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

  Signature: 0    :: c
             1    :: c
             add  :: c -> a -> c
             cons :: a -> b -> b
             fold :: (c -> a -> c) -> b -> c -> c
             mul  :: c -> a -> c
             nil  :: b
             prod :: b -> c
             sum  :: b -> c

  Rules: fold(F, nil, Y) -> Y
         fold(G, cons(V, W), P) -> fold(G, W, G(P, V))
         sum(X1) -> fold(add, X1, 0)
         fold(mul, Y1, 1) -> prod(Y1)

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) fold#(G, cons(V, W), P) => fold#(G, W, G(P, V))
      (2) sum#(X1) => fold#(add, X1, 0)

***** 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: 1 

There is only one SCC, so all DPs not inside the SCC can be removed.

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

  P2. (1) fold#(G, cons(V, W), P) => fold#(G, W, G(P, V))

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

We use the following projection function:

  nu(fold#) = 2

We thus have:

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

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

Processor output: { }.