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

  Signature: l1    :: Int -> Int -> A (private)
             l2    :: Int -> Int -> A (private)
             start :: Int -> A

  Rules: start(x) -> l1(x, 0) | x = x
         l1(x, y) -> l2(x, y) | x > y
         l2(x, y) -> l1(x, y + 1) | 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 UNION R_?, i, c), where:

  P1. (1) start#(x) => l1#(x, 0) | x = x
      (2) l1#(x, y) => l2#(x, y) | x > y              (private)
      (3) l2#(x, y) => l1#(x, y + 1) | x = x /\ y = y (private)

***** 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: 3 
  3: 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) l1#(x, y) => l2#(x, y) | x > y
      (2) l2#(x, y) => l1#(x, y + 1) | x = x /\ y = y

***** We apply the Chaining Processor Processor on D2 = (P2, R UNION R_?, i, c).

We chain DPs according to the following mapping:


  l2#(x, y) => l2#(x, y + 1) | x = x /\ y = y /\ x > y + 1  is obtained by chaining  l2#(x, y) => l1#(x, y + 1) | x = x /\ y = y and l1#(x', y') => l2#(x', y') | x' > y'


The following DPs were deleted:

l2#(x, y) => l1#(x, y + 1) | x = x /\ y = y

l1#(x, y) => l2#(x, y) | x > y


By chaining, we added 1 DPs and removed 2 DPs.

Processor output: { D3 = (P3, R UNION R_?, i, c) }, where:

  P3. (1) l2#(x, y) => l2#(x, y + 1) | x = x /\ y = y /\ x > y + 1

***** We apply the Integer Function Processor on D3 = (P3, R UNION R_?, i, c).

We use the following integer mapping:

  J(l2#) = arg_1 - (arg_2 + 1) - 1

We thus have:

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

All DPs are strictly oriented, and may be removed.  Hence, this DP problem is finite.

Processor output: { }.