Termination proof of /tmp/tmpnndjFk/Log.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 190 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 140 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 970 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 IDP

↳11 IDPNonInfProof (⇒, 70 ms)

↳12 AND

    ↳13 IDP

        ↳14 IDependencyGraphProof (⇔, 0 ms)

        ↳15 TRUE

    ↳16 IDP

        ↳17 IDependencyGraphProof (⇔, 0 ms)

        ↳18 TRUE

(0) Obligation:

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

public class Log{
  public static int half(int x) {

    int res = 0;

    while (x > 1) {

      x = x-2;
      res++;

    }

    return res;

  }


  public static int log(int x) {

    int res = 0;

    while (x > 1) {

      x = half(x);
      res++;

    }

    return res;

  }


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


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:
Log.main([Ljava/lang/String;)V: Graph of 141 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: Log.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 31 rules for P and 0 rules for R.

P rules:
616_0_log_ConstantStackPush(EOS(STATIC_616), i118, i118) → 618_0_log_LE(EOS(STATIC_618), i118, i118, 1)
618_0_log_LE(EOS(STATIC_618), i126, i126, matching1) → 622_0_log_LE(EOS(STATIC_622), i126, i126, 1) | =(matching1, 1)
622_0_log_LE(EOS(STATIC_622), i126, i126, matching1) → 626_0_log_Load(EOS(STATIC_626), i126) | &&(>(i126, 1), =(matching1, 1))
626_0_log_Load(EOS(STATIC_626), i126) → 630_0_log_InvokeMethod(EOS(STATIC_630), i126)
630_0_log_InvokeMethod(EOS(STATIC_630), i126) → 633_0_half_ConstantStackPush(EOS(STATIC_633), i126, i126)
633_0_half_ConstantStackPush(EOS(STATIC_633), i126, i126) → 639_0_half_Store(EOS(STATIC_639), i126, i126, 0)
639_0_half_Store(EOS(STATIC_639), i126, i126, matching1) → 641_0_half_Load(EOS(STATIC_641), i126, i126, 0) | =(matching1, 0)
641_0_half_Load(EOS(STATIC_641), i126, i126, matching1) → 667_0_half_Load(EOS(STATIC_667), i126, i126, 0) | =(matching1, 0)
667_0_half_Load(EOS(STATIC_667), i126, i129, i130) → 710_0_half_Load(EOS(STATIC_710), i126, i129, i130)
710_0_half_Load(EOS(STATIC_710), i126, i144, i145) → 755_0_half_Load(EOS(STATIC_755), i126, i144, i145)
755_0_half_Load(EOS(STATIC_755), i126, i159, i160) → 799_0_half_Load(EOS(STATIC_799), i126, i159, i160)
799_0_half_Load(EOS(STATIC_799), i126, i175, i176) → 802_0_half_ConstantStackPush(EOS(STATIC_802), i126, i175, i176, i175)
802_0_half_ConstantStackPush(EOS(STATIC_802), i126, i175, i176, i175) → 805_0_half_LE(EOS(STATIC_805), i126, i175, i176, i175, 1)
805_0_half_LE(EOS(STATIC_805), i126, i182, i176, i182, matching1) → 807_0_half_LE(EOS(STATIC_807), i126, i182, i176, i182, 1) | =(matching1, 1)
805_0_half_LE(EOS(STATIC_805), i126, i183, i176, i183, matching1) → 808_0_half_LE(EOS(STATIC_808), i126, i183, i176, i183, 1) | =(matching1, 1)
807_0_half_LE(EOS(STATIC_807), i126, i182, i176, i182, matching1) → 810_0_half_Load(EOS(STATIC_810), i126, i176) | &&(<=(i182, 1), =(matching1, 1))
810_0_half_Load(EOS(STATIC_810), i126, i176) → 814_0_half_Return(EOS(STATIC_814), i126, i176)
814_0_half_Return(EOS(STATIC_814), i126, i176) → 818_0_log_Store(EOS(STATIC_818), i176)
818_0_log_Store(EOS(STATIC_818), i176) → 822_0_log_Inc(EOS(STATIC_822), i176)
822_0_log_Inc(EOS(STATIC_822), i176) → 826_0_log_JMP(EOS(STATIC_826), i176)
826_0_log_JMP(EOS(STATIC_826), i176) → 832_0_log_Load(EOS(STATIC_832), i176)
832_0_log_Load(EOS(STATIC_832), i176) → 612_0_log_Load(EOS(STATIC_612), i176)
612_0_log_Load(EOS(STATIC_612), i118) → 616_0_log_ConstantStackPush(EOS(STATIC_616), i118, i118)
808_0_half_LE(EOS(STATIC_808), i126, i183, i176, i183, matching1) → 812_0_half_Load(EOS(STATIC_812), i126, i183, i176) | &&(>(i183, 1), =(matching1, 1))
812_0_half_Load(EOS(STATIC_812), i126, i183, i176) → 816_0_half_ConstantStackPush(EOS(STATIC_816), i126, i176, i183)
816_0_half_ConstantStackPush(EOS(STATIC_816), i126, i176, i183) → 820_0_half_IntArithmetic(EOS(STATIC_820), i126, i176, i183, 2)
820_0_half_IntArithmetic(EOS(STATIC_820), i126, i176, i183, matching1) → 824_0_half_Store(EOS(STATIC_824), i126, i176, -(i183, 2)) | &&(>(i183, 0), =(matching1, 2))
824_0_half_Store(EOS(STATIC_824), i126, i176, i184) → 828_0_half_Inc(EOS(STATIC_828), i126, i184, i176)
828_0_half_Inc(EOS(STATIC_828), i126, i184, i176) → 834_0_half_JMP(EOS(STATIC_834), i126, i184, +(i176, 1)) | >=(i176, 0)
834_0_half_JMP(EOS(STATIC_834), i126, i184, i189) → 838_0_half_Load(EOS(STATIC_838), i126, i184, i189)
838_0_half_Load(EOS(STATIC_838), i126, i184, i189) → 799_0_half_Load(EOS(STATIC_799), i126, i184, i189)
R rules:

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

P rules:
805_0_half_LE(EOS(STATIC_805), x0, x1, x2, x1, 1) → 805_0_half_LE(EOS(STATIC_805), x2, x2, 0, x2, 1) | &&(>(x2, 1), <=(x1, 1))
805_0_half_LE(EOS(STATIC_805), x0, x1, x2, x1, 1) → 805_0_half_LE(EOS(STATIC_805), x0, -(x1, 2), +(x2, 1), -(x1, 2), 1) | &&(>(+(x2, 1), 0), >(x1, 1))
R rules:

Filtered ground terms:

805_0_half_LE(x1, x2, x3, x4, x5, x6) → 805_0_half_LE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_805_0_half_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_805_0_half_LE1(x1, x3, x4, x5, x6)
Cond_805_0_half_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_805_0_half_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:

805_0_half_LE(x1, x2, x3, x4) → 805_0_half_LE(x1, x3, x4)
Cond_805_0_half_LE(x1, x2, x3, x4, x5) → Cond_805_0_half_LE(x1, x2, x4, x5)
Cond_805_0_half_LE1(x1, x2, x3, x4, x5) → Cond_805_0_half_LE1(x1, x2, x4, x5)

Filtered unneeded arguments:

Cond_805_0_half_LE(x1, x2, x3, x4) → Cond_805_0_half_LE(x1, x3)
Cond_805_0_half_LE1(x1, x2, x3, x4) → Cond_805_0_half_LE1(x1, x3, x4)
805_0_half_LE(x1, x2, x3) → 805_0_half_LE(x2, x3)

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

P rules:
805_0_half_LE(x2, x1) → 805_0_half_LE(0, x2) | &&(>(x2, 1), <=(x1, 1))
805_0_half_LE(x2, x1) → 805_0_half_LE(+(x2, 1), -(x1, 2)) | &&(>(x2, -1), >(x1, 1))
R rules:

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

P rules:
805_0_HALF_LE(x2, x1) → COND_805_0_HALF_LE(&&(>(x2, 1), <=(x1, 1)), x2, x1)
COND_805_0_HALF_LE(TRUE, x2, x1) → 805_0_HALF_LE(0, x2)
805_0_HALF_LE(x2, x1) → COND_805_0_HALF_LE1(&&(>(x2, -1), >(x1, 1)), x2, x1)
COND_805_0_HALF_LE1(TRUE, x2, x1) → 805_0_HALF_LE(+(x2, 1), -(x1, 2))
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): 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(x2[0] > 1 && x1[0] <= 1, x2[0], x1[0])
(1): COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])
(2): 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(x2[2] > -1 && x1[2] > 1, x2[2], x1[2])
(3): COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(x2[3] + 1, x1[3] - 2)

(0) -> (1), if (x2[0] > 1 && x1[0] <= 1 ∧x2[0] →^* x2[1]∧x1[0] →^* x1[1])

(1) -> (0), if (0 →^* x2[0]∧x2[1] →^* x1[0])

(1) -> (2), if (0 →^* x2[2]∧x2[1] →^* x1[2])

(2) -> (3), if (x2[2] > -1 && x1[2] > 1 ∧x2[2] →^* x2[3]∧x1[2] →^* x1[3])

(3) -> (0), if (x2[3] + 1 →^* x2[0]∧x1[3] - 2 →^* x1[0])

(3) -> (2), if (x2[3] + 1 →^* x2[2]∧x1[3] - 2 →^* x1[2])

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@4280e6ab 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 805_0_HALF_LE(x2, x1) → COND_805_0_HALF_LE(&&(>(x2, 1), <=(x1, 1)), x2, x1) the following chains were created:

We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) which results in the following constraint:
(1)    (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ 805_0_HALF_LE(x2[0], x1[0])≥NonInfC∧805_0_HALF_LE(x2[0], x1[0])≥COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])∧(U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x2[0], 1)=TRUE∧<=(x1[0], 1)=TRUE ⇒ 805_0_HALF_LE(x2[0], x1[0])≥NonInfC∧805_0_HALF_LE(x2[0], x1[0])≥COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])∧(U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

(8)    (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_805_0_HALF_LE(TRUE, x2, x1) → 805_0_HALF_LE(0, x2) the following chains were created:

We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) which results in the following constraint:
(9) (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1∧&&(>(x2[0]1, 1), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[0]2∧x2[1]1=x1[0]2∧&&(>(x2[0]2, 1), <=(x1[0]2, 1))=TRUE∧x2[0]2=x2[1]2∧x1[0]2=x1[1]2 ⇒ COND_805_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_805_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥805_0_HALF_LE(0, x2[1]1)∧(U^Increasing(805_0_HALF_LE(0, x2[1]1)), ≥))

We solved constraint (9) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(10) (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1∧&&(>(x2[0]1, 1), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[2]∧x2[1]1=x1[2]∧&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ COND_805_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_805_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥805_0_HALF_LE(0, x2[1]1)∧(U^Increasing(805_0_HALF_LE(0, x2[1]1)), ≥))

We solved constraint (10) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) which results in the following constraint:
(11) (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]∧-(x1[3], 2)=x1[0]∧&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1∧&&(>(x2[0]1, 1), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_805_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_805_0_HALF_LE(TRUE, x2[1], x1[1])≥805_0_HALF_LE(0, x2[1])∧(U^Increasing(805_0_HALF_LE(0, x2[1])), ≥))

We solved constraint (11) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(12)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]∧-(x1[3], 2)=x1[0]∧&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]1∧x2[1]=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1 ⇒ COND_805_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_805_0_HALF_LE(TRUE, x2[1], x1[1])≥805_0_HALF_LE(0, x2[1])∧(U^Increasing(805_0_HALF_LE(0, x2[1])), ≥))

We simplified constraint (12) using rules (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
(13)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE∧>(+(x2[2], 1), 1)=TRUE∧<=(-(x1[2], 2), 1)=TRUE ⇒ COND_805_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_805_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥805_0_HALF_LE(0, +(x2[2], 1))∧(U^Increasing(805_0_HALF_LE(0, x2[1])), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [-1] ≥ 0∧[3] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [-1] ≥ 0∧[3] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [-1] ≥ 0∧[3] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    ([1] + x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    ([1] + x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_19 + bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[(-1)bso_20] + x1[2] ≥ 0)

For Pair 805_0_HALF_LE(x2, x1) → COND_805_0_HALF_LE1(&&(>(x2, -1), >(x1, 1)), x2, x1) the following chains were created:

We consider the chain 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(19)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ 805_0_HALF_LE(x2[2], x1[2])≥NonInfC∧805_0_HALF_LE(x2[2], x1[2])≥COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])∧(U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥))

We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(20)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE ⇒ 805_0_HALF_LE(x2[2], x1[2])≥NonInfC∧805_0_HALF_LE(x2[2], x1[2])≥COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])∧(U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(24)    (x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_805_0_HALF_LE1(TRUE, x2, x1) → 805_0_HALF_LE(+(x2, 1), -(x1, 2)) the following chains were created:

We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) which results in the following constraint:
(25) (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧x2[1]=x1[2]∧&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]1∧-(x1[3], 2)=x1[0]1∧&&(>(x2[0]1, 1), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_805_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_805_0_HALF_LE1(TRUE, x2[3], x1[3])≥805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))

We solved constraint (25) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN).
We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(26)    (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧x2[1]=x1[2]∧&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1 ⇒ COND_805_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_805_0_HALF_LE1(TRUE, x2[3], x1[3])≥805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))

We simplified constraint (26) using rules (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
(27)    (>(x2[0], 1)=TRUE∧<=(x1[0], 1)=TRUE∧>(-(x2[0], 2), 1)=TRUE ⇒ COND_805_0_HALF_LE1(TRUE, 0, x2[0])≥NonInfC∧COND_805_0_HALF_LE1(TRUE, 0, x2[0])≥805_0_HALF_LE(+(0, 1), -(x2[0], 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-4] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-4] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-4] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(31)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧[-2] + x2[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32)    ([2] + x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    ([2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
We consider the chain 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) which results in the following constraint:
(34)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1∧+(x2[3]1, 1)=x2[0]∧-(x1[3]1, 2)=x1[0]∧&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ COND_805_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_805_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))

We simplified constraint (34) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(35)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE∧>(+(x2[2], 1), -1)=TRUE∧>(-(x1[2], 2), 1)=TRUE∧>(+(+(x2[2], 1), 1), 1)=TRUE∧<=(-(-(x1[2], 2), 2), 1)=TRUE ⇒ COND_805_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_805_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥805_0_HALF_LE(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(36)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] ≥ 0∧[5] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(37)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] ≥ 0∧[5] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(38)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] ≥ 0∧[5] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(39)    (x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
We consider the chain 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(41)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1∧+(x2[3]1, 1)=x2[2]2∧-(x1[3]1, 2)=x1[2]2∧&&(>(x2[2]2, -1), >(x1[2]2, 1))=TRUE∧x2[2]2=x2[3]2∧x1[2]2=x1[3]2 ⇒ COND_805_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_805_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))

We simplified constraint (41) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(42)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE∧>(+(x2[2], 1), -1)=TRUE∧>(-(x1[2], 2), 1)=TRUE∧>(+(+(x2[2], 1), 1), -1)=TRUE∧>(-(-(x1[2], 2), 2), 1)=TRUE ⇒ COND_805_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_805_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥805_0_HALF_LE(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))

We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(43)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] + [-6] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(44)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] + [-6] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(45)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] + [-6] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(46)    (x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧[-4] + x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(47)    (x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧[-2] + x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(48)    (x2[2] ≥ 0∧[4] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(4)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

805_0_HALF_LE(x2, x1) → COND_805_0_HALF_LE(&&(>(x2, 1), <=(x1, 1)), x2, x1)
- (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)
- (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)
COND_805_0_HALF_LE(TRUE, x2, x1) → 805_0_HALF_LE(0, x2)
- ([1] + x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_19 + bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[(-1)bso_20] + x1[2] ≥ 0)
805_0_HALF_LE(x2, x1) → COND_805_0_HALF_LE1(&&(>(x2, -1), >(x1, 1)), x2, x1)
- (x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)
COND_805_0_HALF_LE1(TRUE, x2, x1) → 805_0_HALF_LE(+(x2, 1), -(x1, 2))
- ([2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
- (x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
- (x2[2] ≥ 0∧[4] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(4)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = [3]
POL(805_0_HALF_LE(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_805_0_HALF_LE(x₁, x₂, x₃)) = [-1] + x₃ + x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(COND_805_0_HALF_LE1(x₁, x₂, x₃)) = [-1] + x₃ + x₂ + [-1]x₁
POL(-1) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(2) = [2]

The following pairs are in P_>:

COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))

The following pairs are in P_bound:

COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])
805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])
COND_805_0_HALF_LE1(TRUE, x2[3], x1[3]) → 805_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))

The following pairs are in P_≥:

805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])
COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])
805_0_HALF_LE(x2[2], x1[2]) → COND_805_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])

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

(1) -> (0), if (0 →^* x2[0]∧x2[1] →^* x1[0])

(0) -> (1), if (x2[0] > 1 && x1[0] <= 1 ∧x2[0] →^* x2[1]∧x1[0] →^* x1[1])

(1) -> (2), if (0 →^* x2[2]∧x2[1] →^* x1[2])

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])
(0): 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(x2[0] > 1 && x1[0] <= 1, x2[0], x1[0])

(1) -> (0), if (0 →^* x2[0]∧x2[1] →^* x1[0])

(0) -> (1), if (x2[0] > 1 && x1[0] <= 1 ∧x2[0] →^* x2[1]∧x1[0] →^* x1[1])

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@1ca806a9 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_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) the following chains were created:

We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]), 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]) which results in the following constraint:
(1)    (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1 ⇒ COND_805_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_805_0_HALF_LE(TRUE, x2[1], x1[1])≥805_0_HALF_LE(0, x2[1])∧(U^Increasing(805_0_HALF_LE(0, x2[1])), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x2[0], 1)=TRUE∧<=(x1[0], 1)=TRUE ⇒ COND_805_0_HALF_LE(TRUE, x2[0], x1[0])≥NonInfC∧COND_805_0_HALF_LE(TRUE, x2[0], x1[0])≥805_0_HALF_LE(0, x2[0])∧(U^Increasing(805_0_HALF_LE(0, x2[1])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)

(8)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)

For Pair 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]) the following chains were created:

We consider the chain 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0]), COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1]) which results in the following constraint:
(9)    (&&(>(x2[0], 1), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ 805_0_HALF_LE(x2[0], x1[0])≥NonInfC∧805_0_HALF_LE(x2[0], x1[0])≥COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])∧(U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥))

We simplified constraint (9) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(10)    (>(x2[0], 1)=TRUE∧<=(x1[0], 1)=TRUE ⇒ 805_0_HALF_LE(x2[0], x1[0])≥NonInfC∧805_0_HALF_LE(x2[0], x1[0])≥COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])∧(U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[-3 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[-3 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (x2[0] + [-2] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[-3 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(15)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)

(16)    (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] + x1[0] + [2]x2[0] ≥ 0)

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

COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])
- (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)
- (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(805_0_HALF_LE(0, x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[3 + (-1)bso_14] ≥ 0)
805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])
- (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)
- (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] + x1[0] + [2]x2[0] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_805_0_HALF_LE(x₁, x₂, x₃)) = [2] + [-1]x₂ + [-1]x₁
POL(805_0_HALF_LE(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(0) = 0
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(<=(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])

The following pairs are in P_bound:

805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])

The following pairs are in P_≥:

805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(&&(>(x2[0], 1), <=(x1[0], 1)), x2[0], x1[0])

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

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

(12) Complex Obligation (AND)

(13) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 805_0_HALF_LE(x2[0], x1[0]) → COND_805_0_HALF_LE(x2[0] > 1 && x1[0] <= 1, x2[0], x1[0])

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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

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

The following domains are used:
none

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_805_0_HALF_LE(TRUE, x2[1], x1[1]) → 805_0_HALF_LE(0, x2[1])

The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) IDPNonInfProof (SOUND transformation)

(12) Complex Obligation (AND)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE