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

  Signature: 0     :: a
             add   :: a -> a -> a
             curry :: (a -> a -> a) -> a -> a -> a
             plus  :: a -> a -> a
             s     :: a -> a

  Rules: plus(0, X) -> X
         plus(s(Y), U) -> s(plus(Y, U))
         curry(H, W, P) -> H(W, P)
         add -> curry(plus)

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) add#(arg1, arg2) => plus#(fresh1, fresh2)
      (3) add#(arg1, arg2) => curry#(plus, arg1, arg2)

***** 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 
  3:

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) plus#(s(Y), U) => plus#(Y, U)

***** 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: { }.