We consider universal computability of the LCTRS with only rule scheme Calc:

  Signature: f  :: Bool -> Int -> o
             ft :: Bool -> Int -> Int -> o

  Rules: f(true, x) -> ft(true, x, y) | y = y
         ft(true, x, y) -> ft(y >= x, x + 1, 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 UNION R_?, i, c), where:

  P1. (1) f#(true, x) => ft#(true, x, y) | y = y
      (2) ft#(true, x, y) => ft#(y >= x, x + 1, y)

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

We compute a graph approximation with the following edges:

  1: 2 
  2: 2 

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

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

  P2. (1) ft#(true, x, y) => ft#(y >= x, x + 1, y)

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