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

  Signature: l1    :: Int -> Int -> A        (private)
             l2    :: Int -> Int -> A        (private)
             l3    :: Int -> Int -> Int -> A (private)
             l4    :: Int -> Int -> Int -> A (private)
             l5    :: Int -> 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) -> l3(x, y, y) | x = x /\ y = y
         l3(x, y, z) -> l4(x, y, z) | z > 0 /\ x = x /\ y = y
         l4(x, y, z) -> l3(x, y, z - y - 1) | x = x /\ y = y /\ z = z
         l4(x, y, z) -> l5(x, y + 1, z) | x = x /\ y = y /\ z = z
         l5(x, y, z) -> l1(x, y) | x = x /\ y = y /\ z = z

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) => l3#(x, y, y) | x = x /\ y = y                     (private)
      (4) l3#(x, y, z) => l4#(x, y, z) | z > 0 /\ x = x /\ y = y         (private)
      (5) l4#(x, y, z) => l3#(x, y, z - y - 1) | x = x /\ y = y /\ z = z (private)
      (6) l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z     (private)
      (7) l5#(x, y, z) => l1#(x, y) | x = x /\ y = y /\ z = z            (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: 4 
  4: 5 6 
  5: 4 
  6: 7 
  7: 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) => l3#(x, y, y) | x = x /\ y = y
      (3) l3#(x, y, z) => l4#(x, y, z) | z > 0 /\ x = x /\ y = y
      (4) l4#(x, y, z) => l3#(x, y, z - y - 1) | x = x /\ y = y /\ z = z
      (5) l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z
      (6) l5#(x, y, z) => l1#(x, y) | x = x /\ y = y /\ z = z

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

We chain DPs according to the following mapping:


  l5#(x, y, z) => l2#(x, y) | x = x /\ y = y /\ z = z /\ x > y  is obtained by chaining  l5#(x, y, z) => l1#(x, y) | x = x /\ y = y /\ z = z and l1#(x', y') => l2#(x', y') | x' > y'


The following DPs were deleted:

l5#(x, y, z) => l1#(x, y) | x = x /\ y = y /\ z = z

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) => l3#(x, y, y) | x = x /\ y = y
      (2) l3#(x, y, z) => l4#(x, y, z) | z > 0 /\ x = x /\ y = y
      (3) l4#(x, y, z) => l3#(x, y, z - y - 1) | x = x /\ y = y /\ z = z
      (4) l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z
      (5) l5#(x, y, z) => l2#(x, y) | x = x /\ y = y /\ z = z /\ x > y

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

We chain DPs according to the following mapping:


  l5#(x, y, z) => l3#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y)  is obtained by chaining  l5#(x, y, z) => l2#(x, y) | x = x /\ y = y /\ z = z /\ x > y and l2#(x', y') => l3#(x', y', y') | x' = x' /\ y' = y'


The following DPs were deleted:

l5#(x, y, z) => l2#(x, y) | x = x /\ y = y /\ z = z /\ x > y

l2#(x, y) => l3#(x, y, y) | x = x /\ y = y


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

Processor output: { D4 = (P4, R UNION R_?, i, c) }, where:

  P4. (1) l3#(x, y, z) => l4#(x, y, z) | z > 0 /\ x = x /\ y = y
      (2) l4#(x, y, z) => l3#(x, y, z - y - 1) | x = x /\ y = y /\ z = z
      (3) l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z
      (4) l5#(x, y, z) => l3#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y)

***** We apply the Chaining Processor Processor on D4 = (P4, R UNION R_?, i, c).

We chain DPs according to the following mapping:


  l4#(x, y, z) => l4#(x, y, z - y - 1) | x = x /\ y = y /\ z = z /\ (z - y - 1 > 0 /\ x = x /\ y = y)               is obtained by chaining  l4#(x, y, z) => l3#(x, y, z - y - 1) | x = x /\ y = y /\ z = z                      and l3#(x', y', z') => l4#(x', y', z') | z' > 0 /\ x' = x' /\ y' = y'
  l5#(x, y, z) => l4#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y)  is obtained by chaining  l5#(x, y, z) => l3#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) and l3#(x', y', z') => l4#(x', y', z') | z' > 0 /\ x' = x' /\ y' = y'


The following DPs were deleted:

l4#(x, y, z) => l3#(x, y, z - y - 1) | x = x /\ y = y /\ z = z

l5#(x, y, z) => l3#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y)

l3#(x, y, z) => l4#(x, y, z) | z > 0 /\ x = x /\ y = y


By chaining, we added 2 DPs and removed 3 DPs.

Processor output: { D5 = (P5, R UNION R_?, i, c) }, where:

  P5. (1) l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z
      (2) l4#(x, y, z) => l4#(x, y, z - y - 1) | x = x /\ y = y /\ z = z /\ (z - y - 1 > 0 /\ x = x /\ y = y)
      (3) l5#(x, y, z) => l4#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y)

***** We apply the Graph Processor on D5 = (P5, R UNION R_?, i, c).

We compute a graph approximation with the following edges:

  1: 3 
  2: 1 2 
  3: 1 

There are 2 SCCs.

Processor output: { D6 = (P6, R UNION R_?, i, c) ; D7 = (P7, R UNION R_?, i, c) }, where:

  P6. (1) l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z
      (2) l5#(x, y, z) => l4#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y)

  P7. (1) l4#(x, y, z) => l4#(x, y, z - y - 1) | x = x /\ y = y /\ z = z /\ (z - y - 1 > 0 /\ x = x /\ y = y)

***** We apply the Chaining Processor Processor on D6 = (P6, R UNION R_?, i, c).

We chain DPs according to the following mapping:


  l5#(x, y, z) => l5#(x, y + 1, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y) /\ (x = x /\ y = y /\ y = y)  is obtained by chaining  l5#(x, y, z) => l4#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y) and l4#(x', y', z') => l5#(x', y' + 1, z') | x' = x' /\ y' = y' /\ z' = z'


The following DPs were deleted:

l5#(x, y, z) => l4#(x, y, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y)

l4#(x, y, z) => l5#(x, y + 1, z) | x = x /\ y = y /\ z = z


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

Processor output: { D8 = (P8, R UNION R_?, i, c) }, where:

  P8. (1) l5#(x, y, z) => l5#(x, y + 1, y) | x = x /\ y = y /\ z = z /\ x > y /\ (x = x /\ y = y) /\ (y > 0 /\ x = x /\ y = y) /\ (x = x /\ y = y /\ y = y)

***** No progress could be made on DP problem D7 = (P7, R UNION R_?, i, c).