Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: AppE

public class AppE {
	AppE next;

	public static void main(String[] args) {
		Random.args = args;
		AppE list = createList();
		list.appE(Random.random());
	}

	public void appE(int i) {
		if (next == null) {
			if (i <= 0) {
				return;
			} else {
				next = new AppE();
			}
			i--;
	 	}
		next.appE(i);
	}

	public static AppE createList() {
		AppE result = null;
		int length = Random.random();
		while (length > 0) {
			result = new AppE(result);
			length--;
		}
		return result;
	}

	public AppE() {
		this.next = null;
	}

	public AppE(AppE n) {
		this.next = n;
	}
}

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:
AppE.main([Ljava/lang/String;)V: Graph of 146 nodes with 0 SCCs.

AppE.createList()LAppE;: Graph of 91 nodes with 1 SCC.

AppE.appE(I)V: Graph of 56 nodes with 0 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 2 SCCss.

(4) Complex Obligation (AND)

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: AppE.appE(I)V
SCC calls the following helper methods: AppE.appE(I)V
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 34 rules for P and 23 rules for R.

P rules:
532_0_appE_FieldAccess(EOS(STATIC_532), java.lang.Object(AppE(EOC, o99)), i67, java.lang.Object(AppE(EOC, o99))) → 536_0_appE_FieldAccess(EOS(STATIC_536), java.lang.Object(AppE(EOC, o99)), i67, java.lang.Object(AppE(EOC, o99)))
536_0_appE_FieldAccess(EOS(STATIC_536), java.lang.Object(AppE(EOC, o99)), i67, java.lang.Object(AppE(EOC, o99))) → 541_0_appE_NONNULL(EOS(STATIC_541), java.lang.Object(AppE(EOC, o99)), i67, o99)
541_0_appE_NONNULL(EOS(STATIC_541), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67, java.lang.Object(o102sub)) → 549_0_appE_NONNULL(EOS(STATIC_549), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67, java.lang.Object(o102sub))
541_0_appE_NONNULL(EOS(STATIC_541), java.lang.Object(AppE(EOC, NULL)), i67, NULL) → 550_0_appE_NONNULL(EOS(STATIC_550), java.lang.Object(AppE(EOC, NULL)), i67, NULL)
549_0_appE_NONNULL(EOS(STATIC_549), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67, java.lang.Object(o102sub)) → 557_0_appE_Load(EOS(STATIC_557), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67)
557_0_appE_Load(EOS(STATIC_557), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67) → 567_0_appE_FieldAccess(EOS(STATIC_567), i67, java.lang.Object(AppE(EOC, java.lang.Object(o102sub))))
567_0_appE_FieldAccess(EOS(STATIC_567), i67, java.lang.Object(AppE(EOC, java.lang.Object(o102sub)))) → 577_0_appE_Load(EOS(STATIC_577), i67, java.lang.Object(o102sub))
577_0_appE_Load(EOS(STATIC_577), i67, java.lang.Object(o102sub)) → 588_0_appE_InvokeMethod(EOS(STATIC_588), java.lang.Object(o102sub), i67)
588_0_appE_InvokeMethod(EOS(STATIC_588), java.lang.Object(o102sub), i67) → 597_1_appE_InvokeMethod(597_0_appE_Load(EOS(STATIC_597), java.lang.Object(o102sub), i67), java.lang.Object(o102sub), i67)
597_0_appE_Load(EOS(STATIC_597), java.lang.Object(o102sub), i67) → 610_0_appE_Load(EOS(STATIC_610), java.lang.Object(o102sub), i67)
610_0_appE_Load(EOS(STATIC_610), java.lang.Object(o102sub), i67) → 528_0_appE_Load(EOS(STATIC_528), java.lang.Object(o102sub), i67)
528_0_appE_Load(EOS(STATIC_528), java.lang.Object(o91sub), i67) → 532_0_appE_FieldAccess(EOS(STATIC_532), java.lang.Object(o91sub), i67, java.lang.Object(o91sub))
550_0_appE_NONNULL(EOS(STATIC_550), java.lang.Object(AppE(EOC, NULL)), i67, NULL) → 558_0_appE_Load(EOS(STATIC_558), java.lang.Object(AppE(EOC, NULL)), i67)
558_0_appE_Load(EOS(STATIC_558), java.lang.Object(AppE(EOC, NULL)), i67) → 569_0_appE_GT(EOS(STATIC_569), java.lang.Object(AppE(EOC, NULL)), i67, i67)
569_0_appE_GT(EOS(STATIC_569), java.lang.Object(AppE(EOC, NULL)), i73, i73) → 581_0_appE_GT(EOS(STATIC_581), java.lang.Object(AppE(EOC, NULL)), i73, i73)
581_0_appE_GT(EOS(STATIC_581), java.lang.Object(AppE(EOC, NULL)), i73, i73) → 592_0_appE_Load(EOS(STATIC_592), java.lang.Object(AppE(EOC, NULL)), i73) | >(i73, 0)
592_0_appE_Load(EOS(STATIC_592), java.lang.Object(AppE(EOC, NULL)), i73) → 601_0_appE_New(EOS(STATIC_601), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)))
601_0_appE_New(EOS(STATIC_601), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL))) → 618_0_appE_Duplicate(EOS(STATIC_618), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
618_0_appE_Duplicate(EOS(STATIC_618), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 622_0_appE_InvokeMethod(EOS(STATIC_622), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
622_0_appE_InvokeMethod(EOS(STATIC_622), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 639_0_<init>_Load(EOS(STATIC_639), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
639_0_<init>_Load(EOS(STATIC_639), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 662_0_<init>_InvokeMethod(EOS(STATIC_662), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
662_0_<init>_InvokeMethod(EOS(STATIC_662), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 670_0_<init>_Load(EOS(STATIC_670), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
670_0_<init>_Load(EOS(STATIC_670), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 687_0_<init>_ConstantStackPush(EOS(STATIC_687), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
687_0_<init>_ConstantStackPush(EOS(STATIC_687), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 694_0_<init>_FieldAccess(EOS(STATIC_694), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), NULL)
694_0_<init>_FieldAccess(EOS(STATIC_694), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), NULL) → 705_0_<init>_Return(EOS(STATIC_705), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
705_0_<init>_Return(EOS(STATIC_705), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 712_0_appE_FieldAccess(EOS(STATIC_712), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
712_0_appE_FieldAccess(EOS(STATIC_712), java.lang.Object(AppE(EOC, NULL)), i73, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 725_0_appE_Inc(EOS(STATIC_725), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i73)
725_0_appE_Inc(EOS(STATIC_725), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i73) → 742_0_appE_Load(EOS(STATIC_742), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), +(i73, -1)) | >(i73, 0)
742_0_appE_Load(EOS(STATIC_742), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i95) → 753_0_appE_FieldAccess(EOS(STATIC_753), i95, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))))
753_0_appE_FieldAccess(EOS(STATIC_753), i95, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL))))) → 758_0_appE_Load(EOS(STATIC_758), i95, java.lang.Object(AppE(EOC, NULL)))
758_0_appE_Load(EOS(STATIC_758), i95, java.lang.Object(AppE(EOC, NULL))) → 772_0_appE_InvokeMethod(EOS(STATIC_772), java.lang.Object(AppE(EOC, NULL)), i95)
772_0_appE_InvokeMethod(EOS(STATIC_772), java.lang.Object(AppE(EOC, NULL)), i95) → 782_1_appE_InvokeMethod(782_0_appE_Load(EOS(STATIC_782), java.lang.Object(AppE(EOC, NULL)), i95), java.lang.Object(AppE(EOC, NULL)), i95)
782_0_appE_Load(EOS(STATIC_782), java.lang.Object(AppE(EOC, NULL)), i95) → 783_0_appE_Load(EOS(STATIC_783), java.lang.Object(AppE(EOC, NULL)), i95)
783_0_appE_Load(EOS(STATIC_783), java.lang.Object(AppE(EOC, NULL)), i95) → 528_0_appE_Load(EOS(STATIC_528), java.lang.Object(AppE(EOC, NULL)), i95)
R rules:
569_0_appE_GT(EOS(STATIC_569), java.lang.Object(AppE(EOC, NULL)), matching1, matching2) → 580_0_appE_GT(EOS(STATIC_580), java.lang.Object(AppE(EOC, NULL)), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
580_0_appE_GT(EOS(STATIC_580), java.lang.Object(AppE(EOC, NULL)), matching1, matching2) → 590_0_appE_Return(EOS(STATIC_590), java.lang.Object(AppE(EOC, NULL)), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
597_1_appE_InvokeMethod(590_0_appE_Return(EOS(STATIC_590), java.lang.Object(AppE(EOC, NULL)), matching1), java.lang.Object(AppE(EOC, NULL)), matching2) → 646_0_appE_Return(EOS(STATIC_646), java.lang.Object(AppE(EOC, NULL)), 0, java.lang.Object(AppE(EOC, NULL)), 0) | &&(=(matching1, 0), =(matching2, 0))
597_1_appE_InvokeMethod(660_0_appE_Return(EOS(STATIC_660)), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), matching1) → 692_0_appE_Return(EOS(STATIC_692), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), 0) | =(matching1, 0)
597_1_appE_InvokeMethod(801_0_appE_Return(EOS(STATIC_801)), java.lang.Object(o102sub), i108) → 838_0_appE_Return(EOS(STATIC_838), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i108)
597_1_appE_InvokeMethod(848_0_appE_Return(EOS(STATIC_848)), java.lang.Object(o102sub), i134) → 900_0_appE_Return(EOS(STATIC_900), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))))), i134)
597_1_appE_InvokeMethod(928_0_appE_Return(EOS(STATIC_928)), java.lang.Object(o102sub), i162) → 955_0_appE_Return(EOS(STATIC_955), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i162)
646_0_appE_Return(EOS(STATIC_646), java.lang.Object(AppE(EOC, NULL)), matching1, java.lang.Object(AppE(EOC, NULL)), matching2) → 660_0_appE_Return(EOS(STATIC_660)) | &&(=(matching1, 0), =(matching2, 0))
692_0_appE_Return(EOS(STATIC_692), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), matching1) → 740_0_appE_Return(EOS(STATIC_740), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), 0) | =(matching1, 0)
740_0_appE_Return(EOS(STATIC_740), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o168)))), matching1) → 839_0_appE_Return(EOS(STATIC_839), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o168)))), 0) | =(matching1, 0)
782_1_appE_InvokeMethod(590_0_appE_Return(EOS(STATIC_590), java.lang.Object(AppE(EOC, NULL)), matching1), java.lang.Object(AppE(EOC, NULL)), matching2) → 795_0_appE_Return(EOS(STATIC_795), java.lang.Object(AppE(EOC, NULL)), 0, java.lang.Object(AppE(EOC, NULL)), 0) | &&(=(matching1, 0), =(matching2, 0))
782_1_appE_InvokeMethod(801_0_appE_Return(EOS(STATIC_801)), java.lang.Object(AppE(EOC, NULL)), i111) → 841_0_appE_Return(EOS(STATIC_841), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i111)
782_1_appE_InvokeMethod(848_0_appE_Return(EOS(STATIC_848)), java.lang.Object(AppE(EOC, NULL)), i136) → 909_0_appE_Return(EOS(STATIC_909), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))))), i136)
782_1_appE_InvokeMethod(928_0_appE_Return(EOS(STATIC_928)), java.lang.Object(AppE(EOC, NULL)), i166) → 960_0_appE_Return(EOS(STATIC_960), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i166)
795_0_appE_Return(EOS(STATIC_795), java.lang.Object(AppE(EOC, NULL)), matching1, java.lang.Object(AppE(EOC, NULL)), matching2) → 801_0_appE_Return(EOS(STATIC_801)) | &&(=(matching1, 0), =(matching2, 0))
838_0_appE_Return(EOS(STATIC_838), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i108) → 839_0_appE_Return(EOS(STATIC_839), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i108)
839_0_appE_Return(EOS(STATIC_839), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))), i124) → 848_0_appE_Return(EOS(STATIC_848))
841_0_appE_Return(EOS(STATIC_841), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i111) → 910_0_appE_Return(EOS(STATIC_910), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i111)
900_0_appE_Return(EOS(STATIC_900), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))))), i134) → 839_0_appE_Return(EOS(STATIC_839), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))))), i134)
909_0_appE_Return(EOS(STATIC_909), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))))), i136) → 910_0_appE_Return(EOS(STATIC_910), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o253)))))), i136)
910_0_appE_Return(EOS(STATIC_910), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))), i151) → 928_0_appE_Return(EOS(STATIC_928))
955_0_appE_Return(EOS(STATIC_955), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i162) → 839_0_appE_Return(EOS(STATIC_839), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i162)
960_0_appE_Return(EOS(STATIC_960), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i166) → 910_0_appE_Return(EOS(STATIC_910), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i166)

Combined rules. Obtained 2 conditional rules for P and 9 conditional rules for R.

P rules:
532_0_appE_FieldAccess(EOS(STATIC_532), java.lang.Object(AppE(EOC, java.lang.Object(x0))), x1, java.lang.Object(AppE(EOC, java.lang.Object(x0)))) → 597_1_appE_InvokeMethod(532_0_appE_FieldAccess(EOS(STATIC_532), java.lang.Object(x0), x1, java.lang.Object(x0)), java.lang.Object(x0), x1)
532_0_appE_FieldAccess(EOS(STATIC_532), java.lang.Object(AppE(EOC, NULL)), x0, java.lang.Object(AppE(EOC, NULL))) → 782_1_appE_InvokeMethod(532_0_appE_FieldAccess(EOS(STATIC_532), java.lang.Object(AppE(EOC, NULL)), +(x0, -1), java.lang.Object(AppE(EOC, NULL))), java.lang.Object(AppE(EOC, NULL)), +(x0, -1)) | >(x0, 0)
R rules:
597_1_appE_InvokeMethod(590_0_appE_Return(EOS(STATIC_590), java.lang.Object(AppE(EOC, NULL)), 0), java.lang.Object(AppE(EOC, NULL)), 0) → 660_0_appE_Return(EOS(STATIC_660))
597_1_appE_InvokeMethod(801_0_appE_Return(EOS(STATIC_801)), java.lang.Object(x0), x1) → 848_0_appE_Return(EOS(STATIC_848))
597_1_appE_InvokeMethod(848_0_appE_Return(EOS(STATIC_848)), java.lang.Object(x0), x1) → 848_0_appE_Return(EOS(STATIC_848))
597_1_appE_InvokeMethod(928_0_appE_Return(EOS(STATIC_928)), java.lang.Object(x0), x1) → 848_0_appE_Return(EOS(STATIC_848))
597_1_appE_InvokeMethod(660_0_appE_Return(EOS(STATIC_660)), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), 0) → 848_0_appE_Return(EOS(STATIC_848))
782_1_appE_InvokeMethod(590_0_appE_Return(EOS(STATIC_590), java.lang.Object(AppE(EOC, NULL)), 0), java.lang.Object(AppE(EOC, NULL)), 0) → 801_0_appE_Return(EOS(STATIC_801))
782_1_appE_InvokeMethod(801_0_appE_Return(EOS(STATIC_801)), java.lang.Object(AppE(EOC, NULL)), x0) → 928_0_appE_Return(EOS(STATIC_928))
782_1_appE_InvokeMethod(848_0_appE_Return(EOS(STATIC_848)), java.lang.Object(AppE(EOC, NULL)), x0) → 928_0_appE_Return(EOS(STATIC_928))
782_1_appE_InvokeMethod(928_0_appE_Return(EOS(STATIC_928)), java.lang.Object(AppE(EOC, NULL)), x0) → 928_0_appE_Return(EOS(STATIC_928))

Filtered ground terms:

782_1_appE_InvokeMethod(x1, x2, x3) → 782_1_appE_InvokeMethod(x1, x3)
AppE(x1, x2) → AppE(x2)
532_0_appE_FieldAccess(x1, x2, x3, x4) → 532_0_appE_FieldAccess(x2, x3, x4)
Cond_532_0_appE_FieldAccess(x1, x2, x3, x4, x5) → Cond_532_0_appE_FieldAccess(x1, x4)
928_0_appE_Return(x1) → 928_0_appE_Return
848_0_appE_Return(x1) → 848_0_appE_Return
801_0_appE_Return(x1) → 801_0_appE_Return
590_0_appE_Return(x1, x2, x3) → 590_0_appE_Return
660_0_appE_Return(x1) → 660_0_appE_Return

Filtered duplicate args:

532_0_appE_FieldAccess(x1, x2, x3) → 532_0_appE_FieldAccess(x2, x3)

Combined rules. Obtained 2 conditional rules for P and 9 conditional rules for R.

P rules:
532_0_appE_FieldAccess(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 597_1_appE_InvokeMethod(532_0_appE_FieldAccess(x1, java.lang.Object(x0)), java.lang.Object(x0), x1)
532_0_appE_FieldAccess(x0, java.lang.Object(AppE(NULL))) → 782_1_appE_InvokeMethod(532_0_appE_FieldAccess(+(x0, -1), java.lang.Object(AppE(NULL))), +(x0, -1)) | >(x0, 0)
R rules:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 660_0_appE_Return
597_1_appE_InvokeMethod(801_0_appE_Return, java.lang.Object(x0), x1) → 848_0_appE_Return
597_1_appE_InvokeMethod(848_0_appE_Return, java.lang.Object(x0), x1) → 848_0_appE_Return
597_1_appE_InvokeMethod(928_0_appE_Return, java.lang.Object(x0), x1) → 848_0_appE_Return
597_1_appE_InvokeMethod(660_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 848_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, 0) → 801_0_appE_Return
782_1_appE_InvokeMethod(801_0_appE_Return, x0) → 928_0_appE_Return
782_1_appE_InvokeMethod(848_0_appE_Return, x0) → 928_0_appE_Return
782_1_appE_InvokeMethod(928_0_appE_Return, x0) → 928_0_appE_Return

Performed bisimulation on rules. Used the following equivalence classes: {[590_0_appE_Return, 660_0_appE_Return, 801_0_appE_Return, 848_0_appE_Return, 928_0_appE_Return]=590_0_appE_Return}

Finished conversion. Obtained 3 rules for P and 5 rules for R. System has predefined symbols.

P rules:
532_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 532_0_APPE_FIELDACCESS(x1, java.lang.Object(x0))
532_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL)))
COND_532_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL)))
R rules:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, x0) → 590_0_appE_Return

(7) 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

The ITRS R consists of the following rules:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, x0) → 590_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))
(1): 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))
(2): COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(x0[2] + -1, java.lang.Object(AppE(NULL)))

(0) -> (0), if (x1[0] →^* x1[0]'∧java.lang.Object(x0[0]) →^* java.lang.Object(AppE(java.lang.Object(x0[0]'))))

(0) -> (1), if (x1[0] →^* x0[1]∧java.lang.Object(x0[0]) →^* java.lang.Object(AppE(NULL)))

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

(2) -> (0), if (x0[2] + -1 →^* x1[0]∧java.lang.Object(AppE(NULL)) →^* java.lang.Object(AppE(java.lang.Object(x0[0]))))

(2) -> (1), if x0[2] + -1 →^* x0[1]

The set Q consists of the following terms:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

(8) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@8e760f2 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 532_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 532_0_APPE_FIELDACCESS(x1, java.lang.Object(x0)) the following chains were created:

• We consider the chain 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])) which results in the following constraint:
  

  (1)    (x1[0]=x1[0]1∧java.lang.Object(x0[0])=java.lang.Object(AppE(java.lang.Object(x0[0]1)))∧x1[0]1=x1[0]2∧java.lang.Object(x0[0]1)=java.lang.Object(AppE(java.lang.Object(x0[0]2))) ⇒ 532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥NonInfC∧532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))∧(U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥))

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

  (2)    (532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(x0[0]2))))))≥NonInfC∧532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(x0[0]2))))))≥532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]2))))∧(U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥))

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

  (3)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧[4 + (-1)bso_22] + [4]x0[0]2 ≥ 0)

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

  (4)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧[4 + (-1)bso_22] + [4]x0[0]2 ≥ 0)

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

  (5)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧[4 + (-1)bso_22] + [4]x0[0]2 ≥ 0)

  
  
  We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
  

  (6)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0∧[1] ≥ 0)

  
  
  
• We consider the chain COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])) which results in the following constraint:
  

  (7)    (+(x0[2], -1)=x1[0]∧java.lang.Object(AppE(NULL))=java.lang.Object(AppE(java.lang.Object(x0[0])))∧x1[0]=x1[0]1∧java.lang.Object(x0[0])=java.lang.Object(AppE(java.lang.Object(x0[0]1))) ⇒ 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥NonInfC∧532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))∧(U^Increasing(532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))), ≥))

  
  
  We solved constraint (7) using rules (I), (II).
• We consider the chain 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) which results in the following constraint:
  

  (8)    (x1[0]=x1[0]1∧java.lang.Object(x0[0])=java.lang.Object(AppE(java.lang.Object(x0[0]1)))∧x1[0]1=x0[1]∧java.lang.Object(x0[0]1)=java.lang.Object(AppE(NULL)) ⇒ 532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥NonInfC∧532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))∧(U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥))

  
  
  We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint:
  

  (9)    (532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(NULL)))))≥NonInfC∧532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(NULL)))))≥532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(NULL)))∧(U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥))

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

  (10)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧[2 + (-1)bso_22] ≥ 0)

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

  (11)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧[2 + (-1)bso_22] ≥ 0)

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

  (12)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧[2 + (-1)bso_22] ≥ 0)

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

  (13)    ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧0 ≥ 0∧[2 + (-1)bso_22] ≥ 0)

  
  
  
• We consider the chain COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) which results in the following constraint:
  

  (14)    (+(x0[2], -1)=x1[0]∧java.lang.Object(AppE(NULL))=java.lang.Object(AppE(java.lang.Object(x0[0])))∧x1[0]=x0[1]∧java.lang.Object(x0[0])=java.lang.Object(AppE(NULL)) ⇒ 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥NonInfC∧532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))∧(U^Increasing(532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))), ≥))

  
  
  We solved constraint (14) using rules (I), (II).

For Pair 532_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))) which results in the following constraint:
  

  (15)    (>(x0[1], 0)=TRUE∧x0[1]=x0[2] ⇒ 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥))

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

  (16)    (>(x0[1], 0)=TRUE ⇒ 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥))

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

  (17)    (0 ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[1] ≥ 0∧[(-1)bso_24] + x0[1] ≥ 0)

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

  (18)    (0 ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[1] ≥ 0∧[(-1)bso_24] + x0[1] ≥ 0)

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

  (19)    (0 ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[1] ≥ 0∧[(-1)bso_24] + x0[1] ≥ 0)

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

  (20)    (0 ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[bni_23] ≥ 0∧[(2)bni_23 + (-1)Bound*bni_23] ≥ 0∧[1] ≥ 0∧[(-1)bso_24] ≥ 0)

  
  
  

For Pair COND_532_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])) which results in the following constraint:
  

  (21)    (>(x0[1], 0)=TRUE∧x0[1]=x0[2]∧+(x0[2], -1)=x1[0]∧java.lang.Object(AppE(NULL))=java.lang.Object(AppE(java.lang.Object(x0[0]))) ⇒ COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))

  
  
  We solved constraint (21) using rules (I), (II).
• We consider the chain 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) which results in the following constraint:
  

  (22)    (>(x0[1], 0)=TRUE∧x0[1]=x0[2]∧+(x0[2], -1)=x0[1]1 ⇒ COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))

  
  
  We simplified constraint (22) using rules (III), (IV) which results in the following new constraint:
  

  (23)    (>(x0[1], 0)=TRUE ⇒ COND_532_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_532_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥532_0_APPE_FIELDACCESS(+(x0[1], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))

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

  (24)    (0 ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_25 + (-1)Bound*bni_25] ≥ 0∧[(-1)bso_26] ≥ 0)

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

  (25)    (0 ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_25 + (-1)Bound*bni_25] ≥ 0∧[(-1)bso_26] ≥ 0)

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

  (26)    (0 ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_25 + (-1)Bound*bni_25] ≥ 0∧[(-1)bso_26] ≥ 0)

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

  (27)    (0 ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧0 ≥ 0∧[(2)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 ≥ 0∧[(-1)bso_26] ≥ 0)

  
  
  

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

• 532_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 532_0_APPE_FIELDACCESS(x1, java.lang.Object(x0))
  
  • ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧0 ≥ 0∧[4 + (-1)bso_22] ≥ 0∧[1] ≥ 0)
    
  • ((U^Increasing(532_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧0 ≥ 0∧[2 + (-1)bso_22] ≥ 0)
    
  
  
• 532_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[bni_23] ≥ 0∧[(2)bni_23 + (-1)Bound*bni_23] ≥ 0∧[1] ≥ 0∧[(-1)bso_24] ≥ 0)
    
  
  
• COND_532_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧0 ≥ 0∧[(2)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 ≥ 0∧[(-1)bso_26] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(597_1_appE_InvokeMethod(x₁, x₂, x₃)) = 0   
POL(590_0_appE_Return) = 0   
POL(java.lang.Object(x₁)) = [1] + [2]x₁   
POL(AppE(x₁)) = x₁   
POL(NULL) = 0   
POL(0) = 0   
POL(782_1_appE_InvokeMethod(x₁, x₂)) = 0   
POL(532_0_APPE_FIELDACCESS(x₁, x₂)) = [1] + x₂ + x₁   
POL(COND_532_0_APPE_FIELDACCESS(x₁, x₂, x₃)) = [2]   
POL(>(x₁, x₂)) = 0   
POL(+(x₁, x₂)) = 0   
POL(-1) = 0   

The following pairs are in P_>:

532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The following pairs are in P_bound:

532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P_≥:

532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

There are no usable rules.

(9) Complex Obligation (AND)

(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

The ITRS R consists of the following rules:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, x0) → 590_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))
(2): COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(x0[2] + -1, java.lang.Object(AppE(NULL)))

(2) -> (1), if x0[2] + -1 →^* x0[1]

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

The set Q consists of the following terms:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

(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@7940eadf 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 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))) which results in the following constraint:
  

  (1)    (>(x0[1], 0)=TRUE∧x0[1]=x0[2] ⇒ 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥))

  
  
  We simplified constraint (1) using rule (IV) which results in the following new constraint:
  

  (2)    (>(x0[1], 0)=TRUE ⇒ 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥))

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

  (3)    (x0[1] + [-1] ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (4)    (x0[1] + [-1] ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (5)    (x0[1] + [-1] ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (6)    (x0[1] ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_24 + bni_24] + [bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  

For Pair COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) which results in the following constraint:
  

  (7)    (>(x0[1], 0)=TRUE∧x0[1]=x0[2]∧+(x0[2], -1)=x0[1]1 ⇒ COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))

  
  
  We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:
  

  (8)    (>(x0[1], 0)=TRUE ⇒ COND_532_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_532_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥532_0_APPE_FIELDACCESS(+(x0[1], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))

  
  
  We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (9)    (x0[1] + [-1] ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[1] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

  (10)    (x0[1] + [-1] ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[1] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

  (11)    (x0[1] + [-1] ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[1] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

  (12)    (x0[1] ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_26 + bni_26] + [bni_26]x0[1] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

  
  
  

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

• 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
  
  • (x0[1] ≥ 0 ⇒ (U^Increasing(COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_24 + bni_24] + [bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)
    
  
  
• COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))
  
  • (x0[1] ≥ 0 ⇒ (U^Increasing(532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_26 + bni_26] + [bni_26]x0[1] ≥ 0∧[1 + (-1)bso_27] ≥ 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) = [3]   
POL(FALSE) = 0   
POL(597_1_appE_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁   
POL(590_0_appE_Return) = [-1]   
POL(java.lang.Object(x₁)) = [-1] + [-1]x₁   
POL(AppE(x₁)) = [-1] + [-1]x₁   
POL(NULL) = [-1]   
POL(0) = 0   
POL(782_1_appE_InvokeMethod(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁   
POL(532_0_APPE_FIELDACCESS(x₁, x₂)) = [-1] + [-1]x₂ + x₁   
POL(COND_532_0_APPE_FIELDACCESS(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂   
POL(>(x₁, x₂)) = [-1]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   

The following pairs are in P_>:

COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P_bound:

532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_532_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 532_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P_≥:

532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))

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

The ITRS R consists of the following rules:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, x0) → 590_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): 532_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_532_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))


The set Q consists of the following terms:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE

(15) 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


The ITRS R consists of the following rules:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, 0) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, x0) → 590_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

(0) -> (0), if (x1[0] →^* x1[0]'∧java.lang.Object(x0[0]) →^* java.lang.Object(AppE(java.lang.Object(x0[0]'))))

The set Q consists of the following terms:
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

(16) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The TRS R consists of the following rules:

597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(NULL)), pos(01)) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1) → 590_0_appE_Return
597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), pos(01)) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, pos(01)) → 590_0_appE_Return
782_1_appE_InvokeMethod(590_0_appE_Return, x0) → 590_0_appE_Return

The set Q consists of the following terms:

597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

We have to consider all minimal (P,Q,R)-chains.

(18) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

R is empty.
The set Q consists of the following terms:

597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

We have to consider all minimal (P,Q,R)-chains.

(20) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

597_1_appE_InvokeMethod(590_0_appE_Return, java.lang.Object(x0), x1)
782_1_appE_InvokeMethod(590_0_appE_Return, x0)

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(22) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 532_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))
  The graph contains the following edges 1 >= 1, 2 > 2

(23) YES

(24) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: AppE.createList()LAppE;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(25) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 17 rules for P and 0 rules for R.

P rules:
332_0_createList_LE(EOS(STATIC_332), i39, i39) → 338_0_createList_LE(EOS(STATIC_338), i39, i39)
338_0_createList_LE(EOS(STATIC_338), i39, i39) → 348_0_createList_New(EOS(STATIC_348), i39) | >(i39, 0)
348_0_createList_New(EOS(STATIC_348), i39) → 369_0_createList_Duplicate(EOS(STATIC_369), i39)
369_0_createList_Duplicate(EOS(STATIC_369), i39) → 381_0_createList_Load(EOS(STATIC_381), i39)
381_0_createList_Load(EOS(STATIC_381), i39) → 390_0_createList_InvokeMethod(EOS(STATIC_390), i39)
390_0_createList_InvokeMethod(EOS(STATIC_390), i39) → 406_0_<init>_Load(EOS(STATIC_406), i39)
406_0_<init>_Load(EOS(STATIC_406), i39) → 412_0_<init>_InvokeMethod(EOS(STATIC_412), i39)
412_0_<init>_InvokeMethod(EOS(STATIC_412), i39) → 415_0_<init>_Load(EOS(STATIC_415), i39)
415_0_<init>_Load(EOS(STATIC_415), i39) → 418_0_<init>_Load(EOS(STATIC_418), i39)
418_0_<init>_Load(EOS(STATIC_418), i39) → 420_0_<init>_FieldAccess(EOS(STATIC_420), i39)
420_0_<init>_FieldAccess(EOS(STATIC_420), i39) → 425_0_<init>_Return(EOS(STATIC_425), i39)
425_0_<init>_Return(EOS(STATIC_425), i39) → 430_0_createList_Store(EOS(STATIC_430), i39)
430_0_createList_Store(EOS(STATIC_430), i39) → 435_0_createList_Inc(EOS(STATIC_435), i39)
435_0_createList_Inc(EOS(STATIC_435), i39) → 441_0_createList_JMP(EOS(STATIC_441), +(i39, -1)) | >(i39, 0)
441_0_createList_JMP(EOS(STATIC_441), i52) → 446_0_createList_Load(EOS(STATIC_446), i52)
446_0_createList_Load(EOS(STATIC_446), i52) → 320_0_createList_Load(EOS(STATIC_320), i52)
320_0_createList_Load(EOS(STATIC_320), i35) → 332_0_createList_LE(EOS(STATIC_332), i35, i35)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
332_0_createList_LE(EOS(STATIC_332), x0, x0) → 332_0_createList_LE(EOS(STATIC_332), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

332_0_createList_LE(x1, x2, x3) → 332_0_createList_LE(x2, x3)
EOS(x1) → EOS
Cond_332_0_createList_LE(x1, x2, x3, x4) → Cond_332_0_createList_LE(x1, x3, x4)

Filtered duplicate args:

332_0_createList_LE(x1, x2) → 332_0_createList_LE(x2)
Cond_332_0_createList_LE(x1, x2, x3) → Cond_332_0_createList_LE(x1, x3)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
332_0_createList_LE(x0) → 332_0_createList_LE(+(x0, -1)) | >(x0, 0)
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
332_0_CREATELIST_LE(x0) → COND_332_0_CREATELIST_LE(>(x0, 0), x0)
COND_332_0_CREATELIST_LE(TRUE, x0) → 332_0_CREATELIST_LE(+(x0, -1))
R rules:

(26) 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:
(0): 332_0_CREATELIST_LE(x0[0]) → COND_332_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_332_0_CREATELIST_LE(TRUE, x0[1]) → 332_0_CREATELIST_LE(x0[1] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

The set Q is empty.

(27) 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.IdpCand1ShapeHeuristic@45446e1f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 332_0_CREATELIST_LE(x0) → COND_332_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:

• We consider the chain 332_0_CREATELIST_LE(x0[0]) → COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_332_0_CREATELIST_LE(TRUE, x0[1]) → 332_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:
  

  (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 332_0_CREATELIST_LE(x0[0])≥NonInfC∧332_0_CREATELIST_LE(x0[0])≥COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥))

  
  
  We simplified constraint (1) using rule (IV) which results in the following new constraint:
  

  (2)    (>(x0[0], 0)=TRUE ⇒ 332_0_CREATELIST_LE(x0[0])≥NonInfC∧332_0_CREATELIST_LE(x0[0])≥COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

  (5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

  (6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

  
  
  

For Pair COND_332_0_CREATELIST_LE(TRUE, x0) → 332_0_CREATELIST_LE(+(x0, -1)) the following chains were created:

• We consider the chain COND_332_0_CREATELIST_LE(TRUE, x0[1]) → 332_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:
  

  (7)    (COND_332_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_332_0_CREATELIST_LE(TRUE, x0[1])≥332_0_CREATELIST_LE(+(x0[1], -1))∧(U^Increasing(332_0_CREATELIST_LE(+(x0[1], -1))), ≥))

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

  (8)    ((U^Increasing(332_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

  
  
  We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (9)    ((U^Increasing(332_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

  (10)    ((U^Increasing(332_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

  (11)    ((U^Increasing(332_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

  
  
  

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

• 332_0_CREATELIST_LE(x0) → COND_332_0_CREATELIST_LE(>(x0, 0), x0)
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
    
  
  
• COND_332_0_CREATELIST_LE(TRUE, x0) → 332_0_CREATELIST_LE(+(x0, -1))
  
  • ((U^Increasing(332_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)
    
  
  

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

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(332_0_CREATELIST_LE(x₁)) = [2]x₁   
POL(COND_332_0_CREATELIST_LE(x₁, x₂)) = [2]x₂   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   

The following pairs are in P_>:

COND_332_0_CREATELIST_LE(TRUE, x0[1]) → 332_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in P_bound:

332_0_CREATELIST_LE(x0[0]) → COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

332_0_CREATELIST_LE(x0[0]) → COND_332_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(28) Complex Obligation (AND)

(29) 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:
(0): 332_0_CREATELIST_LE(x0[0]) → COND_332_0_CREATELIST_LE(x0[0] > 0, x0[0])


The set Q is empty.

(30) IDependencyGraphProof (EQUIVALENT transformation)

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

(31) TRUE

(32) 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_332_0_CREATELIST_LE(TRUE, x0[1]) → 332_0_CREATELIST_LE(x0[1] + -1)


The set Q is empty.

(33) IDependencyGraphProof (EQUIVALENT transformation)

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

(34) TRUE