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

  Signature: f     :: Bool -> Int -> Int -> o
             fNat  :: Bool -> Int -> Int -> o
             if    :: Bool -> Int -> Int -> Int
             trunc :: Int -> Int

  Rules: f(true, x, y) -> fNat(x >= 0 /\ y >= 0, x, y)
         fNat(true, x, y) -> f(x > y, trunc(x), y + 1)
         trunc(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) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)
      (2) fNat#(true, x, y) => trunc#(x)
      (3) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1)
      (4) trunc#(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) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)
      (2) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1)

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

We use the following theory arguments function:

  f#    : [3]
  fNat# : [1, 2, 3]

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

  P3. (1) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)
      (2) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1) | x = x /\ y = y

  P4. (1) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1)

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

We use the following theory arguments function:

  f#    : [3]
  fNat# : [1, 2, 3]

Processor output: { D5 = (P5, R, i, c) ; D6 = (P6, R, i, c) }, where:

  P5. (1) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y) | y = y
      (2) fNat#(true, x, y) => f#(x > y, trunc(x), y + 1) | x = x /\ y = y

  P6. (1) f#(true, x, y) => fNat#(x >= 0 /\ y >= 0, x, y)

***** 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: { }.

***** We apply the Usable Rules Processor on D5 = (P5, R, i, c).

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

Processor output: { D7 = (P5, R2, i, c) }, where:

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

***** We apply the Graph Processor on D6 = (P6, 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 D7 = (P5, R2, i, c).