Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivMinus2

public class DivMinus2 {
  public static int div(int x, int y) {
    int res = 0;
    while (x >= y && y > 0) {
      x = minus(x,y);
      res = res + 1;
    }
    return res;
  }

  public static int minus(int x, int y) {
    while (y != 0) {
      if (y > 0)  {
        y--;
        x--;
      } else  {
        y++;
        x++;
      }
    }
    return x;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    div(x, y);
  }
}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
DivMinus2.main([Ljava/lang/String;)V: Graph of 213 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: DivMinus2.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 31 rules for P and 0 rules for R.

P rules:
3395_0_div_Load(EOS(STATIC_3395), i1383, i1382, i1383, i1382) → 3396_0_div_LT(EOS(STATIC_3396), i1383, i1382, i1383, i1382, i1383)
3396_0_div_LT(EOS(STATIC_3396), i1383, i1382, i1383, i1382, i1383) → 3399_0_div_LT(EOS(STATIC_3399), i1383, i1382, i1383, i1382, i1383)
3399_0_div_LT(EOS(STATIC_3399), i1383, i1382, i1383, i1382, i1383) → 3401_0_div_Load(EOS(STATIC_3401), i1383, i1382, i1383) | >=(i1382, i1383)
3401_0_div_Load(EOS(STATIC_3401), i1383, i1382, i1383) → 3404_0_div_LE(EOS(STATIC_3404), i1383, i1382, i1383, i1383)
3404_0_div_LE(EOS(STATIC_3404), i1393, i1382, i1393, i1393) → 3408_0_div_LE(EOS(STATIC_3408), i1393, i1382, i1393, i1393)
3408_0_div_LE(EOS(STATIC_3408), i1393, i1382, i1393, i1393) → 3412_0_div_Load(EOS(STATIC_3412), i1393, i1382, i1393) | >(i1393, 0)
3412_0_div_Load(EOS(STATIC_3412), i1393, i1382, i1393) → 3416_0_div_Load(EOS(STATIC_3416), i1393, i1393, i1382)
3416_0_div_Load(EOS(STATIC_3416), i1393, i1393, i1382) → 3419_0_div_InvokeMethod(EOS(STATIC_3419), i1393, i1393, i1382, i1393)
3419_0_div_InvokeMethod(EOS(STATIC_3419), i1393, i1393, i1382, i1393) → 3421_0_minus_Load(EOS(STATIC_3421), i1393, i1393, i1382, i1393, i1382, i1393)
3421_0_minus_Load(EOS(STATIC_3421), i1393, i1393, i1382, i1393, i1382, i1393) → 3434_0_minus_Load(EOS(STATIC_3434), i1393, i1393, i1382, i1393, i1382, i1393)
3434_0_minus_Load(EOS(STATIC_3434), i1393, i1393, i1382, i1393, i1397, i1398) → 3436_0_minus_EQ(EOS(STATIC_3436), i1393, i1393, i1382, i1393, i1397, i1398, i1398)
3436_0_minus_EQ(EOS(STATIC_3436), i1393, i1393, i1382, i1393, i1397, i1404, i1404) → 3437_0_minus_EQ(EOS(STATIC_3437), i1393, i1393, i1382, i1393, i1397, i1404, i1404)
3436_0_minus_EQ(EOS(STATIC_3436), i1393, i1393, i1382, i1393, i1397, matching1, matching2) → 3438_0_minus_EQ(EOS(STATIC_3438), i1393, i1393, i1382, i1393, i1397, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
3437_0_minus_EQ(EOS(STATIC_3437), i1393, i1393, i1382, i1393, i1397, i1404, i1404) → 3440_0_minus_Load(EOS(STATIC_3440), i1393, i1393, i1382, i1393, i1397, i1404) | >(i1404, 0)
3440_0_minus_Load(EOS(STATIC_3440), i1393, i1393, i1382, i1393, i1397, i1404) → 3442_0_minus_LE(EOS(STATIC_3442), i1393, i1393, i1382, i1393, i1397, i1404, i1404)
3442_0_minus_LE(EOS(STATIC_3442), i1393, i1393, i1382, i1393, i1397, i1404, i1404) → 3445_0_minus_Inc(EOS(STATIC_3445), i1393, i1393, i1382, i1393, i1397, i1404) | >(i1404, 0)
3445_0_minus_Inc(EOS(STATIC_3445), i1393, i1393, i1382, i1393, i1397, i1404) → 3447_0_minus_Inc(EOS(STATIC_3447), i1393, i1393, i1382, i1393, i1397, +(i1404, -1)) | >(i1404, 0)
3447_0_minus_Inc(EOS(STATIC_3447), i1393, i1393, i1382, i1393, i1397, i1406) → 3450_0_minus_JMP(EOS(STATIC_3450), i1393, i1393, i1382, i1393, +(i1397, -1), i1406)
3450_0_minus_JMP(EOS(STATIC_3450), i1393, i1393, i1382, i1393, i1407, i1406) → 3453_0_minus_Load(EOS(STATIC_3453), i1393, i1393, i1382, i1393, i1407, i1406)
3453_0_minus_Load(EOS(STATIC_3453), i1393, i1393, i1382, i1393, i1407, i1406) → 3434_0_minus_Load(EOS(STATIC_3434), i1393, i1393, i1382, i1393, i1407, i1406)
3438_0_minus_EQ(EOS(STATIC_3438), i1393, i1393, i1382, i1393, i1397, matching1, matching2) → 3441_0_minus_Load(EOS(STATIC_3441), i1393, i1393, i1382, i1393, i1397) | &&(=(matching1, 0), =(matching2, 0))
3441_0_minus_Load(EOS(STATIC_3441), i1393, i1393, i1382, i1393, i1397) → 3444_0_minus_Return(EOS(STATIC_3444), i1393, i1393, i1382, i1393, i1397)
3444_0_minus_Return(EOS(STATIC_3444), i1393, i1393, i1382, i1393, i1397) → 3446_0_div_Store(EOS(STATIC_3446), i1393, i1393, i1397)
3446_0_div_Store(EOS(STATIC_3446), i1393, i1393, i1397) → 3449_0_div_Load(EOS(STATIC_3449), i1393, i1397, i1393)
3449_0_div_Load(EOS(STATIC_3449), i1393, i1397, i1393) → 3451_0_div_ConstantStackPush(EOS(STATIC_3451), i1393, i1397, i1393)
3451_0_div_ConstantStackPush(EOS(STATIC_3451), i1393, i1397, i1393) → 3455_0_div_IntArithmetic(EOS(STATIC_3455), i1393, i1397, i1393)
3455_0_div_IntArithmetic(EOS(STATIC_3455), i1393, i1397, i1393) → 3548_0_div_Store(EOS(STATIC_3548), i1393, i1397, i1393)
3548_0_div_Store(EOS(STATIC_3548), i1393, i1397, i1393) → 3549_0_div_JMP(EOS(STATIC_3549), i1393, i1397, i1393)
3549_0_div_JMP(EOS(STATIC_3549), i1393, i1397, i1393) → 3551_0_div_Load(EOS(STATIC_3551), i1393, i1397, i1393)
3551_0_div_Load(EOS(STATIC_3551), i1393, i1397, i1393) → 3393_0_div_Load(EOS(STATIC_3393), i1393, i1397, i1393)
3393_0_div_Load(EOS(STATIC_3393), i1383, i1382, i1383) → 3395_0_div_Load(EOS(STATIC_3395), i1383, i1382, i1383, i1382)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
3436_0_minus_EQ(EOS(STATIC_3436), x0, x0, x1, x0, x2, x3, x3) → 3436_0_minus_EQ(EOS(STATIC_3436), x0, x0, x1, x0, +(x2, -1), +(x3, -1), +(x3, -1)) | >(x3, 0)
3436_0_minus_EQ(EOS(STATIC_3436), x0, x0, x1, x0, x2, 0, 0) → 3436_0_minus_EQ(EOS(STATIC_3436), x0, x0, x2, x0, x2, x0, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

3436_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → 3436_0_minus_EQ(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_3436_0_minus_EQ1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_3436_0_minus_EQ1(x1, x3, x4, x5, x6, x7)
Cond_3436_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_3436_0_minus_EQ(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

3436_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7) → 3436_0_minus_EQ(x3, x4, x5, x7)
Cond_3436_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_3436_0_minus_EQ(x1, x4, x5, x6, x8)
Cond_3436_0_minus_EQ1(x1, x2, x3, x4, x5, x6) → Cond_3436_0_minus_EQ1(x1, x4, x5, x6)

Filtered unneeded arguments:

Cond_3436_0_minus_EQ(x1, x2, x3, x4, x5) → Cond_3436_0_minus_EQ(x1, x3, x4, x5)
3436_0_minus_EQ(x1, x2, x3, x4) → 3436_0_minus_EQ(x2, x3, x4)
Cond_3436_0_minus_EQ1(x1, x2, x3, x4) → Cond_3436_0_minus_EQ1(x1, x3, x4)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
3436_0_minus_EQ(x0, x2, x3) → 3436_0_minus_EQ(x0, +(x2, -1), +(x3, -1)) | >(x3, 0)
3436_0_minus_EQ(x0, x2, 0) → 3436_0_minus_EQ(x0, x2, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
3436_0_MINUS_EQ(x0, x2, x3) → COND_3436_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
COND_3436_0_MINUS_EQ(TRUE, x0, x2, x3) → 3436_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
3436_0_MINUS_EQ(x0, x2, 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
COND_3436_0_MINUS_EQ1(TRUE, x0, x2, 0) → 3436_0_MINUS_EQ(x0, x2, x0)
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
||	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(2): 3436_0_MINUS_EQ(x0[2], x2[2], 0) → COND_3436_0_MINUS_EQ1(x2[2] >= x0[2] && x0[2] > 0, x0[2], x2[2], 0)
(3): COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 3436_0_MINUS_EQ(x0[3], x2[3], x0[3])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(1) -> (2), if (x0[1] →^* x0[2]∧x2[1] + -1 →^* x2[2]∧x3[1] + -1 →^* 0)

(2) -> (3), if (x2[2] >= x0[2] && x0[2] > 0 ∧x0[2] →^* x0[3]∧x2[2] →^* x2[3])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x2[0]∧x0[3] →^* x3[0])

(3) -> (2), if (x0[3] →^* x0[2]∧x2[3] →^* x2[2]∧x0[3] →^* 0)

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@459af028 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair 3436_0_MINUS_EQ(x0, x2, x3) → COND_3436_0_MINUS_EQ(>(x3, 0), x0, x2, x3) the following chains were created:

• We consider the chain 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
  

  (1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

  
  
  We simplified constraint (1) using rule (IV) which results in the following new constraint:
  

  (2)    (>(x3[0], 0)=TRUE ⇒ 3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

  
  
  We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [(2)bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [(2)bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [(2)bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(2)bni_20] = 0∧0 = 0∧[(2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  
  
  We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (7)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(2)bni_20] = 0∧0 = 0∧[bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  
  
  

For Pair COND_3436_0_MINUS_EQ(TRUE, x0, x2, x3) → 3436_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1)) the following chains were created:

• We consider the chain 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
  

  (8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

  
  
  We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
  

  (9)    (>(x3[0], 0)=TRUE ⇒ COND_3436_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_3436_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥3436_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

  
  
  We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [(2)bni_22]x2[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [(2)bni_22]x2[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [(2)bni_22]x2[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22] = 0∧0 = 0∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (14)    (x3[0] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22] = 0∧0 = 0∧[bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  
• We consider the chain 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 3436_0_MINUS_EQ(x0[2], x2[2], 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0) which results in the following constraint:
  

  (15)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[2]∧+(x2[1], -1)=x2[2]∧+(x3[1], -1)=0 ⇒ COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

  
  
  We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
  

  (16)    (>(x3[0], 0)=TRUE∧+(x3[0], -1)=0 ⇒ COND_3436_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_3436_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥3436_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

  
  
  We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (17)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [(2)bni_22]x2[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (18)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [(2)bni_22]x2[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (19)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [(2)bni_22]x2[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (20)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22] = 0∧0 = 0∧[(2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (21)    (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22] = 0∧0 = 0∧[bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

  
  
  

For Pair 3436_0_MINUS_EQ(x0, x2, 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0) the following chains were created:

• We consider the chain 3436_0_MINUS_EQ(x0[2], x2[2], 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0), COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 3436_0_MINUS_EQ(x0[3], x2[3], x0[3]) which results in the following constraint:
  

  (22)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x0[2]=x0[3]∧x2[2]=x2[3] ⇒ 3436_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧3436_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥))

  
  
  We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (23)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ 3436_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧3436_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥))

  
  
  We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (24)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

  
  
  We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (25)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

  
  
  We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (26)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

  
  
  We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (27)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [(2)bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

  
  
  We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (28)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(4)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [(2)bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)

  
  
  

For Pair COND_3436_0_MINUS_EQ1(TRUE, x0, x2, 0) → 3436_0_MINUS_EQ(x0, x2, x0) the following chains were created:

• We consider the chain COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 3436_0_MINUS_EQ(x0[3], x2[3], x0[3]), 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
  

  (29)    (x0[3]=x0[0]∧x2[3]=x2[0]∧x0[3]=x3[0] ⇒ COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥3436_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

  
  
  We simplified constraint (29) using rule (IV) which results in the following new constraint:
  

  (30)    (COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥3436_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

  
  
  We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (31)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

  
  
  We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (32)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

  
  
  We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (33)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

  
  
  We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (34)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

  
  
  
• We consider the chain COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 3436_0_MINUS_EQ(x0[3], x2[3], x0[3]), 3436_0_MINUS_EQ(x0[2], x2[2], 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0) which results in the following constraint:
  

  (35)    (x0[3]=x0[2]∧x2[3]=x2[2]∧x0[3]=0 ⇒ COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥3436_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

  
  
  We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:
  

  (36)    (COND_3436_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥NonInfC∧COND_3436_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥3436_0_MINUS_EQ(0, x2[3], 0)∧(U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

  
  
  We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (37)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

  
  
  We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (38)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

  
  
  We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (39)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

  
  
  We simplified constraint (39) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (40)    ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• 3436_0_MINUS_EQ(x0, x2, x3) → COND_3436_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
  
  • (x3[0] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(2)bni_20] = 0∧0 = 0∧[bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
    
  
  
• COND_3436_0_MINUS_EQ(TRUE, x0, x2, x3) → 3436_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
  
  • (x3[0] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22] = 0∧0 = 0∧[bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)
    
  • (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(2)bni_22] = 0∧0 = 0∧[bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)
    
  
  
• 3436_0_MINUS_EQ(x0, x2, 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
  
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(4)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [(2)bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)
    
  
  
• COND_3436_0_MINUS_EQ1(TRUE, x0, x2, 0) → 3436_0_MINUS_EQ(x0, x2, x0)
  
  • ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)
    
  • ((U^Increasing(3436_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧[(-1)bso_27] ≥ 0)
    
  
  

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [1]   
POL(FALSE) = 0   
POL(3436_0_MINUS_EQ(x₁, x₂, x₃)) = [2] + [-1]x₃ + [2]x₂   
POL(COND_3436_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [2] + [-1]x₄ + [2]x₃   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(COND_3436_0_MINUS_EQ1(x₁, x₂, x₃, x₄)) = [-1]x₄ + [2]x₃ + [-1]x₂ + [2]x₁   
POL(&&(x₁, x₂)) = [1]   
POL(>=(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
3436_0_MINUS_EQ(x0[2], x2[2], 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

The following pairs are in P_bound:

3436_0_MINUS_EQ(x0[2], x2[2], 0) → COND_3436_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

The following pairs are in P_≥:

3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 3436_0_MINUS_EQ(x0[3], x2[3], x0[3])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(TRUE, TRUE)¹ ↔ TRUE¹
&&(TRUE, FALSE)¹ → FALSE¹
&&(FALSE, TRUE)¹ → FALSE¹
&&(FALSE, FALSE)¹ → FALSE¹

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
||	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(3): COND_3436_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 3436_0_MINUS_EQ(x0[3], x2[3], x0[3])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x2[0]∧x0[3] →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
||	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(0): 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@459af028 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) the following chains were created:

• We consider the chain 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
  

  (1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

  
  
  We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
  

  (2)    (>(x3[0], 0)=TRUE ⇒ COND_3436_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_3436_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥3436_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

  
  
  We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

  
  
  We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

  
  
  We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

  
  
  We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

  
  
  We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (7)    (x3[0] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

  
  
  

For Pair 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) the following chains were created:

• We consider the chain 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
  

  (8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

  
  
  We simplified constraint (8) using rule (IV) which results in the following new constraint:
  

  (9)    (>(x3[0], 0)=TRUE ⇒ 3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧3436_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

  
  
  We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  
  
  We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  
  
  We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  
  
  We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  
  
  We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (14)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
  
  • (x3[0] ≥ 0 ⇒ (U^Increasing(3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)
    
  
  
• 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
  
  • (x3[0] ≥ 0 ⇒ (U^Increasing(COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)
    
  
  

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_3436_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + x₄   
POL(3436_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + x₃   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   

The following pairs are in P_>:

COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))

The following pairs are in P_bound:

COND_3436_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 3436_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

The following pairs are in P_≥:

3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

There are no usable rules.

(12) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
||	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 3436_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_3436_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(14) TRUE