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: GCD4

public class GCD4 {
  public static int mod(int a, int b) {
    while(a>=b && b > 0) {
      a -= b;
    }
    return a;
  }

  public static int gcd(int a, int b) {
    int tmp;
    while(b > 0 && a > 0) {
      tmp = b;
      b = mod(a, b);
      a = tmp;
    }
    return a;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    gcd(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:
GCD4.main([Ljava/lang/String;)V: Graph of 212 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: GCD4.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 33 rules for P and 0 rules for R.

P rules:
630_0_gcd_LE(EOS(STATIC_630), i85, i94, i94) → 634_0_gcd_LE(EOS(STATIC_634), i85, i94, i94)
634_0_gcd_LE(EOS(STATIC_634), i85, i94, i94) → 638_0_gcd_Load(EOS(STATIC_638), i85, i94) | >(i94, 0)
638_0_gcd_Load(EOS(STATIC_638), i85, i94) → 642_0_gcd_LE(EOS(STATIC_642), i85, i94, i85)
642_0_gcd_LE(EOS(STATIC_642), i97, i94, i97) → 647_0_gcd_LE(EOS(STATIC_647), i97, i94, i97)
647_0_gcd_LE(EOS(STATIC_647), i97, i94, i97) → 654_0_gcd_Load(EOS(STATIC_654), i97, i94) | >(i97, 0)
654_0_gcd_Load(EOS(STATIC_654), i97, i94) → 660_0_gcd_Store(EOS(STATIC_660), i97, i94, i94)
660_0_gcd_Store(EOS(STATIC_660), i97, i94, i94) → 665_0_gcd_Load(EOS(STATIC_665), i97, i94, i94)
665_0_gcd_Load(EOS(STATIC_665), i97, i94, i94) → 670_0_gcd_Load(EOS(STATIC_670), i94, i94, i97)
670_0_gcd_Load(EOS(STATIC_670), i94, i94, i97) → 674_0_gcd_InvokeMethod(EOS(STATIC_674), i94, i97, i94)
674_0_gcd_InvokeMethod(EOS(STATIC_674), i94, i97, i94) → 676_0_mod_Load(EOS(STATIC_676), i94, i97, i94, i97, i94)
676_0_mod_Load(EOS(STATIC_676), i94, i97, i94, i97, i94) → 722_0_mod_Load(EOS(STATIC_722), i94, i97, i94, i97, i94)
722_0_mod_Load(EOS(STATIC_722), i94, i97, i94, i106, i94) → 725_0_mod_Load(EOS(STATIC_725), i94, i97, i94, i106, i94, i106)
725_0_mod_Load(EOS(STATIC_725), i94, i97, i94, i106, i94, i106) → 727_0_mod_LT(EOS(STATIC_727), i94, i97, i94, i106, i94, i106, i94)
727_0_mod_LT(EOS(STATIC_727), i94, i97, i94, i106, i94, i106, i94) → 729_0_mod_LT(EOS(STATIC_729), i94, i97, i94, i106, i94, i106, i94)
727_0_mod_LT(EOS(STATIC_727), i94, i97, i94, i106, i94, i106, i94) → 731_0_mod_LT(EOS(STATIC_731), i94, i97, i94, i106, i94, i106, i94)
729_0_mod_LT(EOS(STATIC_729), i94, i97, i94, i106, i94, i106, i94) → 733_0_mod_Load(EOS(STATIC_733), i94, i97, i94, i106) | <(i106, i94)
733_0_mod_Load(EOS(STATIC_733), i94, i97, i94, i106) → 737_0_mod_Return(EOS(STATIC_737), i94, i97, i94, i106)
737_0_mod_Return(EOS(STATIC_737), i94, i97, i94, i106) → 740_0_gcd_Store(EOS(STATIC_740), i94, i106)
740_0_gcd_Store(EOS(STATIC_740), i94, i106) → 744_0_gcd_Load(EOS(STATIC_744), i106, i94)
744_0_gcd_Load(EOS(STATIC_744), i106, i94) → 748_0_gcd_Store(EOS(STATIC_748), i106, i94)
748_0_gcd_Store(EOS(STATIC_748), i106, i94) → 752_0_gcd_JMP(EOS(STATIC_752), i94, i106)
752_0_gcd_JMP(EOS(STATIC_752), i94, i106) → 758_0_gcd_Load(EOS(STATIC_758), i94, i106)
758_0_gcd_Load(EOS(STATIC_758), i94, i106) → 627_0_gcd_Load(EOS(STATIC_627), i94, i106)
627_0_gcd_Load(EOS(STATIC_627), i85, i87) → 630_0_gcd_LE(EOS(STATIC_630), i85, i87, i87)
731_0_mod_LT(EOS(STATIC_731), i94, i97, i94, i106, i94, i106, i94) → 735_0_mod_Load(EOS(STATIC_735), i94, i97, i94, i106, i94) | >=(i106, i94)
735_0_mod_Load(EOS(STATIC_735), i94, i97, i94, i106, i94) → 738_0_mod_LE(EOS(STATIC_738), i94, i97, i94, i106, i94, i94)
738_0_mod_LE(EOS(STATIC_738), i94, i97, i94, i106, i94, i94) → 742_0_mod_Load(EOS(STATIC_742), i94, i97, i94, i106, i94) | >(i94, 0)
742_0_mod_Load(EOS(STATIC_742), i94, i97, i94, i106, i94) → 746_0_mod_Load(EOS(STATIC_746), i94, i97, i94, i94, i106)
746_0_mod_Load(EOS(STATIC_746), i94, i97, i94, i94, i106) → 750_0_mod_IntArithmetic(EOS(STATIC_750), i94, i97, i94, i94, i106, i94)
750_0_mod_IntArithmetic(EOS(STATIC_750), i94, i97, i94, i94, i106, i94) → 754_0_mod_Store(EOS(STATIC_754), i94, i97, i94, i94, -(i106, i94)) | >(i94, 0)
754_0_mod_Store(EOS(STATIC_754), i94, i97, i94, i94, i108) → 760_0_mod_JMP(EOS(STATIC_760), i94, i97, i94, i108, i94)
760_0_mod_JMP(EOS(STATIC_760), i94, i97, i94, i108, i94) → 764_0_mod_Load(EOS(STATIC_764), i94, i97, i94, i108, i94)
764_0_mod_Load(EOS(STATIC_764), i94, i97, i94, i108, i94) → 722_0_mod_Load(EOS(STATIC_722), i94, i97, i94, i108, i94)
R rules:

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

P rules:
727_0_mod_LT(EOS(STATIC_727), x0, x1, x0, x2, x0, x2, x0) → 727_0_mod_LT(EOS(STATIC_727), x2, x0, x2, x0, x2, x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
727_0_mod_LT(EOS(STATIC_727), x0, x1, x0, x2, x0, x2, x0) → 727_0_mod_LT(EOS(STATIC_727), x0, x1, x0, -(x2, x0), x0, -(x2, x0), x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → 727_0_mod_LT(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_727_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_727_0_mod_LT1(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_727_0_mod_LT(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7) → 727_0_mod_LT(x2, x6, x7)
Cond_727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_727_0_mod_LT(x1, x3, x7, x8)
Cond_727_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_727_0_mod_LT1(x1, x3, x7, x8)

Filtered unneeded arguments:

Cond_727_0_mod_LT(x1, x2, x3, x4) → Cond_727_0_mod_LT(x1, x3, x4)
Cond_727_0_mod_LT1(x1, x2, x3, x4) → Cond_727_0_mod_LT1(x1, x3, x4)
727_0_mod_LT(x1, x2, x3) → 727_0_mod_LT(x2, x3)

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

P rules:
727_0_mod_LT(x2, x0) → 727_0_mod_LT(x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
727_0_mod_LT(x2, x0) → 727_0_mod_LT(-(x2, x0), 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:
727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
COND_727_0_MOD_LT(TRUE, x2, x0) → 727_0_MOD_LT(x0, x2)
727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
COND_727_0_MOD_LT1(TRUE, x2, x0) → 727_0_MOD_LT(-(x2, x0), 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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
(2): 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])
(3): COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(x2[3] - x0[3], x0[3])

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

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

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

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

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

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

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@72865d61 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 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0) the following chains were created:

• We consider the chain 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]) which results in the following constraint:
  

  (1)    (&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUE∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

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

  (2)    (>(x0[0], 0)=TRUE∧>(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (5)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (6)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (7)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

  
  
  We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (8)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

  
  
  

For Pair COND_727_0_MOD_LT(TRUE, x2, x0) → 727_0_MOD_LT(x0, x2) the following chains were created:

• We consider the chain COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:
  

  (9)    (x0[1]=x2[0]∧x2[1]=x0[0] ⇒ COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥))

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

  (10)    (COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥))

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

  (11)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

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

  (12)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

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

  (13)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

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

  (14)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  
  
  
• We consider the chain COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:
  

  (15)    (x0[1]=x2[2]∧x2[1]=x0[2] ⇒ COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥))

  
  
  We simplified constraint (15) using rule (IV) which results in the following new constraint:
  

  (16)    (COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥))

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

  (17)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

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

  (18)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

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

  (19)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

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

  (20)    ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  
  
  

For Pair 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0) the following chains were created:

• We consider the chain 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3]) which results in the following constraint:
  

  (21)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 727_0_MOD_LT(x2[2], x0[2])≥NonInfC∧727_0_MOD_LT(x2[2], x0[2])≥COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))

  
  
  We simplified constraint (21) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (22)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ 727_0_MOD_LT(x2[2], x0[2])≥NonInfC∧727_0_MOD_LT(x2[2], x0[2])≥COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))

  
  
  We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (23)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (24)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (25)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (26)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (27)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)Bound*bni_22 + bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)

  
  
  

For Pair COND_727_0_MOD_LT1(TRUE, x2, x0) → 727_0_MOD_LT(-(x2, x0), x0) the following chains were created:

• We consider the chain 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:
  

  (28)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧-(x2[3], x0[3])=x2[0]∧x0[3]=x0[0] ⇒ COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥727_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

  
  
  We simplified constraint (28) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (29)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥727_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

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

  (30)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (31)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (32)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (33)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (34)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)

  
  
  
• We consider the chain 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:
  

  (35)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧-(x2[3], x0[3])=x2[2]1∧x0[3]=x0[2]1 ⇒ COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥727_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

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

  (36)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥727_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

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

  (37)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (38)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (39)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

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

  (40)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

  
  
  We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (41)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)

  
  
  

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

• 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
  
  • ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)
    
  
  
• COND_727_0_MOD_LT(TRUE, x2, x0) → 727_0_MOD_LT(x0, x2)
  
  • ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
    
  • ((U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
    
  
  
• 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
  
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)Bound*bni_22 + bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)
    
  
  
• COND_727_0_MOD_LT1(TRUE, x2, x0) → 727_0_MOD_LT(-(x2, x0), x0)
  
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)
    
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 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) = [1]   
POL(727_0_MOD_LT(x₁, x₂)) = [-1] + x₂ + x₁   
POL(COND_727_0_MOD_LT(x₁, x₂, x₃)) = [-1] + x₃ + x₂   
POL(&&(x₁, x₂)) = [-1]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(<(x₁, x₂)) = [-1]   
POL(COND_727_0_MOD_LT1(x₁, x₂, x₃)) = [-1] + x₃ + x₂   
POL(>=(x₁, x₂)) = [-1]   
POL(-(x₁, x₂)) = x₁ + [-1]x₂   

The following pairs are in P_>:

COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in P_bound:

727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])
COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in P_≥:

727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])

There are no usable rules.

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
(2): 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])

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

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

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

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])

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

(0) -> (1), if (x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0 ∧x2[0] →^* x2[1]∧x0[0] →^* x0[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@72865d61 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_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]) the following chains were created:

• We consider the chain 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:
  

  (1)    (&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUE∧x2[0]=x2[1]∧x0[0]=x0[1]∧x0[1]=x2[0]1∧x2[1]=x0[0]1 ⇒ COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥))

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

  (2)    (>(x0[0], 0)=TRUE∧>(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ COND_727_0_MOD_LT(TRUE, x2[0], x0[0])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[0], x0[0])≥727_0_MOD_LT(x0[0], x2[0])∧(U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[(-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[(-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)

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

  (5)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[(-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)

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

  (6)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)

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

  (7)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 0)

  
  
  We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (8)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 0)

  
  
  

For Pair 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) the following chains were created:

• We consider the chain COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]) which results in the following constraint:
  

  (9)    (x0[1]=x2[0]∧x2[1]=x0[0]∧&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUE∧x2[0]=x2[1]1∧x0[0]=x0[1]1 ⇒ 727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

  
  
  We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (10)    (>(x0[0], 0)=TRUE∧>(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

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

  (11)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-1 + (-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)

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

  (12)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-1 + (-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)

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

  (13)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-1 + (-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)

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

  (14)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)

  
  
  We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (15)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + x0[0] ≥ 0)

  
  
  We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (16)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + x0[0] ≥ 0)

  
  
  

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

• COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
  
  • ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 0)
    
  
  
• 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
  
  • ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + x0[0] ≥ 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_727_0_MOD_LT(x₁, x₂, x₃)) = [-1]x₁   
POL(727_0_MOD_LT(x₁, x₂)) = x₂ + [-1]x₁   
POL(&&(x₁, x₂)) = [-1]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])

The following pairs are in P_bound:

COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])

The following pairs are in P_≥:

727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])

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

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

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[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