We consider termination of the STRS with no additional rule schemes:

  Signature: 0   :: nat
             rec :: nat -> a -> (nat -> a -> a) -> a
             s   :: nat -> nat

  Rules: rec(0, U, X) -> U
         rec(s(x), U, X) -> X(x, rec(x, U, X))

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, i, c), where:

  P1. (1) rec#(s(x), U, X) => rec#(x, U, X)

***** We apply the Subterm Criterion Processor on D1 = (P1, R, i, c).

We use the following projection function:

  nu(rec#) = 1

We thus have:

  (1) s(x) |>| x

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

Processor output: { }.