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

  Signature: cond   :: Bool -> Int -> Int -> Int -> Int
             d      :: Int -> Int -> Int -> Int
             dNat   :: Bool -> Int -> Int -> Int -> Int
             div    :: Int -> Int -> Int
             divNat :: Bool -> Int -> Int -> Int

  Rules: div(x, y) -> divNat(x >= 0 /\ y >= 1, x, y)
         divNat(true, x, y) -> d(x, y, 0)
         d(x, y, z) -> dNat(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
         dNat(true, x, y, z) -> cond(x >= z, x, y - 1, z)
         cond(true, x, y, z) -> 1 + d(x, y + 1, y + 1 + z)
         cond(false, x, y, z) -> 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) div#(x, y) => divNat#(x >= 0 /\ y >= 1, x, y)
      (2) divNat#(true, x, y) => d#(x, y, 0)
      (3) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
      (4) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z)
      (5) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z)

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

We compute a graph approximation with the following edges:

  1: 2 
  2: 3 
  3: 4 
  4: 5 
  5: 3 

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

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

  P2. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
      (2) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z)
      (3) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z)

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

We use the following theory arguments function:

  cond# : [1, 2, 3, 4]
  d#    : [1, 2, 3]
  dNat# : [1, 2, 3, 4]

Processor output: { D3 = (P3, R UNION R_?, f, c) ; D4 = (P4, R UNION R_?, f, c) }, where:

  P3. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)
      (2) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z) { x, y, z }
      (3) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z) { x, y, z }

  P4. (1) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z)
      (2) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z)

***** We apply the Theory Arguments Processor on D3 = (P3, R UNION R_?, f, c).

We use the following theory arguments function:

  cond# : [1, 2, 3, 4]
  d#    : [1, 2, 3]
  dNat# : [1, 2, 3, 4]

Processor output: { D5 = (P5, R UNION R_?, f, c) ; D6 = (P6, R UNION R_?, f, c) }, where:

  P5. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z) { x, y, z }
      (2) dNat#(true, x, y, z) => cond#(x >= z, x, y - 1, z) { x, y, z }
      (3) cond#(true, x, y, z) => d#(x, y + 1, y + 1 + z) { x, y, z }

  P6. (1) d#(x, y, z) => dNat#(x >= 0 /\ y >= 1 /\ z >= 0, x, y, z)

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

We compute a graph approximation with the following edges:

  1: 2 
  2:

As there are no SCCs, this DP problem is removed.

Processor output: { }.

***** No progress could be made on DP problem D5 = (P5, R UNION R_?, f, c).