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

  Signature: f :: Int -> Int -> A
             g :: Int -> Int -> A

  Rules: f(x, y) -> g(x, y - 1) | x = x /\ y >= 0
         g(x, y) -> f(x, y) | x = x /\ y = y

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, f, c), where:

  P1. (1) f#(x, y) => g#(x, y - 1) | x = x /\ y >= 0
      (2) g#(x, y) => f#(x, y) | x = x /\ y = y

***** We apply the Integer Function Processor on D1 = (P1, R, f, c).

We use the following integer mapping:

  J(f#) = arg_2
  J(g#) = arg_2

We thus have:

  (1) x = x /\ y >= 0 |= y >  y - 1 (and y >= 0)
  (2) x = x /\ y = y  |= y >= y

We may remove the strictly oriented DPs, which yields:

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

  P2. (1) g#(x, y) => f#(x, y) | x = x /\ y = y

***** We apply the Graph Processor on D2 = (P2, R, f, c).

We compute a graph approximation with the following edges:

  1:

As there are no SCCs, this DP problem is removed.

Processor output: { }.