Termination proof of /tmp/tmpQTbZOx/GCD3.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 460 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 130 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 450 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 IDP

        ↳12 IDPNonInfProof (⇒, 130 ms)

        ↳13 IDP

        ↳14 IDependencyGraphProof (⇔, 0 ms)

        ↳15 TRUE

    ↳16 IDP

        ↳17 IDependencyGraphProof (⇔, 0 ms)

        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD3

public class GCD3 {
  public static int mod(int a, int b) {
    if(b == 0) {
      return b;
    }
    if(b < 0) {
      a = -a;
    }
    if(a > 0) {
      while(a>=b) {
        a -= b;
      }
      return a;
    } else {
      while(a < 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:
GCD3.main([Ljava/lang/String;)V: Graph of 215 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: GCD3.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 37 rules for P and 0 rules for R.

P rules:
647_0_gcd_LE(EOS(STATIC_647), i66, i97, i97) → 650_0_gcd_LE(EOS(STATIC_650), i66, i97, i97)
650_0_gcd_LE(EOS(STATIC_650), i66, i97, i97) → 654_0_gcd_Load(EOS(STATIC_654), i66, i97) | >(i97, 0)
654_0_gcd_Load(EOS(STATIC_654), i66, i97) → 658_0_gcd_LE(EOS(STATIC_658), i66, i97, i66)
658_0_gcd_LE(EOS(STATIC_658), i99, i97, i99) → 663_0_gcd_LE(EOS(STATIC_663), i99, i97, i99)
663_0_gcd_LE(EOS(STATIC_663), i99, i97, i99) → 670_0_gcd_Load(EOS(STATIC_670), i99, i97) | >(i99, 0)
670_0_gcd_Load(EOS(STATIC_670), i99, i97) → 676_0_gcd_Store(EOS(STATIC_676), i99, i97, i97)
676_0_gcd_Store(EOS(STATIC_676), i99, i97, i97) → 681_0_gcd_Load(EOS(STATIC_681), i99, i97, i97)
681_0_gcd_Load(EOS(STATIC_681), i99, i97, i97) → 686_0_gcd_Load(EOS(STATIC_686), i97, i97, i99)
686_0_gcd_Load(EOS(STATIC_686), i97, i97, i99) → 690_0_gcd_InvokeMethod(EOS(STATIC_690), i97, i99, i97)
690_0_gcd_InvokeMethod(EOS(STATIC_690), i97, i99, i97) → 691_0_mod_Load(EOS(STATIC_691), i97, i99, i97, i99, i97)
691_0_mod_Load(EOS(STATIC_691), i97, i99, i97, i99, i97) → 693_0_mod_NE(EOS(STATIC_693), i97, i99, i97, i99, i97, i97)
693_0_mod_NE(EOS(STATIC_693), i97, i99, i97, i99, i97, i97) → 695_0_mod_Load(EOS(STATIC_695), i97, i99, i97, i99, i97) | >(i97, 0)
695_0_mod_Load(EOS(STATIC_695), i97, i99, i97, i99, i97) → 697_0_mod_GE(EOS(STATIC_697), i97, i99, i97, i99, i97, i97)
697_0_mod_GE(EOS(STATIC_697), i97, i99, i97, i99, i97, i97) → 699_0_mod_Load(EOS(STATIC_699), i97, i99, i97, i99, i97) | >=(i97, 0)
699_0_mod_Load(EOS(STATIC_699), i97, i99, i97, i99, i97) → 701_0_mod_LE(EOS(STATIC_701), i97, i99, i97, i99, i97, i99)
701_0_mod_LE(EOS(STATIC_701), i97, i99, i97, i99, i97, i99) → 704_0_mod_Load(EOS(STATIC_704), i97, i99, i97, i99, i97) | >(i99, 0)
704_0_mod_Load(EOS(STATIC_704), i97, i99, i97, i99, i97) → 739_0_mod_Load(EOS(STATIC_739), i97, i99, i97, i99, i97)
739_0_mod_Load(EOS(STATIC_739), i97, i99, i97, i105, i97) → 742_0_mod_Load(EOS(STATIC_742), i97, i99, i97, i105, i97, i105)
742_0_mod_Load(EOS(STATIC_742), i97, i99, i97, i105, i97, i105) → 744_0_mod_LT(EOS(STATIC_744), i97, i99, i97, i105, i97, i105, i97)
744_0_mod_LT(EOS(STATIC_744), i97, i99, i97, i105, i97, i105, i97) → 746_0_mod_LT(EOS(STATIC_746), i97, i99, i97, i105, i97, i105, i97)
744_0_mod_LT(EOS(STATIC_744), i97, i99, i97, i105, i97, i105, i97) → 747_0_mod_LT(EOS(STATIC_747), i97, i99, i97, i105, i97, i105, i97)
746_0_mod_LT(EOS(STATIC_746), i97, i99, i97, i105, i97, i105, i97) → 749_0_mod_Load(EOS(STATIC_749), i97, i99, i97, i105, i97) | <(i105, i97)
749_0_mod_Load(EOS(STATIC_749), i97, i99, i97, i105, i97) → 753_0_mod_Return(EOS(STATIC_753), i97, i99, i97, i105, i97, i105)
753_0_mod_Return(EOS(STATIC_753), i97, i99, i97, i105, i97, i105) → 756_0_gcd_Store(EOS(STATIC_756), i97, i105)
756_0_gcd_Store(EOS(STATIC_756), i97, i105) → 760_0_gcd_Load(EOS(STATIC_760), i105, i97)
760_0_gcd_Load(EOS(STATIC_760), i105, i97) → 764_0_gcd_Store(EOS(STATIC_764), i105, i97)
764_0_gcd_Store(EOS(STATIC_764), i105, i97) → 768_0_gcd_JMP(EOS(STATIC_768), i97, i105)
768_0_gcd_JMP(EOS(STATIC_768), i97, i105) → 775_0_gcd_Load(EOS(STATIC_775), i97, i105)
775_0_gcd_Load(EOS(STATIC_775), i97, i105) → 530_0_gcd_Load(EOS(STATIC_530), i97, i105)
530_0_gcd_Load(EOS(STATIC_530), i66, i67) → 647_0_gcd_LE(EOS(STATIC_647), i66, i67, i67)
747_0_mod_LT(EOS(STATIC_747), i97, i99, i97, i105, i97, i105, i97) → 751_0_mod_Load(EOS(STATIC_751), i97, i99, i97, i105, i97) | >=(i105, i97)
751_0_mod_Load(EOS(STATIC_751), i97, i99, i97, i105, i97) → 754_0_mod_Load(EOS(STATIC_754), i97, i99, i97, i97, i105)
754_0_mod_Load(EOS(STATIC_754), i97, i99, i97, i97, i105) → 758_0_mod_IntArithmetic(EOS(STATIC_758), i97, i99, i97, i97, i105, i97)
758_0_mod_IntArithmetic(EOS(STATIC_758), i97, i99, i97, i97, i105, i97) → 762_0_mod_Store(EOS(STATIC_762), i97, i99, i97, i97, -(i105, i97)) | >(i97, 0)
762_0_mod_Store(EOS(STATIC_762), i97, i99, i97, i97, i107) → 766_0_mod_JMP(EOS(STATIC_766), i97, i99, i97, i107, i97)
766_0_mod_JMP(EOS(STATIC_766), i97, i99, i97, i107, i97) → 771_0_mod_Load(EOS(STATIC_771), i97, i99, i97, i107, i97)
771_0_mod_Load(EOS(STATIC_771), i97, i99, i97, i107, i97) → 739_0_mod_Load(EOS(STATIC_739), i97, i99, i97, i107, i97)
R rules:

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

P rules:
744_0_mod_LT(EOS(STATIC_744), x0, x1, x0, x2, x0, x2, x0) → 744_0_mod_LT(EOS(STATIC_744), x2, x0, x2, x0, x2, x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
744_0_mod_LT(EOS(STATIC_744), x0, x1, x0, x2, x0, x2, x0) → 744_0_mod_LT(EOS(STATIC_744), x0, x1, x0, -(x2, x0), x0, -(x2, x0), x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

744_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → 744_0_mod_LT(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_744_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_744_0_mod_LT1(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_744_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_744_0_mod_LT(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

744_0_mod_LT(x1, x2, x3, x4, x5, x6, x7) → 744_0_mod_LT(x2, x6, x7)
Cond_744_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_744_0_mod_LT(x1, x3, x7, x8)
Cond_744_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_744_0_mod_LT1(x1, x3, x7, x8)

Filtered unneeded arguments:

Cond_744_0_mod_LT(x1, x2, x3, x4) → Cond_744_0_mod_LT(x1, x3, x4)
Cond_744_0_mod_LT1(x1, x2, x3, x4) → Cond_744_0_mod_LT1(x1, x3, x4)
744_0_mod_LT(x1, x2, x3) → 744_0_mod_LT(x2, x3)

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

P rules:
744_0_mod_LT(x2, x0) → 744_0_mod_LT(x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
744_0_mod_LT(x2, x0) → 744_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:
744_0_MOD_LT(x2, x0) → COND_744_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
COND_744_0_MOD_LT(TRUE, x2, x0) → 744_0_MOD_LT(x0, x2)
744_0_MOD_LT(x2, x0) → COND_744_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
COND_744_0_MOD_LT1(TRUE, x2, x0) → 744_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): 744_0_MOD_LT(x2[0], x0[0]) → COND_744_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_744_0_MOD_LT(TRUE, x2[1], x0[1]) → 744_0_MOD_LT(x0[1], x2[1])
(2): 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])
(3): COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_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@2ac2e1b1 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 744_0_MOD_LT(x2, x0) → COND_744_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0) the following chains were created:

We consider the chain 744_0_MOD_LT(x2[0], x0[0]) → COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_744_0_MOD_LT(TRUE, x2[1], x0[1]) → 744_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] ⇒ 744_0_MOD_LT(x2[0], x0[0])≥NonInfC∧744_0_MOD_LT(x2[0], x0[0])≥COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_744_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 ⇒ 744_0_MOD_LT(x2[0], x0[0])≥NonInfC∧744_0_MOD_LT(x2[0], x0[0])≥COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_744_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_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[-1 + (-1)bso_19] + 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(COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[-1 + (-1)bso_19] + 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(COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[-1 + (-1)bso_19] + 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(COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + 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(COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + 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(COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)

For Pair COND_744_0_MOD_LT(TRUE, x2, x0) → 744_0_MOD_LT(x0, x2) the following chains were created:

We consider the chain COND_744_0_MOD_LT(TRUE, x2[1], x0[1]) → 744_0_MOD_LT(x0[1], x2[1]), 744_0_MOD_LT(x2[0], x0[0]) → COND_744_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_744_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_744_0_MOD_LT(TRUE, x2[1], x0[1])≥744_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (9) using rule (IV) which results in the following new constraint:
(10)    (COND_744_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_744_0_MOD_LT(TRUE, x2[1], x0[1])≥744_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(744_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(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14)    ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)
We consider the chain COND_744_0_MOD_LT(TRUE, x2[1], x0[1]) → 744_0_MOD_LT(x0[1], x2[1]), 744_0_MOD_LT(x2[2], x0[2]) → COND_744_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_744_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_744_0_MOD_LT(TRUE, x2[1], x0[1])≥744_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (15) using rule (IV) which results in the following new constraint:
(16)    (COND_744_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_744_0_MOD_LT(TRUE, x2[1], x0[1])≥744_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(744_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(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

For Pair 744_0_MOD_LT(x2, x0) → COND_744_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0) the following chains were created:

We consider the chain 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_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] ⇒ 744_0_MOD_LT(x2[2], x0[2])≥NonInfC∧744_0_MOD_LT(x2[2], x0[2])≥COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_744_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 ⇒ 744_0_MOD_LT(x2[2], x0[2])≥NonInfC∧744_0_MOD_LT(x2[2], x0[2])≥COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_744_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_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x0[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_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x0[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_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x0[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_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x0[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_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

For Pair COND_744_0_MOD_LT1(TRUE, x2, x0) → 744_0_MOD_LT(-(x2, x0), x0) the following chains were created:

We consider the chain 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 744_0_MOD_LT(x2[0], x0[0]) → COND_744_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_744_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_744_0_MOD_LT1(TRUE, x2[3], x0[3])≥744_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(744_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_744_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_744_0_MOD_LT1(TRUE, x2[2], x0[2])≥744_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(744_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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)
We consider the chain 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 744_0_MOD_LT(x2[2], x0[2]) → COND_744_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_744_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_744_0_MOD_LT1(TRUE, x2[3], x0[3])≥744_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(744_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_744_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_744_0_MOD_LT1(TRUE, x2[2], x0[2])≥744_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(744_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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

744_0_MOD_LT(x2, x0) → COND_744_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_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)
COND_744_0_MOD_LT(TRUE, x2, x0) → 744_0_MOD_LT(x0, x2)
- ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)
- ((U^Increasing(744_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)
744_0_MOD_LT(x2, x0) → COND_744_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(2)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)
COND_744_0_MOD_LT1(TRUE, x2, x0) → 744_0_MOD_LT(-(x2, x0), x0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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) = [3]
POL(744_0_MOD_LT(x₁, x₂)) = [1] + x₂
POL(COND_744_0_MOD_LT(x₁, x₂, x₃)) = [2] + x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(<(x₁, x₂)) = [-1]
POL(COND_744_0_MOD_LT1(x₁, x₂, x₃)) = [1] + x₃
POL(>=(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

COND_744_0_MOD_LT(TRUE, x2[1], x0[1]) → 744_0_MOD_LT(x0[1], x2[1])

The following pairs are in P_bound:

744_0_MOD_LT(x2[0], x0[0]) → COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])
COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in P_≥:

744_0_MOD_LT(x2[0], x0[0]) → COND_744_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])
COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3])

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

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

(8) Complex Obligation (AND)

(9) 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): 744_0_MOD_LT(x2[0], x0[0]) → COND_744_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(2): 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])
(3): COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(x2[3] - x0[3], 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])

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

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) 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:
(3): COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(x2[3] - x0[3], x0[3])
(2): 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])

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

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

The set Q is empty.

(12) 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@2ac2e1b1 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_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3]) the following chains were created:

We consider the chain 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:
(1)    (&&(>=(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_744_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_744_0_MOD_LT1(TRUE, x2[3], x0[3])≥744_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_744_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_744_0_MOD_LT1(TRUE, x2[2], x0[2])≥744_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x0[2] + [bni_13]x2[2] ≥ 0∧[(-1)bso_14] + x0[2] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x0[2] + [bni_13]x2[2] ≥ 0∧[(-1)bso_14] + x0[2] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x0[2] + [bni_13]x2[2] ≥ 0∧[(-1)bso_14] + x0[2] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[2] ≥ 0∧[(-1)bso_14] + x0[2] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[2] ≥ 0∧[1 + (-1)bso_14] + x0[2] ≥ 0)

For Pair 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) the following chains were created:

We consider the chain 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3]) which results in the following constraint:
(8)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 744_0_MOD_LT(x2[2], x0[2])≥NonInfC∧744_0_MOD_LT(x2[2], x0[2])≥COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))

We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ 744_0_MOD_LT(x2[2], x0[2])≥NonInfC∧744_0_MOD_LT(x2[2], x0[2])≥COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[2] + [bni_15]x2[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[2] + [bni_15]x2[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[2] + [bni_15]x2[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x2[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x2[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3])
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(744_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[2] ≥ 0∧[1 + (-1)bso_14] + x0[2] ≥ 0)
744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x2[2] ≥ 0∧[(-1)bso_16] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_744_0_MOD_LT1(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂ + [-1]x₁
POL(744_0_MOD_LT(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(&&(x₁, x₂)) = 0
POL(>=(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in P_bound:

COND_744_0_MOD_LT1(TRUE, x2[3], x0[3]) → 744_0_MOD_LT(-(x2[3], x0[3]), x0[3])
744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])

The following pairs are in P_≥:

744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])

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

(13) 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:
(2): 744_0_MOD_LT(x2[2], x0[2]) → COND_744_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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:
none

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_744_0_MOD_LT(TRUE, x2[1], x0[1]) → 744_0_MOD_LT(x0[1], x2[1])

The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) Obligation:

(12) IDPNonInfProof (SOUND transformation)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE