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

  Signature: mult :: N -> N -> N
             plus :: N -> N -> N
             s    :: N -> N
             z    :: N

  Rules: plus(z, X) -> X
         plus(s(Y), U) -> plus(Y, s(U))
         plus(plus(V, W), P) -> plus(V, plus(W, P))
         mult(z, X1) -> z
         mult(s(Y1), U1) -> plus(mult(Y1, U1), U1)
         mult(plus(V1, W1), P1) -> plus(mult(V1, P1), mult(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) plus#(s(Y), U) => plus#(Y, s(U))
      (2) plus#(plus(V, W), P) => plus#(W, P)
      (3) plus#(plus(V, W), P) => plus#(V, plus(W, P))
      (4) mult#(s(Y1), U1) => mult#(Y1, U1)
      (5) mult#(s(Y1), U1) => plus#(mult(Y1, U1), U1)
      (6) mult#(plus(V1, W1), P1) => mult#(V1, P1)
      (7) mult#(plus(V1, W1), P1) => mult#(W1, P1)
      (8) mult#(plus(V1, W1), P1) => plus#(mult(V1, P1), mult(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 3 
  2: 1 2 3 
  3: 1 2 3 
  4: 4 5 6 7 8 
  5: 1 2 3 
  6: 4 5 6 7 8 
  7: 4 5 6 7 8 
  8: 1 2 3 

There are 2 SCCs.

Processor output: { D2 = (P2, R UNION R_?, i, c) ; D3 = (P3, R UNION R_?, i, c) }, where:

  P2. (1) plus#(s(Y), U) => plus#(Y, s(U))
      (2) plus#(plus(V, W), P) => plus#(W, P)
      (3) plus#(plus(V, W), P) => plus#(V, plus(W, P))

  P3. (1) mult#(s(Y1), U1) => mult#(Y1, U1)
      (2) mult#(plus(V1, W1), P1) => mult#(V1, P1)
      (3) mult#(plus(V1, W1), P1) => mult#(W1, P1)

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

We use the following projection function:

  nu(plus#) = 1

We thus have:

  (1) s(Y)       |>| Y
  (2) plus(V, W) |>| W
  (3) plus(V, W) |>| 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(mult#) = 1

We thus have:

  (1) s(Y1)        |>| Y1
  (2) plus(V1, W1) |>| V1
  (3) plus(V1, W1) |>| W1

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

Processor output: { }.