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

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

  Rules: g(F, Y) -> Y
         f(G) -> f(g(G))

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#(G) => g#(G, fresh1)
      (2) f#(G) => f#(g(G))

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

We compute a graph approximation with the following edges:

  1:
  2: 1 2 

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

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

  P2. (1) f#(G) => f#(g(G))

***** We apply the Usable Rules Processor on D2 = (P2, R, i, c).

We obtain 0 usable rules (out of 2 rules in the input problem).

Processor output: { D3 = (P2, {}, i, c) }.

***** No progress could be made on DP problem D3 = (P2, {}, i, c).