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

  Signature: 0       :: b
             cons    :: a -> b -> b
             nil     :: b
             plus    :: b -> b -> b
             s       :: b -> b
             sumwith :: (a -> b) -> b -> b

  Rules: plus(0, X) -> X
         plus(s(Y), U) -> s(plus(Y, U))
         sumwith(H, nil) -> nil
         sumwith(I, cons(P, X1)) -> plus(I(P), sumwith(I, X1))

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, U)
      (2) sumwith#(I, cons(P, X1)) => sumwith#(I, X1)
      (3) sumwith#(I, cons(P, X1)) => plus#(I(P), sumwith(I, X1))

***** 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 

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, U)

  P3. (1) sumwith#(I, cons(P, X1)) => sumwith#(I, X1)

***** 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

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(sumwith#) = 2

We thus have:

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

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

Processor output: { }.