Termination proof of /tmp/tmpftzhlr/GCD2.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 150 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 140 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 790 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 IDP

↳11 IDPNonInfProof (⇒, 560 ms)

↳12 IDP

↳13 IDependencyGraphProof (⇔, 0 ms)

↳14 IDP

↳15 IDPNonInfProof (⇒, 340 ms)

↳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: GCD2

public class GCD2 {
  public static int mod(int a, int b) {
    if (a == b) {
      return 0;
    }
    while(a>b) {
      a -= b;
    }
    return a;
  }

  public static int gcd(int a, int b) {
    int tmp;
    while(b != 0 && a >= 0 && b >= 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:
GCD2.main([Ljava/lang/String;)V: Graph of 218 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: GCD2.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 43 rules for P and 0 rules for R.

P rules:
618_0_gcd_EQ(EOS(STATIC_618), i89, i96, i96) → 620_0_gcd_EQ(EOS(STATIC_620), i89, i96, i96)
620_0_gcd_EQ(EOS(STATIC_620), i89, i96, i96) → 623_0_gcd_Load(EOS(STATIC_623), i89, i96) | !(=(i96, 0))
623_0_gcd_Load(EOS(STATIC_623), i89, i96) → 626_0_gcd_LT(EOS(STATIC_626), i89, i96, i89)
626_0_gcd_LT(EOS(STATIC_626), i89, i96, i89) → 630_0_gcd_Load(EOS(STATIC_630), i89, i96) | >=(i89, 0)
630_0_gcd_Load(EOS(STATIC_630), i89, i96) → 635_0_gcd_LT(EOS(STATIC_635), i89, i96, i96)
635_0_gcd_LT(EOS(STATIC_635), i89, i103, i103) → 641_0_gcd_LT(EOS(STATIC_641), i89, i103, i103)
641_0_gcd_LT(EOS(STATIC_641), i89, i103, i103) → 651_0_gcd_Load(EOS(STATIC_651), i89, i103) | >=(i103, 0)
651_0_gcd_Load(EOS(STATIC_651), i89, i103) → 655_0_gcd_Store(EOS(STATIC_655), i89, i103, i103)
655_0_gcd_Store(EOS(STATIC_655), i89, i103, i103) → 659_0_gcd_Load(EOS(STATIC_659), i89, i103, i103)
659_0_gcd_Load(EOS(STATIC_659), i89, i103, i103) → 664_0_gcd_Load(EOS(STATIC_664), i103, i103, i89)
664_0_gcd_Load(EOS(STATIC_664), i103, i103, i89) → 668_0_gcd_InvokeMethod(EOS(STATIC_668), i103, i89, i103)
668_0_gcd_InvokeMethod(EOS(STATIC_668), i103, i89, i103) → 670_0_mod_Load(EOS(STATIC_670), i103, i89, i103, i89, i103)
670_0_mod_Load(EOS(STATIC_670), i103, i89, i103, i89, i103) → 672_0_mod_Load(EOS(STATIC_672), i103, i89, i103, i89, i103, i89)
672_0_mod_Load(EOS(STATIC_672), i103, i89, i103, i89, i103, i89) → 675_0_mod_NE(EOS(STATIC_675), i103, i89, i103, i89, i103, i89, i103)
675_0_mod_NE(EOS(STATIC_675), i103, i89, i103, i89, i103, i89, i103) → 677_0_mod_NE(EOS(STATIC_677), i103, i89, i103, i89, i103, i89, i103)
675_0_mod_NE(EOS(STATIC_675), i103, i103, i103, i103, i103, i103, i103) → 678_0_mod_NE(EOS(STATIC_678), i103, i103, i103, i103, i103, i103, i103)
677_0_mod_NE(EOS(STATIC_677), i103, i89, i103, i89, i103, i89, i103) → 681_0_mod_Load(EOS(STATIC_681), i103, i89, i103, i89, i103) | !(=(i89, i103))
681_0_mod_Load(EOS(STATIC_681), i103, i89, i103, i89, i103) → 743_0_mod_Load(EOS(STATIC_743), i103, i89, i103, i89, i103)
743_0_mod_Load(EOS(STATIC_743), i103, i89, i103, i107, i103) → 751_0_mod_Load(EOS(STATIC_751), i103, i89, i103, i107, i103, i107)
751_0_mod_Load(EOS(STATIC_751), i103, i89, i103, i107, i103, i107) → 754_0_mod_LE(EOS(STATIC_754), i103, i89, i103, i107, i103, i107, i103)
754_0_mod_LE(EOS(STATIC_754), i103, i89, i103, i107, i103, i107, i103) → 756_0_mod_LE(EOS(STATIC_756), i103, i89, i103, i107, i103, i107, i103)
754_0_mod_LE(EOS(STATIC_754), i103, i89, i103, i107, i103, i107, i103) → 757_0_mod_LE(EOS(STATIC_757), i103, i89, i103, i107, i103, i107, i103)
756_0_mod_LE(EOS(STATIC_756), i103, i89, i103, i107, i103, i107, i103) → 759_0_mod_Load(EOS(STATIC_759), i103, i89, i103, i107) | <=(i107, i103)
759_0_mod_Load(EOS(STATIC_759), i103, i89, i103, i107) → 764_0_mod_Return(EOS(STATIC_764), i103, i89, i103, i107)
764_0_mod_Return(EOS(STATIC_764), i103, i89, i103, i107) → 770_0_gcd_Store(EOS(STATIC_770), i103, i107)
770_0_gcd_Store(EOS(STATIC_770), i103, i107) → 776_0_gcd_Load(EOS(STATIC_776), i107, i103)
776_0_gcd_Load(EOS(STATIC_776), i107, i103) → 780_0_gcd_Store(EOS(STATIC_780), i107, i103)
780_0_gcd_Store(EOS(STATIC_780), i107, i103) → 784_0_gcd_JMP(EOS(STATIC_784), i103, i107)
784_0_gcd_JMP(EOS(STATIC_784), i103, i107) → 794_0_gcd_Load(EOS(STATIC_794), i103, i107)
794_0_gcd_Load(EOS(STATIC_794), i103, i107) → 614_0_gcd_Load(EOS(STATIC_614), i103, i107)
614_0_gcd_Load(EOS(STATIC_614), i89, i90) → 618_0_gcd_EQ(EOS(STATIC_618), i89, i90, i90)
757_0_mod_LE(EOS(STATIC_757), i103, i89, i103, i107, i103, i107, i103) → 762_0_mod_Load(EOS(STATIC_762), i103, i89, i103, i107, i103) | >(i107, i103)
762_0_mod_Load(EOS(STATIC_762), i103, i89, i103, i107, i103) → 766_0_mod_Load(EOS(STATIC_766), i103, i89, i103, i103, i107)
766_0_mod_Load(EOS(STATIC_766), i103, i89, i103, i103, i107) → 774_0_mod_IntArithmetic(EOS(STATIC_774), i103, i89, i103, i103, i107, i103)
774_0_mod_IntArithmetic(EOS(STATIC_774), i103, i89, i103, i103, i107, i103) → 778_0_mod_Store(EOS(STATIC_778), i103, i89, i103, i103, -(i107, i103)) | >(i103, 0)
778_0_mod_Store(EOS(STATIC_778), i103, i89, i103, i103, i113) → 782_0_mod_JMP(EOS(STATIC_782), i103, i89, i103, i113, i103)
782_0_mod_JMP(EOS(STATIC_782), i103, i89, i103, i113, i103) → 789_0_mod_Load(EOS(STATIC_789), i103, i89, i103, i113, i103)
789_0_mod_Load(EOS(STATIC_789), i103, i89, i103, i113, i103) → 743_0_mod_Load(EOS(STATIC_743), i103, i89, i103, i113, i103)
678_0_mod_NE(EOS(STATIC_678), i103, i103, i103, i103, i103, i103, i103) → 683_0_mod_ConstantStackPush(EOS(STATIC_683), i103, i103, i103, i103, i103)
683_0_mod_ConstantStackPush(EOS(STATIC_683), i103, i103, i103, i103, i103) → 687_0_mod_Return(EOS(STATIC_687), i103, i103, i103, i103, i103, 0)
687_0_mod_Return(EOS(STATIC_687), i103, i103, i103, i103, i103, matching1) → 692_0_gcd_Store(EOS(STATIC_692), i103, 0) | =(matching1, 0)
692_0_gcd_Store(EOS(STATIC_692), i103, matching1) → 717_0_gcd_Store(EOS(STATIC_717), i103, 0) | =(matching1, 0)
717_0_gcd_Store(EOS(STATIC_717), i103, i89) → 770_0_gcd_Store(EOS(STATIC_770), i103, i89)
R rules:

Combined rules. Obtained 4 conditional rules for P and 0 conditional rules for R.

P rules:
618_0_gcd_EQ(EOS(STATIC_618), x0, x1, x1) → 754_0_mod_LE(EOS(STATIC_754), x1, x0, x1, x0, x1, x0, x1) | &&(&&(>(x1, 0), >(+(x0, 1), 0)), !(=(x0, x1)))
754_0_mod_LE(EOS(STATIC_754), x0, x1, x0, x2, x0, x2, x0) → 618_0_gcd_EQ(EOS(STATIC_618), x0, x2, x2) | <=(x2, x0)
754_0_mod_LE(EOS(STATIC_754), x0, x1, x0, x2, x0, x2, x0) → 754_0_mod_LE(EOS(STATIC_754), x0, x1, x0, -(x2, x0), x0, -(x2, x0), x0) | &&(>(x2, x0), >(x0, 0))
618_0_gcd_EQ(EOS(STATIC_618), x0, x0, x0) → 618_0_gcd_EQ(EOS(STATIC_618), x0, 0, 0) | >(x0, 0)
R rules:

Filtered ground terms:

618_0_gcd_EQ(x1, x2, x3, x4) → 618_0_gcd_EQ(x2, x3, x4)
Cond_618_0_gcd_EQ1(x1, x2, x3, x4, x5) → Cond_618_0_gcd_EQ1(x1, x3, x4, x5)
754_0_mod_LE(x1, x2, x3, x4, x5, x6, x7, x8) → 754_0_mod_LE(x2, x3, x4, x5, x6, x7, x8)
Cond_754_0_mod_LE1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_754_0_mod_LE1(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_754_0_mod_LE(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_754_0_mod_LE(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_618_0_gcd_EQ(x1, x2, x3, x4, x5) → Cond_618_0_gcd_EQ(x1, x3, x4, x5)

Filtered duplicate args:

618_0_gcd_EQ(x1, x2, x3) → 618_0_gcd_EQ(x1, x3)
Cond_618_0_gcd_EQ(x1, x2, x3, x4) → Cond_618_0_gcd_EQ(x1, x2, x4)
754_0_mod_LE(x1, x2, x3, x4, x5, x6, x7) → 754_0_mod_LE(x2, x4, x6, x7)
Cond_754_0_mod_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_754_0_mod_LE(x1, x3, x7, x8)
Cond_754_0_mod_LE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_754_0_mod_LE1(x1, x3, x7, x8)
Cond_618_0_gcd_EQ1(x1, x2, x3, x4) → Cond_618_0_gcd_EQ1(x1, x4)

Filtered unneeded arguments:

754_0_mod_LE(x1, x2, x3, x4) → 754_0_mod_LE(x2, x3, x4)
Cond_754_0_mod_LE(x1, x2, x3, x4) → Cond_754_0_mod_LE(x1, x3, x4)
Cond_754_0_mod_LE1(x1, x2, x3, x4) → Cond_754_0_mod_LE1(x1, x3, x4)

Combined rules. Obtained 4 conditional rules for P and 0 conditional rules for R.

P rules:
618_0_gcd_EQ(x0, x1) → 754_0_mod_LE(x0, x0, x1) | &&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1)))
754_0_mod_LE(x2, x2, x0) → 618_0_gcd_EQ(x0, x2) | <=(x2, x0)
754_0_mod_LE(x2, x2, x0) → 754_0_mod_LE(-(x2, x0), -(x2, x0), x0) | &&(>(x2, x0), >(x0, 0))
618_0_gcd_EQ(x0, x0) → 618_0_gcd_EQ(x0, 0) | >(x0, 0)
R rules:

Finished conversion. Obtained 8 rules for P and 0 rules for R. System has predefined symbols.

P rules:
618_0_GCD_EQ(x0, x1) → COND_618_0_GCD_EQ(&&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1))), x0, x1)
COND_618_0_GCD_EQ(TRUE, x0, x1) → 754_0_MOD_LE(x0, x0, x1)
754_0_MOD_LE(x2, x2, x0) → COND_754_0_MOD_LE(<=(x2, x0), x2, x2, x0)
COND_754_0_MOD_LE(TRUE, x2, x2, x0) → 618_0_GCD_EQ(x0, x2)
754_0_MOD_LE(x2, x2, x0) → COND_754_0_MOD_LE1(&&(>(x2, x0), >(x0, 0)), x2, x2, x0)
COND_754_0_MOD_LE1(TRUE, x2, x2, x0) → 754_0_MOD_LE(-(x2, x0), -(x2, x0), x0)
618_0_GCD_EQ(x0, x0) → COND_618_0_GCD_EQ1(>(x0, 0), x0, x0)
COND_618_0_GCD_EQ1(TRUE, x0, x0) → 618_0_GCD_EQ(x0, 0)
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): 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])
(1): COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
(2): 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(3): COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
(4): 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(x2[4] > x0[4] && x0[4] > 0, x2[4], x2[4], x0[4])
(5): COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(x2[5] - x0[5], x2[5] - x0[5], x0[5])
(6): 618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(x0[6] > 0, x0[6], x0[6])
(7): COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 618_0_GCD_EQ(x0[7], 0)

(0) -> (1), if (x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]) ∧x0[0] →^* x0[1]∧x1[0] →^* x1[1])

(1) -> (2), if (x0[1] →^* x2[2]∧x1[1] →^* x0[2])

(1) -> (4), if (x0[1] →^* x2[4]∧x1[1] →^* x0[4])

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

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x1[0])

(3) -> (6), if (x0[3] →^* x0[6]∧x2[3] →^* x0[6])

(4) -> (5), if (x2[4] > x0[4] && x0[4] > 0 ∧x2[4] →^* x2[5]∧x0[4] →^* x0[5])

(5) -> (2), if (x2[5] - x0[5] →^* x2[2]∧x0[5] →^* x0[2])

(5) -> (4), if (x2[5] - x0[5] →^* x2[4]∧x0[5] →^* x0[4])

(6) -> (7), if (x0[6] > 0 ∧x0[6] →^* x0[7])

(7) -> (0), if (x0[7] →^* x0[0]∧0 →^* x1[0])

(7) -> (6), if (x0[7] →^* x0[6]∧0 →^* x0[6])

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@7d6dcf80 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 618_0_GCD_EQ(x0, x1) → COND_618_0_GCD_EQ(&&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1))), x0, x1) the following chains were created:

We consider the chain 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]) which results in the following constraint:
(1)    (&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:
(2)    (>(x1[0], 0)=TRUE∧>(x0[0], -1)=TRUE∧<(x0[0], x1[0])=TRUE ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

(3)    (>(x1[0], 0)=TRUE∧>(x0[0], -1)=TRUE∧>(x0[0], x1[0])=TRUE ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(6)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(7)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(8)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(10)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]x0[0] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

For Pair COND_618_0_GCD_EQ(TRUE, x0, x1) → 754_0_MOD_LE(x0, x0, x1) the following chains were created:

We consider the chain COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:
(14)    (x0[1]=x2[2]∧x1[1]=x0[2] ⇒ COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (14) using rule (IV) which results in the following new constraint:
(15)    (COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)
We consider the chain COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:
(20)    (x0[1]=x2[4]∧x1[1]=x0[4] ⇒ COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (20) using rule (IV) which results in the following new constraint:
(21)    (COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

For Pair 754_0_MOD_LE(x2, x2, x0) → COND_754_0_MOD_LE(<=(x2, x0), x2, x2, x0) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]) which results in the following constraint:
(26)    (<=(x2[2], x0[2])=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 754_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧754_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))

We simplified constraint (26) using rule (IV) which results in the following new constraint:
(27)    (<=(x2[2], x0[2])=TRUE ⇒ 754_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧754_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] + [-1]x2[2] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] + [-1]x2[2] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] + [-1]x2[2] ≥ 0)

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31)    (x0[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(32)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [(-1)bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)

(33)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)

For Pair COND_754_0_MOD_LE(TRUE, x2, x2, x0) → 618_0_GCD_EQ(x0, x2) the following chains were created:

We consider the chain COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:
(34)    (x0[3]=x0[0]∧x2[3]=x1[0] ⇒ COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥618_0_GCD_EQ(x0[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (34) using rule (IV) which results in the following new constraint:
(35)    (COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥618_0_GCD_EQ(x0[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(39)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
We consider the chain COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]) which results in the following constraint:
(40)    (x0[3]=x0[6]∧x2[3]=x0[6] ⇒ COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥618_0_GCD_EQ(x0[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (40) using rule (III) which results in the following new constraint:
(41)    (COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x2[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x2[3])≥618_0_GCD_EQ(x2[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (44) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(45)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

For Pair 754_0_MOD_LE(x2, x2, x0) → COND_754_0_MOD_LE1(&&(>(x2, x0), >(x0, 0)), x2, x2, x0) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]) which results in the following constraint:
(46)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUE∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 754_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧754_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))

We simplified constraint (46) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(47)    (>(x2[4], x0[4])=TRUE∧>(x0[4], 0)=TRUE ⇒ 754_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧754_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))

We simplified constraint (47) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(48)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (48) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(49)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(50)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(51)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)

We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(52)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41 + bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)

For Pair COND_754_0_MOD_LE1(TRUE, x2, x2, x0) → 754_0_MOD_LE(-(x2, x0), -(x2, x0), x0) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:
(53)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUE∧x2[4]=x2[5]∧x0[4]=x0[5]∧-(x2[5], x0[5])=x2[2]∧x0[5]=x0[2] ⇒ COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (53) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(54)    (>(x2[4], x0[4])=TRUE∧>(x0[4], 0)=TRUE ⇒ COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥754_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (54) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(55)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (55) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(56)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(57)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(58)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(59)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43 + bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)
We consider the chain 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:
(60)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUE∧x2[4]=x2[5]∧x0[4]=x0[5]∧-(x2[5], x0[5])=x2[4]1∧x0[5]=x0[4]1 ⇒ COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (60) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(61)    (>(x2[4], x0[4])=TRUE∧>(x0[4], 0)=TRUE ⇒ COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥754_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (61) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(62)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (62) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(63)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (63) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(64)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(65)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (65) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(66)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43 + bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

For Pair 618_0_GCD_EQ(x0, x0) → COND_618_0_GCD_EQ1(>(x0, 0), x0, x0) the following chains were created:

We consider the chain 618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]), COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 618_0_GCD_EQ(x0[7], 0) which results in the following constraint:
(67)    (>(x0[6], 0)=TRUE∧x0[6]=x0[7] ⇒ 618_0_GCD_EQ(x0[6], x0[6])≥NonInfC∧618_0_GCD_EQ(x0[6], x0[6])≥COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])∧(U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥))

We simplified constraint (67) using rule (IV) which results in the following new constraint:
(68)    (>(x0[6], 0)=TRUE ⇒ 618_0_GCD_EQ(x0[6], x0[6])≥NonInfC∧618_0_GCD_EQ(x0[6], x0[6])≥COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])∧(U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥))

We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(69)    (x0[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] ≥ 0)

We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(70)    (x0[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] ≥ 0)

We simplified constraint (70) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(71)    (x0[6] + [-1] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] ≥ 0)

We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(72)    (x0[6] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45 + bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] ≥ 0)

For Pair COND_618_0_GCD_EQ1(TRUE, x0, x0) → 618_0_GCD_EQ(x0, 0) the following chains were created:

We consider the chain 618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]), COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 618_0_GCD_EQ(x0[7], 0), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:
(73)    (>(x0[6], 0)=TRUE∧x0[6]=x0[7]∧x0[7]=x0[0]∧0=x1[0] ⇒ COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7])≥NonInfC∧COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7])≥618_0_GCD_EQ(x0[7], 0)∧(U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥))

We simplified constraint (73) using rules (III), (IV) which results in the following new constraint:
(74)    (>(x0[6], 0)=TRUE ⇒ COND_618_0_GCD_EQ1(TRUE, x0[6], x0[6])≥NonInfC∧COND_618_0_GCD_EQ1(TRUE, x0[6], x0[6])≥618_0_GCD_EQ(x0[6], 0)∧(U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥))

We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(75)    (x0[6] + [-1] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] + [bni_47]x0[6] ≥ 0∧[(-1)bso_48] + x0[6] ≥ 0)

We simplified constraint (75) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(76)    (x0[6] + [-1] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] + [bni_47]x0[6] ≥ 0∧[(-1)bso_48] + x0[6] ≥ 0)

We simplified constraint (76) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(77)    (x0[6] + [-1] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] + [bni_47]x0[6] ≥ 0∧[(-1)bso_48] + x0[6] ≥ 0)

We simplified constraint (77) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(78)    (x0[6] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47 + bni_47] + [bni_47]x0[6] ≥ 0∧[1 + (-1)bso_48] + x0[6] ≥ 0)
We consider the chain 618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]), COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 618_0_GCD_EQ(x0[7], 0), 618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]) which results in the following constraint:
(79) (>(x0[6], 0)=TRUE∧x0[6]=x0[7]∧x0[7]=x0[6]1∧0=x0[6]1 ⇒ COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7])≥NonInfC∧COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7])≥618_0_GCD_EQ(x0[7], 0)∧(U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥))

We solved constraint (79) using rules (I), (II), (III), (IDP_CONSTANT_FOLD).

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

618_0_GCD_EQ(x0, x1) → COND_618_0_GCD_EQ(&&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1))), x0, x1)
- (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]x0[0] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)
- (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)
COND_618_0_GCD_EQ(TRUE, x0, x1) → 754_0_MOD_LE(x0, x0, x1)
- ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)
- ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)
754_0_MOD_LE(x2, x2, x0) → COND_754_0_MOD_LE(<=(x2, x0), x2, x2, x0)
- (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [(-1)bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)
- (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)
COND_754_0_MOD_LE(TRUE, x2, x2, x0) → 618_0_GCD_EQ(x0, x2)
- ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
- ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
754_0_MOD_LE(x2, x2, x0) → COND_754_0_MOD_LE1(&&(>(x2, x0), >(x0, 0)), x2, x2, x0)
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41 + bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)
COND_754_0_MOD_LE1(TRUE, x2, x2, x0) → 754_0_MOD_LE(-(x2, x0), -(x2, x0), x0)
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43 + bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43 + bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)
618_0_GCD_EQ(x0, x0) → COND_618_0_GCD_EQ1(>(x0, 0), x0, x0)
- (x0[6] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45 + bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] ≥ 0)
COND_618_0_GCD_EQ1(TRUE, x0, x0) → 618_0_GCD_EQ(x0, 0)
- (x0[6] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47 + bni_47] + [bni_47]x0[6] ≥ 0∧[1 + (-1)bso_48] + x0[6] ≥ 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(618_0_GCD_EQ(x₁, x₂)) = x₂
POL(COND_618_0_GCD_EQ(x₁, x₂, x₃)) = x₃
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(-1) = [-1]
POL(!(x₁)) = [-1]
POL(=(x₁, x₂)) = [-1]
POL(754_0_MOD_LE(x₁, x₂, x₃)) = x₃ + [-1]x₂ + x₁
POL(COND_754_0_MOD_LE(x₁, x₂, x₃, x₄)) = x₃
POL(<=(x₁, x₂)) = [-1]
POL(COND_754_0_MOD_LE1(x₁, x₂, x₃, x₄)) = x₄
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(COND_618_0_GCD_EQ1(x₁, x₂, x₃)) = [-1]x₃ + [2]x₂

The following pairs are in P_>:

COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 618_0_GCD_EQ(x0[7], 0)

The following pairs are in P_bound:

618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])
COND_618_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 618_0_GCD_EQ(x0[7], 0)

The following pairs are in P_≥:

618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
618_0_GCD_EQ(x0[6], x0[6]) → COND_618_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])

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

(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

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x1[0])

(0) -> (1), if (x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]) ∧x0[0] →^* x0[1]∧x1[0] →^* x1[1])

(1) -> (2), if (x0[1] →^* x2[2]∧x1[1] →^* x0[2])

(5) -> (2), if (x2[5] - x0[5] →^* x2[2]∧x0[5] →^* x0[2])

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

(1) -> (4), if (x0[1] →^* x2[4]∧x1[1] →^* x0[4])

(5) -> (4), if (x2[5] - x0[5] →^* x2[4]∧x0[5] →^* x0[4])

(4) -> (5), if (x2[4] > x0[4] && x0[4] > 0 ∧x2[4] →^* x2[5]∧x0[4] →^* x0[5])

(3) -> (6), if (x0[3] →^* x0[6]∧x2[3] →^* x0[6])

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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(x2[5] - x0[5], x2[5] - x0[5], x0[5])
(4): 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(x2[4] > x0[4] && x0[4] > 0, x2[4], x2[4], x0[4])
(3): COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
(2): 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(1): COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
(0): 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x1[0])

(0) -> (1), if (x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]) ∧x0[0] →^* x0[1]∧x1[0] →^* x1[1])

(1) -> (2), if (x0[1] →^* x2[2]∧x1[1] →^* x0[2])

(5) -> (2), if (x2[5] - x0[5] →^* x2[2]∧x0[5] →^* x0[2])

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

(1) -> (4), if (x0[1] →^* x2[4]∧x1[1] →^* x0[4])

(5) -> (4), if (x2[5] - x0[5] →^* x2[4]∧x0[5] →^* x0[4])

(4) -> (5), if (x2[4] > x0[4] && x0[4] > 0 ∧x2[4] →^* x2[5]∧x0[4] →^* x0[5])

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@7d6dcf80 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_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:
(1)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUE∧x2[4]=x2[5]∧x0[4]=x0[5]∧-(x2[5], x0[5])=x2[2]∧x0[5]=x0[2] ⇒ COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x2[4], x0[4])=TRUE∧>(x0[4], 0)=TRUE ⇒ COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥754_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)
We consider the chain 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:
(8)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUE∧x2[4]=x2[5]∧x0[4]=x0[5]∧-(x2[5], x0[5])=x2[4]1∧x0[5]=x0[4]1 ⇒ COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x2[4], x0[4])=TRUE∧>(x0[4], 0)=TRUE ⇒ COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_754_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥754_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)

For Pair 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]) which results in the following constraint:
(15)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUE∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 754_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧754_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))

We simplified constraint (15) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(16)    (>(x2[4], x0[4])=TRUE∧>(x0[4], 0)=TRUE ⇒ 754_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧754_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[(-1)bso_32] + x0[4] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[(-1)bso_32] + x0[4] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[(-1)bso_32] + x0[4] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [(2)bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[(-1)bso_32] + x0[4] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31 + (3)bni_31] + [(2)bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[1 + (-1)bso_32] + x0[4] ≥ 0)

For Pair COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]) the following chains were created:

We consider the chain COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:
(22)    (x0[3]=x0[0]∧x2[3]=x1[0] ⇒ COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥618_0_GCD_EQ(x0[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (22) using rule (IV) which results in the following new constraint:
(23)    (COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥618_0_GCD_EQ(x0[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (26) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(27)    ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]) which results in the following constraint:
(28)    (<=(x2[2], x0[2])=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 754_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧754_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))

We simplified constraint (28) using rule (IV) which results in the following new constraint:
(29)    (<=(x2[2], x0[2])=TRUE ⇒ 754_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧754_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))

We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(30)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[2] + [bni_35]x2[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(31)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[2] + [bni_35]x2[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(32)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[2] + [bni_35]x2[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    (x0[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(34)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)

(35)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [(-2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)

For Pair COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]) the following chains were created:

We consider the chain COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:
(36)    (x0[1]=x2[2]∧x1[1]=x0[2] ⇒ COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (36) using rule (IV) which results in the following new constraint:
(37)    (COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(39)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(40)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (40) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(41)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
We consider the chain COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:
(42)    (x0[1]=x2[4]∧x1[1]=x0[4] ⇒ COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (42) using rule (IV) which results in the following new constraint:
(43)    (COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(44)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(45)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (45) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(46)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (46) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(47)    ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) the following chains were created:

We consider the chain 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]) which results in the following constraint:
(48)    (&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

We simplified constraint (48) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:
(49)    (>(x1[0], 0)=TRUE∧>(x0[0], -1)=TRUE∧<(x0[0], x1[0])=TRUE ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

(50)    (>(x1[0], 0)=TRUE∧>(x0[0], -1)=TRUE∧>(x0[0], x1[0])=TRUE ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

We simplified constraint (49) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(51)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(52)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(53)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(54)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(55)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (54) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (55) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(57)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (56) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(58)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(59)    (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + bni_39] + [(2)bni_39]x0[0] + [bni_39]x1[0] ≥ 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(60)    (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + (3)bni_39] + [(2)bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

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

COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[(-1)bso_30] ≥ 0)
754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
- (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31 + (3)bni_31] + [(2)bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[1 + (-1)bso_32] + x0[4] ≥ 0)
COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
- ((U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)
754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
- (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)
- (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [(-2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)
COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
- ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
- ((U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
- (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + bni_39] + [(2)bni_39]x0[0] + [bni_39]x1[0] ≥ 0∧[(-1)bso_40] ≥ 0)
- (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + (3)bni_39] + [(2)bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)

POL(TRUE) = [2]
POL(FALSE) = [1]
POL(COND_754_0_MOD_LE1(x₁, x₂, x₃, x₄)) = x₂
POL(754_0_MOD_LE(x₁, x₂, x₃)) = x₃ + x₂
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(COND_754_0_MOD_LE(x₁, x₂, x₃, x₄)) = x₄ + [2]x₃ + [-1]x₂
POL(618_0_GCD_EQ(x₁, x₂)) = x₂ + x₁
POL(<=(x₁, x₂)) = [-1]
POL(COND_618_0_GCD_EQ(x₁, x₂, x₃)) = x₃ + x₂
POL(-1) = [-1]
POL(!(x₁)) = [-1]
POL(=(x₁, x₂)) = [-1]

The following pairs are in P_>:

754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])

The following pairs are in P_bound:

COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
754_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_754_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])

The following pairs are in P_≥:

COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])

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

FALSE¹ → &&(TRUE, FALSE)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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(5): COND_754_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 754_0_MOD_LE(x2[5] - x0[5], x2[5] - x0[5], x0[5])
(3): COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
(2): 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(1): COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
(0): 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x1[0])

(0) -> (1), if (x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]) ∧x0[0] →^* x0[1]∧x1[0] →^* x1[1])

(1) -> (2), if (x0[1] →^* x2[2]∧x1[1] →^* x0[2])

(5) -> (2), if (x2[5] - x0[5] →^* x2[2]∧x0[5] →^* x0[2])

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

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) 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_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
(0): 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])
(3): COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
(2): 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x1[0])

(0) -> (1), if (x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]) ∧x0[0] →^* x0[1]∧x1[0] →^* x1[1])

(1) -> (2), if (x0[1] →^* x2[2]∧x1[1] →^* x0[2])

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

The set Q is empty.

(15) 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@2e89630 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 3 Max Right Steps: 2

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_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]) the following chains were created:

We consider the chain 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]) which results in the following constraint:
(1)    (<=(x2[2], x0[2])=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧x0[3]=x0[0]∧x2[3]=x1[0]∧&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x2[2]1∧x1[1]=x0[2]1∧<=(x2[2]1, x0[2]1)=TRUE∧x2[2]1=x2[3]1∧x0[2]1=x0[3]1 ⇒ COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[1], x1[1])≥754_0_MOD_LE(x0[1], x0[1], x1[1])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

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], x2[2])=TRUE∧>(x2[2], 0)=TRUE∧>(x0[2], -1)=TRUE∧<(x0[2], x2[2])=TRUE ⇒ COND_618_0_GCD_EQ(TRUE, x0[2], x2[2])≥NonInfC∧COND_618_0_GCD_EQ(TRUE, x0[2], x2[2])≥754_0_MOD_LE(x0[2], x0[2], x2[2])∧(U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] + [(-1)bni_24]x0[2] ≥ 0∧[-3 + (-1)bso_25] + [-2]x2[2] + [-1]x0[2] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] + [(-1)bni_24]x0[2] ≥ 0∧[-3 + (-1)bso_25] + [-2]x2[2] + [-1]x0[2] ≥ 0)

We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(6)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] + [(-1)bni_24]x0[2] ≥ 0∧[-3 + (-1)bso_25] + [-2]x2[2] + [-1]x0[2] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(7)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] + [(-1)bni_24]x0[2] ≥ 0∧[-3 + (-1)bso_25] + [-2]x2[2] + [-1]x0[2] ≥ 0)

We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(8)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(754_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] + [(-1)bni_24]x0[2] ≥ 0∧[-3 + (-1)bso_25] + [-2]x2[2] + [-1]x0[2] ≥ 0)

We solved constraint (7) using rule (IDP_SMT_SPLIT).We solved constraint (8) using rule (IDP_SMT_SPLIT).

For Pair 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) the following chains were created:

We consider the chain COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:
(9)    (x0[1]=x2[2]∧x1[1]=x0[2]∧<=(x2[2], x0[2])=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧x0[3]=x0[0]∧x2[3]=x1[0]∧&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUE∧x0[0]=x0[1]1∧x1[0]=x1[1]1∧x0[1]1=x2[2]1∧x1[1]1=x0[2]1 ⇒ 618_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧618_0_GCD_EQ(x0[0], x1[0])≥COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(10)    (<=(x2[2], x0[2])=TRUE∧>(x2[2], 0)=TRUE∧>(x0[2], -1)=TRUE∧<(x0[2], x2[2])=TRUE ⇒ 618_0_GCD_EQ(x0[2], x2[2])≥NonInfC∧618_0_GCD_EQ(x0[2], x2[2])≥COND_618_0_GCD_EQ(&&(&&(>(x2[2], 0), >(x0[2], -1)), !(=(x0[2], x2[2]))), x0[2], x2[2])∧(U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[-1 + (-1)bso_27] + x0[2] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[-1 + (-1)bso_27] + x0[2] ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(14)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[-1 + (-1)bso_27] + x0[2] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(15)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[-1 + (-1)bso_27] + x0[2] ≥ 0)

We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[-1 + (-1)bso_27] + x0[2] ≥ 0)

We solved constraint (15) using rule (IDP_SMT_SPLIT).We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    (x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x2[2] + x0[2] ≥ 0∧[-1] + x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[-1 + (-1)bso_27] + x2[2] + x0[2] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x0[2] ≥ 0∧x2[2] ≥ 0∧[1] + x2[2] + x0[2] ≥ 0∧[-1] + x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[(-1)bso_27] + x2[2] + x0[2] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    ([1] + x0[2] ≥ 0∧x2[2] ≥ 0∧[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[1 + (-1)bso_27] + x2[2] + x0[2] ≥ 0)

For Pair COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]) the following chains were created:

We consider the chain 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]) which results in the following constraint:
(20)    (&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x2[2]∧x1[1]=x0[2]∧<=(x2[2], x0[2])=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧x0[3]=x0[0]1∧x2[3]=x1[0]1∧&&(&&(>(x1[0]1, 0), >(x0[0]1, -1)), !(=(x0[0]1, x1[0]1)))=TRUE∧x0[0]1=x0[1]1∧x1[0]1=x1[1]1 ⇒ COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥618_0_GCD_EQ(x0[3], x2[3])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (20) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(21)    (<=(x0[0], x1[0])=TRUE∧>(x1[0], 0)=TRUE∧>(x0[0], -1)=TRUE∧>(x0[0], 0)=TRUE∧>(x1[0], -1)=TRUE∧<(x0[0], x1[0])=TRUE∧<(x1[0], x0[0])=TRUE ⇒ COND_754_0_MOD_LE(TRUE, x0[0], x0[0], x1[0])≥NonInfC∧COND_754_0_MOD_LE(TRUE, x0[0], x0[0], x1[0])≥618_0_GCD_EQ(x1[0], x0[0])∧(U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (26) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(27)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (27) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(32)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We solved constraint (30) using rule (IDP_SMT_SPLIT).We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (x1[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x0[0] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We solved constraint (32) using rule (IDP_SMT_SPLIT).We solved constraint (33) using rule (IDP_SMT_SPLIT).We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35)    (x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(36)    ([1] + x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

For Pair 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) the following chains were created:

We consider the chain COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1]), 754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3]), 618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:
(37)    (x0[3]=x0[0]∧x2[3]=x1[0]∧&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x2[2]∧x1[1]=x0[2]∧<=(x2[2], x0[2])=TRUE∧x2[2]=x2[3]1∧x0[2]=x0[3]1∧x0[3]1=x0[0]1∧x2[3]1=x1[0]1 ⇒ 754_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧754_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))

We simplified constraint (37) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(38)    (<=(x0[0], x1[0])=TRUE∧>(x1[0], 0)=TRUE∧>(x0[0], -1)=TRUE∧<(x0[0], x1[0])=TRUE ⇒ 754_0_MOD_LE(x0[0], x0[0], x1[0])≥NonInfC∧754_0_MOD_LE(x0[0], x0[0], x1[0])≥COND_754_0_MOD_LE(<=(x0[0], x1[0]), x0[0], x0[0], x1[0])∧(U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))

We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(39)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[0] ≥ 0∧[2 + (-1)bso_31] + x1[0] + x0[0] ≥ 0)

We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(40)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[0] ≥ 0∧[2 + (-1)bso_31] + x1[0] + x0[0] ≥ 0)

We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(42)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[0] ≥ 0∧[2 + (-1)bso_31] + x1[0] + x0[0] ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(43)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[0] ≥ 0∧[2 + (-1)bso_31] + x1[0] + x0[0] ≥ 0)

We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[0] ≥ 0∧[2 + (-1)bso_31] + x1[0] + x0[0] ≥ 0)

We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45)    (x1[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x0[0] + [bni_30]x1[0] ≥ 0∧[2 + (-1)bso_31] + [2]x0[0] + x1[0] ≥ 0)

We solved constraint (44) using rule (IDP_SMT_SPLIT).We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(46)    ([1] + x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[0] + [bni_30]x1[0] ≥ 0∧[3 + (-1)bso_31] + [2]x0[0] + x1[0] ≥ 0)

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

COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
- ([1] + x0[2] ≥ 0∧x2[2] ≥ 0∧[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x2[2] ≥ 0∧[1 + (-1)bso_27] + x2[2] + x0[2] ≥ 0)
COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])
- ([1] + x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(618_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)
754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
- ([1] + x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[0] + [bni_30]x1[0] ≥ 0∧[3 + (-1)bso_31] + [2]x0[0] + x1[0] ≥ 0)

POL(TRUE) = [1]
POL(FALSE) = [2]
POL(COND_618_0_GCD_EQ(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁
POL(754_0_MOD_LE(x₁, x₂, x₃)) = [1] + x₃ + x₂ + [-1]x₁
POL(618_0_GCD_EQ(x₁, x₂)) = [-1] + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(-1) = [-1]
POL(!(x₁)) = [-1]
POL(=(x₁, x₂)) = [-1]
POL(COND_754_0_MOD_LE(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₂
POL(<=(x₁, x₂)) = [2]

The following pairs are in P_>:

COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
618_0_GCD_EQ(x0[0], x1[0]) → COND_618_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])

The following pairs are in P_bound:

COND_618_0_GCD_EQ(TRUE, x0[1], x1[1]) → 754_0_MOD_LE(x0[1], x0[1], x1[1])
754_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_754_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])

The following pairs are in P_≥:

COND_754_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 618_0_GCD_EQ(x0[3], x2[3])

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

(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:

Boolean, Integer

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x1[0])

The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(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) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) IDPNonInfProof (SOUND transformation)

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE