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

  Signature: app     :: list -> list -> list
             cons    :: nat -> list -> list
             map     :: (nat -> nat) -> list -> list
             nil     :: list
             reverse :: list -> list
             shuffle :: list -> list

  Rules: app(nil, X) -> X
         app(cons(Y, U), V) -> cons(Y, app(U, V))
         reverse(nil) -> nil
         reverse(cons(W, P)) -> app(reverse(P), cons(W, nil))
         shuffle(nil) -> nil
         shuffle(cons(X1, Y1)) -> cons(X1, shuffle(reverse(Y1)))
         map(G1, nil) -> nil
         map(H1, cons(W1, P1)) -> cons(H1(W1), map(H1, 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) app#(cons(Y, U), V) => app#(U, V)
      (2) reverse#(cons(W, P)) => reverse#(P)
      (3) reverse#(cons(W, P)) => app#(reverse(P), cons(W, nil))
      (4) shuffle#(cons(X1, Y1)) => reverse#(Y1)
      (5) shuffle#(cons(X1, Y1)) => shuffle#(reverse(Y1))
      (6) map#(H1, cons(W1, P1)) => map#(H1, 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 
  3: 1 
  4: 2 3 
  5: 4 5 
  6: 6 

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) app#(cons(Y, U), V) => app#(U, V)

  P3. (1) reverse#(cons(W, P)) => reverse#(P)

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

  P5. (1) map#(H1, cons(W1, P1)) => map#(H1, P1)

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

We use the following projection function:

  nu(app#) = 1

We thus have:

  (1) cons(Y, 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(reverse#) = 1

We thus have:

  (1) cons(W, P) |>| P

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

Processor output: { }.

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

We obtain 4 usable rules (out of 8 rules in the input problem).

Processor output: { D6 = (P4, R2, i, c) }, where:

  R2. (1) app(nil, X) -> X
      (2) app(cons(Y, U), V) -> cons(Y, app(U, V))
      (3) reverse(nil) -> nil
      (4) reverse(cons(W, P)) -> app(reverse(P), cons(W, nil))

***** 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(W1, P1) |>| P1

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 D6 = (P4, R2, i, c).