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)
         l1(x, y) -> l2(x, y) | x > y
         l2(x, y) -> l3(x, y, y)
         l3(x, y, z) -> l4(x, y, z) | z > 0
         l4(x, y, z) -> l3(x, y, z - 1)
         l4(x, y, z) -> l5(x, y + 1, z)
         l5(x, y, z) -> l1(x, 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_?, f, c), where:

  P1. (1) start#(x) => l1#(x, 0)
      (2) l1#(x, y) => l2#(x, y) | x > y       (private)
      (3) l2#(x, y) => l3#(x, y, y)            (private)
      (4) l3#(x, y, z) => l4#(x, y, z) | z > 0 (private)
      (5) l4#(x, y, z) => l3#(x, y, z - 1)     (private)
      (6) l4#(x, y, z) => l5#(x, y + 1, z)     (private)
      (7) l5#(x, y, z) => l1#(x, y)            (private)

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

We use the following theory arguments function, which fixes all public dependency pairs:

  l1#    : [1, 2]
  l2#    : [1, 2]
  l3#    : [1, 2, 3]
  l4#    : [1, 2, 3]
  l5#    : [1, 2, 3]
  start# : []

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

  P2. (1) start#(x) => l1#(x, 0)
      (2) l1#(x, y) => l2#(x, y) | x > y                   (private)
      (3) l2#(x, y) => l3#(x, y, y) { x, y }               (private)
      (4) l3#(x, y, z) => l4#(x, y, z) | z > 0 { x, y, z } (private)
      (5) l4#(x, y, z) => l3#(x, y, z - 1) { x, y, z }     (private)
      (6) l4#(x, y, z) => l5#(x, y + 1, z) { x, y, z }     (private)
      (7) l5#(x, y, z) => l1#(x, y) { x, y, z }            (private)

***** We apply the Graph Processor on D2 = (P2, R UNION R_?, f, 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: { D3 = (P3, R UNION R_?, f, c) }, where:

  P3. (1) l1#(x, y) => l2#(x, y) | x > y
      (2) l2#(x, y) => l3#(x, y, y) { x, y }
      (3) l3#(x, y, z) => l4#(x, y, z) | z > 0 { x, y, z }
      (4) l4#(x, y, z) => l3#(x, y, z - 1) { x, y, z }
      (5) l4#(x, y, z) => l5#(x, y + 1, z) { x, y, z }
      (6) l5#(x, y, z) => l1#(x, y) { x, y, z }

***** No progress could be made on DP problem D3 = (P3, R UNION R_?, f, c).