append(nil, U1) -> U1
         append(cons(V1, W1), P1) -> cons(V1, append(W1, P1))

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) map#(Z, cons(U, V)) => map#(Z, V)
      (2) flatten#(node(W, P)) => flatten#(fresh1)
      (3) flatten#(node(W, P)) => map#(flatten, P)
      (4) flatten#(node(W, P)) => concat#(map(flatten, P))
      (5) concat#(cons(X1, Y1)) => concat#(Y1)
      (6) concat#(cons(X1, Y1)) => append#(X1, concat(Y1))
      (7) append#(cons(V1, W1), P1) => append#(W1, P1)

***** 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 4 
  3: 1 
  4: 5 6 
  5: 5 6 
  6: 7 
  7: 7 

There are 4 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) }, where:

  P2. (1) map#(Z, cons(U, V)) => map#(Z, V)

  P3. (1) append#(cons(V1, W1), P1) => append#(W1, P1)

  P4. (1) concat#(cons(X1, Y1)) => concat#(Y1)

  P5. (1) flatten#(node(W, P)) => flatten#(fresh1)

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

We use the following projection function:

  nu(map#) = 2

We thus have:

  (1) cons(U, V) |>| V

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(append#) = 1

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

We use the following projection function:

  nu(concat#) = 1

We thus have:

  (1) cons(X1, Y1) |>| Y1

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

Processor output: { }.

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

We obtain 0 usable rules (out of 7 rules in the input problem).

Processor output: { D6 = (P5, {}, i, c) }.

***** No progress could be made on DP problem D6 = (P5, {}, i, c).