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

  Signature: sif :: Bool -> Int -> Int -> Int
             sum :: Int -> Int -> Int

  Rules: sum(x, y) -> sif(x >= y, x, y)
         sif(true, x, y) -> y + sum(x, y + 1)
         sif(false, x, y) -> 0

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_?, f, c), where:

  P1. (1) sum#(x, y) => sif#(x >= y, x, y)
      (2) sif#(true, x, y) => sum#(x, y + 1)

***** We apply the Theory Arguments Processor on D1 = (P1, R UNION R_?, f, c).

We use the following theory arguments function:

  sif# : [1, 2, 3]
  sum# : [1, 2]

Processor output: { D2 = (P2, R UNION R_?, f, c) ; D3 = (P3, R UNION R_?, f, c) }, where:

  P2. (1) sum#(x, y) => sif#(x >= y, x, y)
      (2) sif#(true, x, y) => sum#(x, y + 1) { x, y }

  P3. (1) sif#(true, x, y) => sum#(x, y + 1)

***** We apply the Theory Arguments Processor on D2 = (P2, R UNION R_?, f, c).

We use the following theory arguments function:

  sif# : [1, 2, 3]
  sum# : [1, 2]

Processor output: { D4 = (P4, R UNION R_?, f, c) ; D5 = (P5, R UNION R_?, f, c) }, where:

  P4. (1) sum#(x, y) => sif#(x >= y, x, y) { x, y }
      (2) sif#(true, x, y) => sum#(x, y + 1) { x, y }

  P5. (1) sum#(x, y) => sif#(x >= y, x, y)

***** We apply the Graph Processor on D3 = (P3, R UNION 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: { }.

***** No progress could be made on DP problem D4 = (P4, R UNION R_?, f, c).