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

  Signature: f     :: Int -> Int -> o
             round :: Int -> Int

  Rules: f(x, y) -> f(x, round(x)) | x >= 1 /\ y = x - 1
         round(x) -> x | x = x
         round(x) -> x + 1 | 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) => round#(x) | x >= 1 /\ y = x - 1
      (2) f#(x, y) => f#(x, round(x)) | x >= 1 /\ y = x - 1

***** 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#(x, y) => f#(x, round(x)) | x >= 1 /\ y = x - 1

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

We obtain 2 usable rules (out of 3 rules in the input problem).

Processor output: { D3 = (P2, R2, i, c) }, where:

  R2. (1) round(x) -> x | x = x
      (2) round(x) -> x + 1 | x = x

***** No progress could be made on DP problem D3 = (P2, R2, i, c).