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

  Signature: cond1 :: Bool -> Int -> Int -> Int
             cond2 :: Bool -> Int -> Int -> Int
             diff  :: Int -> Int -> Int

  Rules: diff(x, y) -> cond1(x = y, x, y)
         cond1(true, x, y) -> 0
         cond1(false, x, y) -> cond2(x > y, x, y)
         cond2(true, x, y) -> 1 + diff(x, y + 1)
         cond2(false, x, y) -> 1 + diff(x + 1, y)

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) diff#(x, y) => cond1#(x = y, x, y)
      (2) cond1#(false, x, y) => cond2#(x > y, x, y)
      (3) cond2#(true, x, y) => diff#(x, y + 1)
      (4) cond2#(false, x, y) => diff#(x + 1, y)

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

We chain DPs according to the following mapping:


  cond2#(true, x, y) => cond1#(x = y + 1, x, y + 1) | true /\ true   is obtained by chaining  cond2#(true, x, y) => diff#(x, y + 1)  and diff#(x', y') => cond1#(x' = y', x', y')
  cond2#(false, x, y) => cond1#(x + 1 = y, x + 1, y) | true /\ true  is obtained by chaining  cond2#(false, x, y) => diff#(x + 1, y) and diff#(x', y') => cond1#(x' = y', x', y')


The following DPs were deleted:

cond2#(true, x, y) => diff#(x, y + 1)

cond2#(false, x, y) => diff#(x + 1, y)

diff#(x, y) => cond1#(x = y, x, y)


By chaining, we added 2 DPs and removed 3 DPs.

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

  P2. (1) cond1#(false, x, y) => cond2#(x > y, x, y)
      (2) cond2#(true, x, y) => cond1#(x = y + 1, x, y + 1) | true /\ true
      (3) cond2#(false, x, y) => cond1#(x + 1 = y, x + 1, y) | true /\ true

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

We use the following theory arguments function:

  cond1# : [1, 2, 3]
  cond2# : [1, 2, 3]

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

  P3. (1) cond1#(false, x, y) => cond2#(x > y, x, y)
      (2) cond2#(true, x, y) => cond1#(x = y + 1, x, y + 1) | true /\ true /\ x = x /\ y = y
      (3) cond2#(false, x, y) => cond1#(x + 1 = y, x + 1, y) | true /\ true /\ x = x /\ y = y

  P4. (1) cond2#(true, x, y) => cond1#(x = y + 1, x, y + 1) | true /\ true
      (2) cond2#(false, x, y) => cond1#(x + 1 = y, x + 1, y) | true /\ true

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

We use the following theory arguments function:

  cond1# : [1, 2, 3]
  cond2# : [1, 2, 3]

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

  P5. (1) cond1#(false, x, y) => cond2#(x > y, x, y) | x = x /\ y = y
      (2) cond2#(true, x, y) => cond1#(x = y + 1, x, y + 1) | true /\ true /\ x = x /\ y = y
      (3) cond2#(false, x, y) => cond1#(x + 1 = y, x + 1, y) | true /\ true /\ x = x /\ y = y

  P6. (1) cond1#(false, x, y) => cond2#(x > y, x, y)

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

We compute a graph approximation with the following edges:

  1:
  2:

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 0 usable rules (out of 5 rules in the input problem).

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

***** 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, {}, i, c).