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

  Signature: 0    :: a
             add  :: a -> a -> a
             fact :: a -> a
             mult :: a -> a -> a
             rec  :: (a -> a -> a) -> a -> a -> a
             s    :: a -> a

  Rules: add(0, X) -> X
         add(s(Y), U) -> s(add(Y, U))
         mult(0, V) -> 0
         mult(s(W), P) -> add(mult(W, P), P)
         rec(F1, Y1, 0) -> Y1
         rec(G1, V1, s(W1)) -> G1(s(W1), rec(G1, V1, W1))
         fact -> rec(mult, s(0))

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) add#(s(Y), U) => add#(Y, U)
      (2) mult#(s(W), P) => mult#(W, P)
      (3) mult#(s(W), P) => add#(mult(W, P), P)
      (4) rec#(G1, V1, s(W1)) => rec#(G1, V1, W1)
      (5) fact#(arg1) => mult#(fresh1, fresh2)
      (6) fact#(arg1) => rec#(mult, s(0), arg1)

***** 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: 2 3 
  3: 1 
  4: 4 
  5: 2 3 
  6: 4 

There are 3 SCCs.

Processor output: { D2 = (P2, R UNION R_?, f, c) ; D3 = (P3, R UNION R_?, f, c) ; D4 = (P4, R UNION R_?, f, c) }, where:

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

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

  P4. (1) rec#(G1, V1, s(W1)) => rec#(G1, V1, W1)

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

We use the following projection function:

  nu(add#) = 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 Subterm Criterion Processor on D3 = (P3, R UNION R_?, f, c).

We use the following projection function:

  nu(mult#) = 1

We thus have:

  (1) s(W) |>| W

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_?, f, c).

We use the following projection function:

  nu(rec#) = 3

We thus have:

  (1) s(W1) |>| W1

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

Processor output: { }.