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

  Signature: cond  :: Int -> Int -> Int -> Int
             if    :: Bool -> Int -> Int -> Int
             min   :: Int -> Int -> Int
             minus :: Int -> Int -> Int

  Rules: minus(x, x) -> 0
         minus(x, y) -> cond(min(x, y), x, y)
         cond(y, x, y) -> 1 + minus(x, y + 1)
         min(u, v) -> if(u < v, u, v)
         if(true, u, v) -> u
         if(false, u, v) -> v

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) minus#(x, y) => min#(x, y)
      (2) minus#(x, y) => cond#(min(x, y), x, y)
      (3) cond#(y, x, y) => minus#(x, y + 1)
      (4) min#(u, v) => if#(u < v, u, v)

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

We compute a graph approximation with the following edges:

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

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) minus#(x, y) => cond#(min(x, y), x, y)
      (2) cond#(y, x, y) => minus#(x, y + 1)

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

We use the following theory arguments function:

  cond#  : [2, 3]
  minus# : [1, 2]

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

  P3. (1) minus#(x, y) => cond#(min(x, y), x, y)
      (2) cond#(y, x, y) => minus#(x, y + 1) { y, x }

  P4. (1) cond#(y, x, y) => minus#(x, y + 1)

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

We use the following theory arguments function:

  cond#  : [2, 3]
  minus# : [1, 2]

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

  P5. (1) minus#(x, y) => cond#(min(x, y), x, y) { x, y }
      (2) cond#(y, x, y) => minus#(x, y + 1) { y, x }

  P6. (1) minus#(x, y) => cond#(min(x, y), x, y)

***** We apply the Graph Processor on D4 = (P4, 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 D5 = (P5, R UNION R_?, f, c).