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

  Signature: condLoop :: Bool -> Int -> Int -> Int -> Int
             condMod  :: Bool -> Int -> Int -> Int -> Int
             halfExp  :: Int -> Int -> Int -> Int
             pow      :: Int -> Int -> Int
             sqBase   :: Int -> Int -> Int -> Int

  Rules: pow(b, e) -> condLoop(e > 0, b, e, 1)
         condLoop(false, b, e, r) -> r
         condLoop(true, b, e, r) -> condMod(e % 2 = 1, b, e, r)
         condMod(false, b, e, r) -> sqBase(b, e, r)
         condMod(true, b, e, r) -> sqBase(b, e, r * b)
         sqBase(b, e, r) -> halfExp(b * b, e, r)
         halfExp(b, e, r) -> condLoop(e > 0, b, e / 2, r)

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_?, i, c), where:

  P1. (1) pow#(b, e) => condLoop#(e > 0, b, e, 1)
      (2) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (3) condMod#(false, b, e, r) => sqBase#(b, e, r)
      (4) condMod#(true, b, e, r) => sqBase#(b, e, r * b)
      (5) sqBase#(b, e, r) => halfExp#(b * b, e, r)
      (6) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r)

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

We compute a graph approximation with the following edges:

  1: 2 
  2: 3 4 
  3: 5 
  4: 5 
  5: 6 
  6: 2 

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

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

  P2. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (2) condMod#(false, b, e, r) => sqBase#(b, e, r)
      (3) condMod#(true, b, e, r) => sqBase#(b, e, r * b)
      (4) sqBase#(b, e, r) => halfExp#(b * b, e, r)
      (5) halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r)

***** We apply the Chaining Processor Processor on D2 = (P2, R UNION R_?, i, c).

We chain DPs according to the following mapping:


  sqBase#(b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ true  is obtained by chaining  sqBase#(b, e, r) => halfExp#(b * b, e, r) and halfExp#(b', e', r') => condLoop#(e' > 0, b', e' / 2, r')


The following DPs were deleted:

sqBase#(b, e, r) => halfExp#(b * b, e, r)

halfExp#(b, e, r) => condLoop#(e > 0, b, e / 2, r)


By chaining, we added 1 DPs and removed 2 DPs.

Processor output: { D3 = (P3, R UNION R_?, i, c) }, where:

  P3. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (2) condMod#(false, b, e, r) => sqBase#(b, e, r)
      (3) condMod#(true, b, e, r) => sqBase#(b, e, r * b)
      (4) sqBase#(b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ true

***** We apply the Chaining Processor Processor on D3 = (P3, R UNION R_?, i, c).

We chain DPs according to the following mapping:


  condMod#(false, b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ (true /\ true)     is obtained by chaining  condMod#(false, b, e, r) => sqBase#(b, e, r)    and sqBase#(b', e', r') => condLoop#(e' > 0, b' * b', e' / 2, r') | true /\ true
  condMod#(true, b, e, r) => condLoop#(e > 0, b * b, e / 2, r * b) | true /\ (true /\ true)  is obtained by chaining  condMod#(true, b, e, r) => sqBase#(b, e, r * b) and sqBase#(b', e', r') => condLoop#(e' > 0, b' * b', e' / 2, r') | true /\ true


The following DPs were deleted:

condMod#(false, b, e, r) => sqBase#(b, e, r)

condMod#(true, b, e, r) => sqBase#(b, e, r * b)

sqBase#(b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ true


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

Processor output: { D4 = (P4, R UNION R_?, i, c) }, where:

  P4. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (2) condMod#(false, b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ (true /\ true)
      (3) condMod#(true, b, e, r) => condLoop#(e > 0, b * b, e / 2, r * b) | true /\ (true /\ true)

***** We apply the Theory Arguments Processor on D4 = (P4, R UNION R_?, i, c).

We use the following theory arguments function:

  condLoop# : [1, 2, 3, 4]
  condMod#  : [1, 2, 3, 4]

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

  P5. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)
      (2) condMod#(false, b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ (true /\ true) /\ b = b /\ e = e /\ r = r
      (3) condMod#(true, b, e, r) => condLoop#(e > 0, b * b, e / 2, r * b) | true /\ (true /\ true) /\ b = b /\ e = e /\ r = r

  P6. (1) condMod#(false, b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ (true /\ true)
      (2) condMod#(true, b, e, r) => condLoop#(e > 0, b * b, e / 2, r * b) | true /\ (true /\ true)

***** We apply the Theory Arguments Processor on D5 = (P5, R UNION R_?, i, c).

We use the following theory arguments function:

  condLoop# : [1, 2, 3, 4]
  condMod#  : [1, 2, 3, 4]

Processor output: { D7 = (P7, R UNION R_?, i, c) ; D8 = (P8, R UNION R_?, i, c) }, where:

  P7. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r) | b = b /\ e = e /\ r = r
      (2) condMod#(false, b, e, r) => condLoop#(e > 0, b * b, e / 2, r) | true /\ (true /\ true) /\ b = b /\ e = e /\ r = r
      (3) condMod#(true, b, e, r) => condLoop#(e > 0, b * b, e / 2, r * b) | true /\ (true /\ true) /\ b = b /\ e = e /\ r = r

  P8. (1) condLoop#(true, b, e, r) => condMod#(e % 2 = 1, b, e, r)

***** We apply the Graph Processor on D6 = (P6, R UNION 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 D7 = (P7, R UNION R_?, i, c).

We obtain 0 usable rules (out of 7 rules in the input problem).

Processor output: { D9 = (P7, {}, i, c) }.

***** We apply the Graph Processor on D8 = (P8, R UNION 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 D9 = (P7, {}, i, c).