We consider termination of the LCSTRS with only rule scheme Calc:

  Signature: cons :: Int -> list -> list
             f    :: Int -> Int -> list -> list
             nil  :: list

  Rules: f(x, y, z) -> f(y - 2, x + 1, cons(0, z)) | y > 0 /\ x = x
         f(x, y, z) -> z | y <= 0 /\ x = 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) f#(x, y, z) => f#(y - 2, x + 1, cons(0, z)) | y > 0 /\ x = x

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

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

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

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