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

  Signature: a :: o
             b :: o
             f :: o -> o -> o
             g :: o -> o -> o
             h :: (o -> o) -> o

  Rules: f(x, b) -> g(x, x)
         g(x, a) -> f(x, x)
         h(F) -> F(a)
         h(F) -> F(b)

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) f#(x, b) => g#(x, x)
      (2) g#(x, a) => f#(x, x)

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

We obtain 0 usable rules (out of 4 rules in the input problem).

Processor output: { D2 = (P1, {}, i, c) }.

***** No progress could be made on DP problem D2 = (P1, {}, i, c).