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

  Signature: const :: a -> b -> a
             fix   :: ((f -> g) -> f -> g) -> f -> g
             subst :: (c -> d -> e) -> (c -> d) -> c -> e

  Rules: const(X, Y) -> X
         subst(G, H, W) -> G(W, H(W))
         fix(J, X1) -> J(fix(J), X1)

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) fix#(J, X1) => fix#(J, fresh1)

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

We obtain 0 usable rules (out of 3 rules in the input problem).

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

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