Termination proof of /tmp/tmpnTnieG/PastaB18.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 70 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 120 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 460 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 IDP

↳11 IDPNonInfProof (⇒, 90 ms)

↳12 IDP

↳13 IDependencyGraphProof (⇔, 0 ms)

↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB18

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB18 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > 0 && y > 0) {
            if (x > y) {
                while (x > 0) {
                    x--;
                }
            } else {
                while (y > 0) {
                    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:
PastaB18.main([Ljava/lang/String;)V: Graph of 198 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: PastaB18.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 32 rules for P and 0 rules for R.

P rules:
303_0_main_LE(EOS(STATIC_303), i51, i47, i51) → 308_0_main_LE(EOS(STATIC_308), i51, i47, i51)
308_0_main_LE(EOS(STATIC_308), i51, i47, i51) → 320_0_main_Load(EOS(STATIC_320), i51, i47) | >(i51, 0)
320_0_main_Load(EOS(STATIC_320), i51, i47) → 330_0_main_LE(EOS(STATIC_330), i51, i47, i47)
330_0_main_LE(EOS(STATIC_330), i51, i54, i54) → 333_0_main_LE(EOS(STATIC_333), i51, i54, i54)
333_0_main_LE(EOS(STATIC_333), i51, i54, i54) → 347_0_main_Load(EOS(STATIC_347), i51, i54) | >(i54, 0)
347_0_main_Load(EOS(STATIC_347), i51, i54) → 358_0_main_Load(EOS(STATIC_358), i51, i54, i51)
358_0_main_Load(EOS(STATIC_358), i51, i54, i51) → 367_0_main_LE(EOS(STATIC_367), i51, i54, i51, i54)
367_0_main_LE(EOS(STATIC_367), i51, i54, i51, i54) → 375_0_main_LE(EOS(STATIC_375), i51, i54, i51, i54)
367_0_main_LE(EOS(STATIC_367), i51, i54, i51, i54) → 376_0_main_LE(EOS(STATIC_376), i51, i54, i51, i54)
375_0_main_LE(EOS(STATIC_375), i51, i54, i51, i54) → 389_0_main_Load(EOS(STATIC_389), i51, i54) | <=(i51, i54)
389_0_main_Load(EOS(STATIC_389), i51, i54) → 439_0_main_Load(EOS(STATIC_439), i51, i54)
439_0_main_Load(EOS(STATIC_439), i51, i68) → 456_0_main_LE(EOS(STATIC_456), i51, i68, i68)
456_0_main_LE(EOS(STATIC_456), i51, matching1, matching2) → 464_0_main_LE(EOS(STATIC_464), i51, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
456_0_main_LE(EOS(STATIC_456), i51, i79, i79) → 465_0_main_LE(EOS(STATIC_465), i51, i79, i79)
464_0_main_LE(EOS(STATIC_464), i51, matching1, matching2) → 477_0_main_Load(EOS(STATIC_477), i51, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
477_0_main_Load(EOS(STATIC_477), i51, matching1) → 297_0_main_Load(EOS(STATIC_297), i51, 0) | =(matching1, 0)
297_0_main_Load(EOS(STATIC_297), i18, i47) → 303_0_main_LE(EOS(STATIC_303), i18, i47, i18)
465_0_main_LE(EOS(STATIC_465), i51, i79, i79) → 479_0_main_Inc(EOS(STATIC_479), i51, i79) | >(i79, 0)
479_0_main_Inc(EOS(STATIC_479), i51, i79) → 491_0_main_JMP(EOS(STATIC_491), i51, +(i79, -1)) | >(i79, 0)
491_0_main_JMP(EOS(STATIC_491), i51, i88) → 499_0_main_Load(EOS(STATIC_499), i51, i88)
499_0_main_Load(EOS(STATIC_499), i51, i88) → 439_0_main_Load(EOS(STATIC_439), i51, i88)
376_0_main_LE(EOS(STATIC_376), i51, i54, i51, i54) → 391_0_main_Load(EOS(STATIC_391), i51, i54) | >(i51, i54)
391_0_main_Load(EOS(STATIC_391), i51, i54) → 450_0_main_Load(EOS(STATIC_450), i51, i54)
450_0_main_Load(EOS(STATIC_450), i72, i54) → 459_0_main_LE(EOS(STATIC_459), i72, i54, i72)
459_0_main_LE(EOS(STATIC_459), matching1, i54, matching2) → 467_0_main_LE(EOS(STATIC_467), 0, i54, 0) | &&(=(matching1, 0), =(matching2, 0))
459_0_main_LE(EOS(STATIC_459), i81, i54, i81) → 468_0_main_LE(EOS(STATIC_468), i81, i54, i81)
467_0_main_LE(EOS(STATIC_467), matching1, i54, matching2) → 484_0_main_Load(EOS(STATIC_484), 0, i54) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
484_0_main_Load(EOS(STATIC_484), matching1, i54) → 297_0_main_Load(EOS(STATIC_297), 0, i54) | =(matching1, 0)
468_0_main_LE(EOS(STATIC_468), i81, i54, i81) → 486_0_main_Inc(EOS(STATIC_486), i81, i54) | >(i81, 0)
486_0_main_Inc(EOS(STATIC_486), i81, i54) → 494_0_main_JMP(EOS(STATIC_494), +(i81, -1), i54) | >(i81, 0)
494_0_main_JMP(EOS(STATIC_494), i89, i54) → 503_0_main_Load(EOS(STATIC_503), i89, i54)
503_0_main_Load(EOS(STATIC_503), i89, i54) → 450_0_main_Load(EOS(STATIC_450), i89, i54)
R rules:

Combined rules. Obtained 6 conditional rules for P and 0 conditional rules for R.

P rules:
303_0_main_LE(EOS(STATIC_303), x0, x1, x0) → 456_0_main_LE(EOS(STATIC_456), x0, x1, x1) | &&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0))
456_0_main_LE(EOS(STATIC_456), x0, 0, 0) → 303_0_main_LE(EOS(STATIC_303), x0, 0, x0)
456_0_main_LE(EOS(STATIC_456), x0, x1, x1) → 456_0_main_LE(EOS(STATIC_456), x0, +(x1, -1), +(x1, -1)) | >(x1, 0)
303_0_main_LE(EOS(STATIC_303), x0, x1, x0) → 459_0_main_LE(EOS(STATIC_459), x0, x1, x0) | &&(&&(>(x1, 0), <(x1, x0)), >(x0, 0))
459_0_main_LE(EOS(STATIC_459), 0, x1, 0) → 303_0_main_LE(EOS(STATIC_303), 0, x1, 0)
459_0_main_LE(EOS(STATIC_459), x0, x1, x0) → 459_0_main_LE(EOS(STATIC_459), +(x0, -1), x1, +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

459_0_main_LE(x1, x2, x3, x4) → 459_0_main_LE(x2, x3, x4)
Cond_459_0_main_LE(x1, x2, x3, x4, x5) → Cond_459_0_main_LE(x1, x3, x4, x5)
303_0_main_LE(x1, x2, x3, x4) → 303_0_main_LE(x2, x3, x4)
Cond_303_0_main_LE1(x1, x2, x3, x4, x5) → Cond_303_0_main_LE1(x1, x3, x4, x5)
456_0_main_LE(x1, x2, x3, x4) → 456_0_main_LE(x2, x3, x4)
Cond_456_0_main_LE(x1, x2, x3, x4, x5) → Cond_456_0_main_LE(x1, x3, x4, x5)
Cond_303_0_main_LE(x1, x2, x3, x4, x5) → Cond_303_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

303_0_main_LE(x1, x2, x3) → 303_0_main_LE(x2, x3)
Cond_303_0_main_LE(x1, x2, x3, x4) → Cond_303_0_main_LE(x1, x3, x4)
456_0_main_LE(x1, x2, x3) → 456_0_main_LE(x1, x3)
Cond_456_0_main_LE(x1, x2, x3, x4) → Cond_456_0_main_LE(x1, x2, x4)
Cond_303_0_main_LE1(x1, x2, x3, x4) → Cond_303_0_main_LE1(x1, x3, x4)
459_0_main_LE(x1, x2, x3) → 459_0_main_LE(x2, x3)
Cond_459_0_main_LE(x1, x2, x3, x4) → Cond_459_0_main_LE(x1, x3, x4)

Combined rules. Obtained 6 conditional rules for P and 0 conditional rules for R.

P rules:
303_0_main_LE(x1, x0) → 456_0_main_LE(x0, x1) | &&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0))
456_0_main_LE(x0, 0) → 303_0_main_LE(0, x0)
456_0_main_LE(x0, x1) → 456_0_main_LE(x0, +(x1, -1)) | >(x1, 0)
303_0_main_LE(x1, x0) → 459_0_main_LE(x1, x0) | &&(&&(>(x1, 0), <(x1, x0)), >(x0, 0))
459_0_main_LE(x1, 0) → 303_0_main_LE(x1, 0)
459_0_main_LE(x1, x0) → 459_0_main_LE(x1, +(x0, -1)) | >(x0, 0)
R rules:

Performed bisimulation on rules. Used the following equivalence classes: {[456_0_main_LE_2, 459_0_main_LE_2]=456_0_main_LE_2, [Cond_456_0_main_LE_3, Cond_459_0_main_LE_3]=Cond_456_0_main_LE_3}

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

P rules:
303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0)
COND_303_0_MAIN_LE(TRUE, x1, x0) → 456_0_MAIN_LE(x0, x1)
456_0_MAIN_LE(x0, 0) → 303_0_MAIN_LE(0, x0)
456_0_MAIN_LE(x0, x1) → COND_456_0_MAIN_LE(>(x1, 0), x0, x1)
COND_456_0_MAIN_LE(TRUE, x0, x1) → 456_0_MAIN_LE(x0, +(x1, -1))
303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0)
COND_303_0_MAIN_LE1(TRUE, x1, x0) → 456_0_MAIN_LE(x1, x0)
456_0_MAIN_LE(x1, 0) → 303_0_MAIN_LE(x1, 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): 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(x1[0] >= x0[0] && x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1])
(2): 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2])
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], x1[4] + -1)
(5): 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0, x1[5], x0[5])
(6): COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6])
(7): 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0)

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

(1) -> (2), if (x0[1] →^* x0[2]∧x1[1] →^* 0)

(1) -> (3), if (x0[1] →^* x0[3]∧x1[1] →^* x1[3])

(1) -> (7), if (x0[1] →^* x1[7]∧x1[1] →^* 0)

(2) -> (0), if (0 →^* x1[0]∧x0[2] →^* x0[0])

(2) -> (5), if (0 →^* x1[5]∧x0[2] →^* x0[5])

(3) -> (4), if (x1[3] > 0 ∧x0[3] →^* x0[4]∧x1[3] →^* x1[4])

(4) -> (2), if (x0[4] →^* x0[2]∧x1[4] + -1 →^* 0)

(4) -> (3), if (x0[4] →^* x0[3]∧x1[4] + -1 →^* x1[3])

(4) -> (7), if (x0[4] →^* x1[7]∧x1[4] + -1 →^* 0)

(5) -> (6), if (x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0 ∧x1[5] →^* x1[6]∧x0[5] →^* x0[6])

(6) -> (2), if (x1[6] →^* x0[2]∧x0[6] →^* 0)

(6) -> (3), if (x1[6] →^* x0[3]∧x0[6] →^* x1[3])

(6) -> (7), if (x1[6] →^* x1[7]∧x0[6] →^* 0)

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

(7) -> (5), if (x1[7] →^* x1[5]∧0 →^* x0[5])

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@31068b41 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 303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]), COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]) which results in the following constraint:
(1)    (&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 303_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧303_0_MAIN_LE(x1[0], x0[0])≥COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE∧>=(x1[0], x0[0])=TRUE∧>(x1[0], 0)=TRUE ⇒ 303_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧303_0_MAIN_LE(x1[0], x0[0])≥COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_25 + bni_25] + [(2)bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[1 + (-1)bso_26] + x0[0] + x1[0] ≥ 0)

For Pair COND_303_0_MAIN_LE(TRUE, x1, x0) → 456_0_MAIN_LE(x0, x1) the following chains were created:

We consider the chain COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]), 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]) which results in the following constraint:
(8)    (x0[1]=x0[2]∧x1[1]=0 ⇒ COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9)    (COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥456_0_MAIN_LE(x0[1], 0)∧(U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
We consider the chain COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:
(14)    (x0[1]=x0[3]∧x1[1]=x1[3] ⇒ COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (14) using rule (IV) which results in the following new constraint:
(15)    (COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
We consider the chain COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]), 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0) which results in the following constraint:
(20)    (x0[1]=x1[7]∧x1[1]=0 ⇒ COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (20) using rules (III), (IV) which results in the following new constraint:
(21)    (COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥456_0_MAIN_LE(x0[1], 0)∧(U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(22)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(23)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(24)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(25)    ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

For Pair 456_0_MAIN_LE(x0, 0) → 303_0_MAIN_LE(0, x0) the following chains were created:

We consider the chain 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]), 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:
(26)    (0=x1[0]∧x0[2]=x0[0] ⇒ 456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (26) using rule (IV) which results in the following new constraint:
(27)    (456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(28)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(29)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(30)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(31)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)
We consider the chain 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]), 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:
(32)    (0=x1[5]∧x0[2]=x0[5] ⇒ 456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (32) using rule (IV) which results in the following new constraint:
(33)    (456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(35)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(36)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(37)    ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

For Pair 456_0_MAIN_LE(x0, x1) → COND_456_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:

We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:
(38)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

We simplified constraint (38) using rule (IV) which results in the following new constraint:
(39)    (>(x1[3], 0)=TRUE ⇒ 456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(40)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[3] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(41)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[3] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(42)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[3] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(43)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_31] = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(44)    (x1[3] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_31] = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧[(-1)bso_32] ≥ 0)

For Pair COND_456_0_MAIN_LE(TRUE, x0, x1) → 456_0_MAIN_LE(x0, +(x1, -1)) the following chains were created:

We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]) which results in the following constraint:
(45)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x0[2]∧+(x1[4], -1)=0 ⇒ COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (45) using rules (III), (IV) which results in the following new constraint:
(46)    (>(x1[3], 0)=TRUE∧+(x1[3], -1)=0 ⇒ COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (46) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(47)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (47) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(48)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (48) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(49)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (49) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(50)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(51)    (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:
(52)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x0[3]1∧+(x1[4], -1)=x1[3]1 ⇒ COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (52) using rules (III), (IV) which results in the following new constraint:
(53)    (>(x1[3], 0)=TRUE ⇒ COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(54)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(55)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(56)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(57)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(58)    (x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0) which results in the following constraint:
(59)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x1[7]∧+(x1[4], -1)=0 ⇒ COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (59) using rules (III), (IV) which results in the following new constraint:
(60)    (>(x1[3], 0)=TRUE∧+(x1[3], -1)=0 ⇒ COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (60) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(61)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (61) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(62)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (62) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(63)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (63) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(64)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(65)    (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0) the following chains were created:

We consider the chain 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]), COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]) which results in the following constraint:
(66)    (&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0))=TRUE∧x1[5]=x1[6]∧x0[5]=x0[6] ⇒ 303_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧303_0_MAIN_LE(x1[5], x0[5])≥COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))

We simplified constraint (66) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(67)    (>(x0[5], 0)=TRUE∧>(x1[5], 0)=TRUE∧<(x1[5], x0[5])=TRUE ⇒ 303_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧303_0_MAIN_LE(x1[5], x0[5])≥COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))

We simplified constraint (67) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(68)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[(-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (68) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(69)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[(-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (69) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(70)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[(-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(71)    (x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1]x1[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[1 + (-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(72)    (x1[5] + x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x1[5] + [bni_35]x0[5] ≥ 0∧[1 + (-1)bso_36] + x1[5] + x0[5] ≥ 0)

We simplified constraint (72) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(73)    ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(2)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x1[5] + [bni_35]x0[5] ≥ 0∧[2 + (-1)bso_36] + x1[5] + x0[5] ≥ 0)

For Pair COND_303_0_MAIN_LE1(TRUE, x1, x0) → 456_0_MAIN_LE(x1, x0) the following chains were created:

We consider the chain COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]), 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]) which results in the following constraint:
(74)    (x1[6]=x0[2]∧x0[6]=0 ⇒ COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (74) using rules (III), (IV) which results in the following new constraint:
(75)    (COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥456_0_MAIN_LE(x1[6], 0)∧(U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (75) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(76)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (76) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(77)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (77) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(78)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (78) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(79)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
We consider the chain COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:
(80)    (x1[6]=x0[3]∧x0[6]=x1[3] ⇒ COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (80) using rule (IV) which results in the following new constraint:
(81)    (COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (81) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(82)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (82) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(83)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (83) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(84)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (84) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(85)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
We consider the chain COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]), 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0) which results in the following constraint:
(86)    (x1[6]=x1[7]∧x0[6]=0 ⇒ COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (86) using rules (III), (IV) which results in the following new constraint:
(87)    (COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥456_0_MAIN_LE(x1[6], 0)∧(U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (87) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(88)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (88) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(89)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (89) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(90)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (90) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(91)    ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair 456_0_MAIN_LE(x1, 0) → 303_0_MAIN_LE(x1, 0) the following chains were created:

We consider the chain 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0), 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:
(92)    (x1[7]=x1[0]∧0=x0[0] ⇒ 456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (92) using rule (IV) which results in the following new constraint:
(93)    (456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (93) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(94)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (94) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(95)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (95) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(96)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (96) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(97)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
We consider the chain 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0), 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:
(98)    (x1[7]=x1[5]∧0=x0[5] ⇒ 456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (98) using rule (IV) which results in the following new constraint:
(99)    (456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (99) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(100)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (100) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(101)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (101) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(102)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (102) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(103)    ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

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

303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0)
- (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_25 + bni_25] + [(2)bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[1 + (-1)bso_26] + x0[0] + x1[0] ≥ 0)
COND_303_0_MAIN_LE(TRUE, x1, x0) → 456_0_MAIN_LE(x0, x1)
- ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
- ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
- ((U^Increasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
456_0_MAIN_LE(x0, 0) → 303_0_MAIN_LE(0, x0)
- ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)
- ((U^Increasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)
456_0_MAIN_LE(x0, x1) → COND_456_0_MAIN_LE(>(x1, 0), x0, x1)
- (x1[3] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_31] = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧[(-1)bso_32] ≥ 0)
COND_456_0_MAIN_LE(TRUE, x0, x1) → 456_0_MAIN_LE(x0, +(x1, -1))
- (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
- (x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
- (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0)
- ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (U^Increasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(2)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x1[5] + [bni_35]x0[5] ≥ 0∧[2 + (-1)bso_36] + x1[5] + x0[5] ≥ 0)
COND_303_0_MAIN_LE1(TRUE, x1, x0) → 456_0_MAIN_LE(x1, x0)
- ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
- ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
- ((U^Increasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
456_0_MAIN_LE(x1, 0) → 303_0_MAIN_LE(x1, 0)
- ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
- ((U^Increasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 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(303_0_MAIN_LE(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_303_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + x₃
POL(&&(x₁, x₂)) = 0
POL(>=(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(456_0_MAIN_LE(x₁, x₂)) = [-1] + x₁
POL(COND_456_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_303_0_MAIN_LE1(x₁, x₂, x₃)) = [-1] + x₂ + [-1]x₁
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])

The following pairs are in P_bound:

303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])

The following pairs are in P_≥:

COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1])
456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2])
456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))
COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6])
456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0)

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

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

(8) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1])
(2): 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2])
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], x1[4] + -1)
(6): COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6])
(7): 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0)

(1) -> (2), if (x0[1] →^* x0[2]∧x1[1] →^* 0)

(4) -> (2), if (x0[4] →^* x0[2]∧x1[4] + -1 →^* 0)

(6) -> (2), if (x1[6] →^* x0[2]∧x0[6] →^* 0)

(1) -> (3), if (x0[1] →^* x0[3]∧x1[1] →^* x1[3])

(4) -> (3), if (x0[4] →^* x0[3]∧x1[4] + -1 →^* x1[3])

(6) -> (3), if (x1[6] →^* x0[3]∧x0[6] →^* x1[3])

(3) -> (4), if (x1[3] > 0 ∧x0[3] →^* x0[4]∧x1[3] →^* x1[4])

(1) -> (7), if (x0[1] →^* x1[7]∧x1[1] →^* 0)

(4) -> (7), if (x0[4] →^* x1[7]∧x1[4] + -1 →^* 0)

(6) -> (7), if (x1[6] →^* x1[7]∧x0[6] →^* 0)

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(10) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(4): COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], x1[4] + -1)
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

(4) -> (3), if (x0[4] →^* x0[3]∧x1[4] + -1 →^* x1[3])

(3) -> (4), if (x1[3] > 0 ∧x0[3] →^* x0[4]∧x1[3] →^* x1[4])

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@31068b41 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_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)) the following chains were created:

We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:
(1)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x0[3]1∧+(x1[4], -1)=x1[3]1 ⇒ COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>(x1[3], 0)=TRUE ⇒ COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) the following chains were created:

We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:
(8)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

We simplified constraint (8) using rule (IV) which results in the following new constraint:
(9)    (>(x1[3], 0)=TRUE ⇒ 456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x1[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[3] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))
- (x1[3] ≥ 0 ⇒ (U^Increasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)
456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
- (x1[3] ≥ 0 ⇒ (U^Increasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_456_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + x₃
POL(456_0_MAIN_LE(x₁, x₂)) = [-1] + x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))

The following pairs are in P_bound:

COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))
456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

The following pairs are in P_≥:

456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

There are no usable rules.

(12) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(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) TRUE