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

  Signature: cond  :: Bool -> 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) = y, x, y)
         cond(true, 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) = y, x, y)
      (3) cond#(true, 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: 1 2 
  4:

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

Processor output: { }.