We consider termination of the LCTRS with only rule scheme Calc:

  Signature: if       :: Bool -> Int -> Int -> Int
             minus    :: Int -> Int -> o
             minusNat :: Bool -> Int -> Int -> o
             round    :: Int -> Int

  Rules: minus(x, y) -> minusNat(y >= 0 /\ x = y + 1, x, y)
         minusNat(true, x, y) -> minus(x, round(y))
         round(x) -> if(x % 2 = 0, x, x + 1)
         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, i, c), where:

  P1. (1) minus#(x, y) => minusNat#(y >= 0 /\ x = y + 1, x, y)
      (2) minusNat#(true, x, y) => round#(y)
      (3) minusNat#(true, x, y) => minus#(x, round(y))
      (4) round#(x) => if#(x % 2 = 0, x, x + 1)

***** We apply the Graph Processor on D1 = (P1, R, i, c).

We compute a graph approximation with the following edges:

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

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

Processor output: { D2 = (P2, R, i, c) }, where:

  P2. (1) minus#(x, y) => minusNat#(y >= 0 /\ x = y + 1, x, y)
      (2) minusNat#(true, x, y) => minus#(x, round(y))

***** We apply the Theory Arguments Processor on D2 = (P2, R, i, c).

We use the following theory arguments function:

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

Processor output: { D3 = (P3, R, i, c) ; D4 = (P4, R, i, c) }, where:

  P3. (1) minus#(x, y) => minusNat#(y >= 0 /\ x = y + 1, x, y)
      (2) minusNat#(true, x, y) => minus#(x, round(y)) | x = x /\ y = y

  P4. (1) minusNat#(true, x, y) => minus#(x, round(y))

***** We apply the Usable Rules Processor on D3 = (P3, R, i, c).

We obtain 3 usable rules (out of 5 rules in the input problem).

Processor output: { D5 = (P3, R2, i, c) }, where:

  R2. (1) if(true, u, v) -> u
      (2) if(false, u, v) -> v
      (3) round(x) -> if(x % 2 = 0, x, x + 1)

***** We apply the Graph Processor on D4 = (P4, R, i, 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 = (P3, R2, i, c).