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:
554_0_appE_FieldAccess(EOS(STATIC_554), java.lang.Object(AppE(EOC, o99)), i67, java.lang.Object(AppE(EOC, o99))) → 562_0_appE_FieldAccess(EOS(STATIC_562), java.lang.Object(AppE(EOC, o99)), i67, java.lang.Object(AppE(EOC, o99)))
562_0_appE_FieldAccess(EOS(STATIC_562), java.lang.Object(AppE(EOC, o99)), i67, java.lang.Object(AppE(EOC, o99))) → 570_0_appE_NONNULL(EOS(STATIC_570), java.lang.Object(AppE(EOC, o99)), i67, o99)
570_0_appE_NONNULL(EOS(STATIC_570), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67, java.lang.Object(o102sub)) → 580_0_appE_NONNULL(EOS(STATIC_580), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67, java.lang.Object(o102sub))
570_0_appE_NONNULL(EOS(STATIC_570), java.lang.Object(AppE(EOC, NULL)), i67, NULL) → 581_0_appE_NONNULL(EOS(STATIC_581), java.lang.Object(AppE(EOC, NULL)), i67, NULL)
580_0_appE_NONNULL(EOS(STATIC_580), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67, java.lang.Object(o102sub)) → 589_0_appE_Load(EOS(STATIC_589), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67)
589_0_appE_Load(EOS(STATIC_589), java.lang.Object(AppE(EOC, java.lang.Object(o102sub))), i67) → 597_0_appE_FieldAccess(EOS(STATIC_597), i67, java.lang.Object(AppE(EOC, java.lang.Object(o102sub))))
597_0_appE_FieldAccess(EOS(STATIC_597), i67, java.lang.Object(AppE(EOC, java.lang.Object(o102sub)))) → 609_0_appE_Load(EOS(STATIC_609), i67, java.lang.Object(o102sub))
609_0_appE_Load(EOS(STATIC_609), i67, java.lang.Object(o102sub)) → 621_0_appE_InvokeMethod(EOS(STATIC_621), java.lang.Object(o102sub), i67)
621_0_appE_InvokeMethod(EOS(STATIC_621), java.lang.Object(o102sub), i67) → 630_1_appE_InvokeMethod(630_0_appE_Load(EOS(STATIC_630), java.lang.Object(o102sub), i67), java.lang.Object(o102sub), i67)
630_0_appE_Load(EOS(STATIC_630), java.lang.Object(o102sub), i67) → 641_0_appE_Load(EOS(STATIC_641), java.lang.Object(o102sub), i67)
641_0_appE_Load(EOS(STATIC_641), java.lang.Object(o102sub), i67) → 546_0_appE_Load(EOS(STATIC_546), java.lang.Object(o102sub), i67)
546_0_appE_Load(EOS(STATIC_546), java.lang.Object(o93sub), i67) → 554_0_appE_FieldAccess(EOS(STATIC_554), java.lang.Object(o93sub), i67, java.lang.Object(o93sub))
581_0_appE_NONNULL(EOS(STATIC_581), java.lang.Object(AppE(EOC, NULL)), i67, NULL) → 591_0_appE_Load(EOS(STATIC_591), java.lang.Object(AppE(EOC, NULL)), i67)
591_0_appE_Load(EOS(STATIC_591), java.lang.Object(AppE(EOC, NULL)), i67) → 600_0_appE_GT(EOS(STATIC_600), java.lang.Object(AppE(EOC, NULL)), i67, i67)
600_0_appE_GT(EOS(STATIC_600), java.lang.Object(AppE(EOC, NULL)), i74, i74) → 612_0_appE_GT(EOS(STATIC_612), java.lang.Object(AppE(EOC, NULL)), i74, i74)
612_0_appE_GT(EOS(STATIC_612), java.lang.Object(AppE(EOC, NULL)), i74, i74) → 624_0_appE_Load(EOS(STATIC_624), java.lang.Object(AppE(EOC, NULL)), i74) | >(i74, 0)
624_0_appE_Load(EOS(STATIC_624), java.lang.Object(AppE(EOC, NULL)), i74) → 634_0_appE_New(EOS(STATIC_634), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)))
634_0_appE_New(EOS(STATIC_634), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL))) → 648_0_appE_Duplicate(EOS(STATIC_648), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
648_0_appE_Duplicate(EOS(STATIC_648), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 657_0_appE_InvokeMethod(EOS(STATIC_657), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
657_0_appE_InvokeMethod(EOS(STATIC_657), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 673_0_<init>_Load(EOS(STATIC_673), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
673_0_<init>_Load(EOS(STATIC_673), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 695_0_<init>_InvokeMethod(EOS(STATIC_695), java.lang.Object(AppE(EOC, NULL)), i74, 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)))
695_0_<init>_InvokeMethod(EOS(STATIC_695), java.lang.Object(AppE(EOC, NULL)), i74, 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))) → 702_0_<init>_Load(EOS(STATIC_702), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
702_0_<init>_Load(EOS(STATIC_702), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 711_0_<init>_ConstantStackPush(EOS(STATIC_711), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
711_0_<init>_ConstantStackPush(EOS(STATIC_711), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 718_0_<init>_FieldAccess(EOS(STATIC_718), java.lang.Object(AppE(EOC, NULL)), i74, 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)
718_0_<init>_FieldAccess(EOS(STATIC_718), java.lang.Object(AppE(EOC, NULL)), i74, 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) → 729_0_<init>_Return(EOS(STATIC_729), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
729_0_<init>_Return(EOS(STATIC_729), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 736_0_appE_FieldAccess(EOS(STATIC_736), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL)))
736_0_appE_FieldAccess(EOS(STATIC_736), java.lang.Object(AppE(EOC, NULL)), i74, java.lang.Object(AppE(EOC, NULL)), java.lang.Object(AppE(EOC, NULL))) → 754_0_appE_Inc(EOS(STATIC_754), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i74)
754_0_appE_Inc(EOS(STATIC_754), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i74) → 771_0_appE_Load(EOS(STATIC_771), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), +(i74, -1)) | >(i74, 0)
771_0_appE_Load(EOS(STATIC_771), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i97) → 782_0_appE_FieldAccess(EOS(STATIC_782), i97, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))))
782_0_appE_FieldAccess(EOS(STATIC_782), i97, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL))))) → 788_0_appE_Load(EOS(STATIC_788), i97, java.lang.Object(AppE(EOC, NULL)))
788_0_appE_Load(EOS(STATIC_788), i97, java.lang.Object(AppE(EOC, NULL))) → 801_0_appE_InvokeMethod(EOS(STATIC_801), java.lang.Object(AppE(EOC, NULL)), i97)
801_0_appE_InvokeMethod(EOS(STATIC_801), java.lang.Object(AppE(EOC, NULL)), i97) → 811_1_appE_InvokeMethod(811_0_appE_Load(EOS(STATIC_811), java.lang.Object(AppE(EOC, NULL)), i97), java.lang.Object(AppE(EOC, NULL)), i97)
811_0_appE_Load(EOS(STATIC_811), java.lang.Object(AppE(EOC, NULL)), i97) → 814_0_appE_Load(EOS(STATIC_814), java.lang.Object(AppE(EOC, NULL)), i97)
814_0_appE_Load(EOS(STATIC_814), java.lang.Object(AppE(EOC, NULL)), i97) → 546_0_appE_Load(EOS(STATIC_546), java.lang.Object(AppE(EOC, NULL)), i97)
R rules:
600_0_appE_GT(EOS(STATIC_600), java.lang.Object(AppE(EOC, NULL)), matching1, matching2) → 611_0_appE_GT(EOS(STATIC_611), java.lang.Object(AppE(EOC, NULL)), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
611_0_appE_GT(EOS(STATIC_611), java.lang.Object(AppE(EOC, NULL)), matching1, matching2) → 623_0_appE_Return(EOS(STATIC_623), java.lang.Object(AppE(EOC, NULL)), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
630_1_appE_InvokeMethod(623_0_appE_Return(EOS(STATIC_623), java.lang.Object(AppE(EOC, NULL)), matching1), java.lang.Object(AppE(EOC, NULL)), matching2) → 680_0_appE_Return(EOS(STATIC_680), java.lang.Object(AppE(EOC, NULL)), 0, java.lang.Object(AppE(EOC, NULL)), 0) | &&(=(matching1, 0), =(matching2, 0))
630_1_appE_InvokeMethod(693_0_appE_Return(EOS(STATIC_693)), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), matching1) → 716_0_appE_Return(EOS(STATIC_716), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), 0) | =(matching1, 0)
630_1_appE_InvokeMethod(832_0_appE_Return(EOS(STATIC_832)), java.lang.Object(o102sub), i115) → 869_0_appE_Return(EOS(STATIC_869), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i115)
630_1_appE_InvokeMethod(878_0_appE_Return(EOS(STATIC_878)), java.lang.Object(o102sub), i135) → 929_0_appE_Return(EOS(STATIC_929), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))))), i135)
630_1_appE_InvokeMethod(955_0_appE_Return(EOS(STATIC_955)), java.lang.Object(o102sub), i165) → 998_0_appE_Return(EOS(STATIC_998), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i165)
680_0_appE_Return(EOS(STATIC_680), java.lang.Object(AppE(EOC, NULL)), matching1, java.lang.Object(AppE(EOC, NULL)), matching2) → 693_0_appE_Return(EOS(STATIC_693)) | &&(=(matching1, 0), =(matching2, 0))
716_0_appE_Return(EOS(STATIC_716), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), matching1) → 770_0_appE_Return(EOS(STATIC_770), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), 0) | =(matching1, 0)
770_0_appE_Return(EOS(STATIC_770), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o168)))), matching1) → 870_0_appE_Return(EOS(STATIC_870), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o168)))), 0) | =(matching1, 0)
811_1_appE_InvokeMethod(623_0_appE_Return(EOS(STATIC_623), java.lang.Object(AppE(EOC, NULL)), matching1), java.lang.Object(AppE(EOC, NULL)), matching2) → 827_0_appE_Return(EOS(STATIC_827), java.lang.Object(AppE(EOC, NULL)), 0, java.lang.Object(AppE(EOC, NULL)), 0) | &&(=(matching1, 0), =(matching2, 0))
811_1_appE_InvokeMethod(832_0_appE_Return(EOS(STATIC_832)), java.lang.Object(AppE(EOC, NULL)), i118) → 872_0_appE_Return(EOS(STATIC_872), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i118)
811_1_appE_InvokeMethod(878_0_appE_Return(EOS(STATIC_878)), java.lang.Object(AppE(EOC, NULL)), i137) → 937_0_appE_Return(EOS(STATIC_937), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))))), i137)
811_1_appE_InvokeMethod(955_0_appE_Return(EOS(STATIC_955)), java.lang.Object(AppE(EOC, NULL)), i168) → 1003_0_appE_Return(EOS(STATIC_1003), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i168)
827_0_appE_Return(EOS(STATIC_827), java.lang.Object(AppE(EOC, NULL)), matching1, java.lang.Object(AppE(EOC, NULL)), matching2) → 832_0_appE_Return(EOS(STATIC_832)) | &&(=(matching1, 0), =(matching2, 0))
869_0_appE_Return(EOS(STATIC_869), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i115) → 870_0_appE_Return(EOS(STATIC_870), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i115)
870_0_appE_Return(EOS(STATIC_870), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))), i125) → 878_0_appE_Return(EOS(STATIC_878))
872_0_appE_Return(EOS(STATIC_872), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i118) → 938_0_appE_Return(EOS(STATIC_938), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), i118)
929_0_appE_Return(EOS(STATIC_929), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))))), i135) → 870_0_appE_Return(EOS(STATIC_870), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))))), i135)
937_0_appE_Return(EOS(STATIC_937), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))))), i137) → 938_0_appE_Return(EOS(STATIC_938), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o255)))))), i137)
938_0_appE_Return(EOS(STATIC_938), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))), i152) → 955_0_appE_Return(EOS(STATIC_955))
998_0_appE_Return(EOS(STATIC_998), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i165) → 870_0_appE_Return(EOS(STATIC_870), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i165)
1003_0_appE_Return(EOS(STATIC_1003), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i168) → 938_0_appE_Return(EOS(STATIC_938), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, o321)))))), i168)

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

P rules:
554_0_appE_FieldAccess(EOS(STATIC_554), java.lang.Object(AppE(EOC, java.lang.Object(x0))), x1, java.lang.Object(AppE(EOC, java.lang.Object(x0)))) → 630_1_appE_InvokeMethod(554_0_appE_FieldAccess(EOS(STATIC_554), java.lang.Object(x0), x1, java.lang.Object(x0)), java.lang.Object(x0), x1)
554_0_appE_FieldAccess(EOS(STATIC_554), java.lang.Object(AppE(EOC, NULL)), x0, java.lang.Object(AppE(EOC, NULL))) → 811_1_appE_InvokeMethod(554_0_appE_FieldAccess(EOS(STATIC_554), 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:
630_1_appE_InvokeMethod(623_0_appE_Return(EOS(STATIC_623), java.lang.Object(AppE(EOC, NULL)), 0), java.lang.Object(AppE(EOC, NULL)), 0) → 693_0_appE_Return(EOS(STATIC_693))
630_1_appE_InvokeMethod(832_0_appE_Return(EOS(STATIC_832)), java.lang.Object(x0), x1) → 878_0_appE_Return(EOS(STATIC_878))
630_1_appE_InvokeMethod(878_0_appE_Return(EOS(STATIC_878)), java.lang.Object(x0), x1) → 878_0_appE_Return(EOS(STATIC_878))
630_1_appE_InvokeMethod(955_0_appE_Return(EOS(STATIC_955)), java.lang.Object(x0), x1) → 878_0_appE_Return(EOS(STATIC_878))
630_1_appE_InvokeMethod(693_0_appE_Return(EOS(STATIC_693)), java.lang.Object(AppE(EOC, java.lang.Object(AppE(EOC, NULL)))), 0) → 878_0_appE_Return(EOS(STATIC_878))
811_1_appE_InvokeMethod(623_0_appE_Return(EOS(STATIC_623), java.lang.Object(AppE(EOC, NULL)), 0), java.lang.Object(AppE(EOC, NULL)), 0) → 832_0_appE_Return(EOS(STATIC_832))
811_1_appE_InvokeMethod(832_0_appE_Return(EOS(STATIC_832)), java.lang.Object(AppE(EOC, NULL)), x0) → 955_0_appE_Return(EOS(STATIC_955))
811_1_appE_InvokeMethod(878_0_appE_Return(EOS(STATIC_878)), java.lang.Object(AppE(EOC, NULL)), x0) → 955_0_appE_Return(EOS(STATIC_955))
811_1_appE_InvokeMethod(955_0_appE_Return(EOS(STATIC_955)), java.lang.Object(AppE(EOC, NULL)), x0) → 955_0_appE_Return(EOS(STATIC_955))

Filtered ground terms:

811_1_appE_InvokeMethod(x1, x2, x3) → 811_1_appE_InvokeMethod(x1, x3)
AppE(x1, x2) → AppE(x2)
554_0_appE_FieldAccess(x1, x2, x3, x4) → 554_0_appE_FieldAccess(x2, x3, x4)
Cond_554_0_appE_FieldAccess(x1, x2, x3, x4, x5) → Cond_554_0_appE_FieldAccess(x1, x4)
955_0_appE_Return(x1) → 955_0_appE_Return
878_0_appE_Return(x1) → 878_0_appE_Return
832_0_appE_Return(x1) → 832_0_appE_Return
623_0_appE_Return(x1, x2, x3) → 623_0_appE_Return
693_0_appE_Return(x1) → 693_0_appE_Return

Filtered duplicate args:

554_0_appE_FieldAccess(x1, x2, x3) → 554_0_appE_FieldAccess(x2, x3)

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

P rules:
554_0_appE_FieldAccess(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 630_1_appE_InvokeMethod(554_0_appE_FieldAccess(x1, java.lang.Object(x0)), java.lang.Object(x0), x1)
554_0_appE_FieldAccess(x0, java.lang.Object(AppE(NULL))) → 811_1_appE_InvokeMethod(554_0_appE_FieldAccess(+(x0, -1), java.lang.Object(AppE(NULL))), +(x0, -1)) | >(x0, 0)
R rules:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 693_0_appE_Return
630_1_appE_InvokeMethod(832_0_appE_Return, java.lang.Object(x0), x1) → 878_0_appE_Return
630_1_appE_InvokeMethod(878_0_appE_Return, java.lang.Object(x0), x1) → 878_0_appE_Return
630_1_appE_InvokeMethod(955_0_appE_Return, java.lang.Object(x0), x1) → 878_0_appE_Return
630_1_appE_InvokeMethod(693_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 878_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, 0) → 832_0_appE_Return
811_1_appE_InvokeMethod(832_0_appE_Return, x0) → 955_0_appE_Return
811_1_appE_InvokeMethod(878_0_appE_Return, x0) → 955_0_appE_Return
811_1_appE_InvokeMethod(955_0_appE_Return, x0) → 955_0_appE_Return

Performed bisimulation on rules. Used the following equivalence classes: {[623_0_appE_Return, 693_0_appE_Return, 832_0_appE_Return, 878_0_appE_Return, 955_0_appE_Return]=623_0_appE_Return}

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

P rules:
554_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 554_0_APPE_FIELDACCESS(x1, java.lang.Object(x0))
554_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL)))
COND_554_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL)))
R rules:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, x0) → 623_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, x0) → 623_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))
(1): 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))
(2): COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_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@b345419 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 554_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 554_0_APPE_FIELDACCESS(x1, java.lang.Object(x0)) the following chains were created:

• We consider the chain 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_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))) ⇒ 554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥NonInfC∧554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))∧(U^Increasing(554_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)    (554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(x0[0]2))))))≥NonInfC∧554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(x0[0]2))))))≥554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]2))))∧(U^Increasing(554_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(554_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(554_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(554_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(554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_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))) ⇒ 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥NonInfC∧554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))∧(U^Increasing(554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))), ≥))

  
  
  We solved constraint (7) using rules (I), (II).
• We consider the chain 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_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)) ⇒ 554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥NonInfC∧554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))∧(U^Increasing(554_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)    (554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(NULL)))))≥NonInfC∧554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(NULL)))))≥554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(NULL)))∧(U^Increasing(554_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(554_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(554_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(554_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(554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_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)) ⇒ 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥NonInfC∧554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))∧(U^Increasing(554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))), ≥))

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

For Pair 554_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_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] ⇒ 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_554_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 ⇒ 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_554_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_554_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_554_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_554_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_554_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_554_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))

  
  
  We solved constraint (21) using rules (I), (II).
• We consider the chain 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(554_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_554_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_554_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥554_0_APPE_FIELDACCESS(+(x0[1], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(554_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(554_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(554_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(554_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(554_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.

• 554_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 554_0_APPE_FIELDACCESS(x1, java.lang.Object(x0))
  
  • ((U^Increasing(554_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(554_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[bni_21] = 0∧0 ≥ 0∧[2 + (-1)bso_22] ≥ 0)
    
  
  
• 554_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_554_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_554_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(554_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(630_1_appE_InvokeMethod(x₁, x₂, x₃)) = 0   
POL(623_0_appE_Return) = 0   
POL(java.lang.Object(x₁)) = [1] + [2]x₁   
POL(AppE(x₁)) = x₁   
POL(NULL) = 0   
POL(0) = 0   
POL(811_1_appE_InvokeMethod(x₁, x₂)) = 0   
POL(554_0_APPE_FIELDACCESS(x₁, x₂)) = [1] + x₂ + x₁   
POL(COND_554_0_APPE_FIELDACCESS(x₁, x₂, x₃)) = [2]   
POL(>(x₁, x₂)) = 0   
POL(+(x₁, x₂)) = 0   
POL(-1) = 0   

The following pairs are in P_>:

554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The following pairs are in P_bound:

554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P_≥:

554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, x0) → 623_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))
(2): COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_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@187d3b0e 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 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_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] ⇒ 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_554_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 ⇒ 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(U^Increasing(COND_554_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_554_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_554_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_554_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_554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))) the following chains were created:

• We consider the chain 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(554_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_554_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_554_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥554_0_APPE_FIELDACCESS(+(x0[1], -1), java.lang.Object(AppE(NULL)))∧(U^Increasing(554_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(554_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(554_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(554_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(554_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.

• 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
  
  • (x0[1] ≥ 0 ⇒ (U^Increasing(COND_554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))
  
  • (x0[1] ≥ 0 ⇒ (U^Increasing(554_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(630_1_appE_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁   
POL(623_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(811_1_appE_InvokeMethod(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁   
POL(554_0_APPE_FIELDACCESS(x₁, x₂)) = [-1] + [-1]x₂ + x₁   
POL(COND_554_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_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P_bound:

554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_554_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 554_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P_≥:

554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, x0) → 623_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): 554_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_554_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))


The set Q consists of the following terms:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, 0) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, x0) → 623_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_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:
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_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:

554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The TRS R consists of the following rules:

630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(NULL)), pos(01)) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1) → 623_0_appE_Return
630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), pos(01)) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, pos(01)) → 623_0_appE_Return
811_1_appE_InvokeMethod(623_0_appE_Return, x0) → 623_0_appE_Return

The set Q consists of the following terms:

630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_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:

554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

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

630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_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].

630_1_appE_InvokeMethod(623_0_appE_Return, java.lang.Object(x0), x1)
811_1_appE_InvokeMethod(623_0_appE_Return, x0)

(21) Obligation:

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

554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_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:

• 554_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 554_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) → 347_0_createList_New(EOS(STATIC_347), i39) | >(i39, 0)
347_0_createList_New(EOS(STATIC_347), i39) → 368_0_createList_Duplicate(EOS(STATIC_368), i39)
368_0_createList_Duplicate(EOS(STATIC_368), i39) → 380_0_createList_Load(EOS(STATIC_380), i39)
380_0_createList_Load(EOS(STATIC_380), i39) → 390_0_createList_InvokeMethod(EOS(STATIC_390), i39)
390_0_createList_InvokeMethod(EOS(STATIC_390), i39) → 405_0_<init>_Load(EOS(STATIC_405), i39)
405_0_<init>_Load(EOS(STATIC_405), i39) → 410_0_<init>_InvokeMethod(EOS(STATIC_410), i39)
410_0_<init>_InvokeMethod(EOS(STATIC_410), i39) → 414_0_<init>_Load(EOS(STATIC_414), i39)
414_0_<init>_Load(EOS(STATIC_414), i39) → 417_0_<init>_Load(EOS(STATIC_417), i39)
417_0_<init>_Load(EOS(STATIC_417), i39) → 419_0_<init>_FieldAccess(EOS(STATIC_419), i39)
419_0_<init>_FieldAccess(EOS(STATIC_419), i39) → 424_0_<init>_Return(EOS(STATIC_424), i39)
424_0_<init>_Return(EOS(STATIC_424), i39) → 428_0_createList_Store(EOS(STATIC_428), i39)
428_0_createList_Store(EOS(STATIC_428), i39) → 433_0_createList_Inc(EOS(STATIC_433), i39)
433_0_createList_Inc(EOS(STATIC_433), i39) → 439_0_createList_JMP(EOS(STATIC_439), +(i39, -1)) | >(i39, 0)
439_0_createList_JMP(EOS(STATIC_439), i52) → 445_0_createList_Load(EOS(STATIC_445), i52)
445_0_createList_Load(EOS(STATIC_445), 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@45b020b1 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