Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 AND
    ↳5 IDP
        ↳6 IDPNonInfProof (⇒)
        ↳7 AND
            ↳8 IDP
                ↳9 IDPNonInfProof (⇒)
                ↳10 IDP
                ↳11 IDependencyGraphProof (⇔)
                ↳12 TRUE
            ↳13 IDP
                ↳14 IDPtoQDPProof (⇒)
                ↳15 QDP
                ↳16 UsableRulesProof (⇔)
                ↳17 QDP
                ↳18 QReductionProof (⇔)
                ↳19 QDP
                ↳20 QDPSizeChangeProof (⇔)
                ↳21 YES
    ↳22 IDP
        ↳23 IDPNonInfProof (⇒)
        ↳24 AND
            ↳25 IDP
                ↳26 IDependencyGraphProof (⇔)
                ↳27 TRUE
            ↳28 IDP
                ↳29 IDependencyGraphProof (⇔)
                ↳30 TRUE

(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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
AppE.main([Ljava/lang/String;)V: Graph of 122 nodes with 0 SCCs.

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

AppE.appE(I)V: Graph of 55 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 36 rules for P and 22 rules for R.


Combined rules. Obtained 2 rules for P and 7 rules for R.


Filtered ground terms:


978_1_appE_InvokeMethod(x1, x2, x3) → 978_1_appE_InvokeMethod(x1, x3)
AppE(x1, x2) → AppE(x2)
534_0_appE_FieldAccess(x1, x2, x3, x4) → 534_0_appE_FieldAccess(x2, x3, x4)
Cond_534_0_appE_FieldAccess(x1, x2, x3, x4, x5) → Cond_534_0_appE_FieldAccess(x1, x4)
1042_0_appE_Return(x1) → 1042_0_appE_Return
1000_0_appE_Return(x1) → 1000_0_appE_Return
672_0_appE_Return(x1, x2, x3) → 672_0_appE_Return
759_0_appE_Return(x1) → 759_0_appE_Return

Filtered duplicate args:


534_0_appE_FieldAccess(x1, x2, x3) → 534_0_appE_FieldAccess(x2, x3)

Combined rules. Obtained 2 rules for P and 7 rules for R.


Finished conversion. Obtained 2 rules for P and 7 rules for R. System has predefined symbols.



Log for SCC 1:

Generated 17 rules for P and 3 rules for R.


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


Filtered ground terms:


352_0_createList_LE(x1, x2, x3) → 352_0_createList_LE(x2, x3)
Cond_352_0_createList_LE(x1, x2, x3, x4) → Cond_352_0_createList_LE(x1, x3, x4)
378_0_createList_Return(x1) → 378_0_createList_Return

Filtered duplicate args:


352_0_createList_LE(x1, x2) → 352_0_createList_LE(x2)
Cond_352_0_createList_LE(x1, x2, x3) → Cond_352_0_createList_LE(x1, x3)

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


Finished conversion. Obtained 1 rules for P and 1 rules for R. System has predefined symbols.


(4) Complex Obligation (AND)

(5) 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:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 759_0_appE_Return
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
978_1_appE_InvokeMethod(672_0_appE_Return, 0) → 1000_0_appE_Return
978_1_appE_InvokeMethod(1000_0_appE_Return, x0) → 1042_0_appE_Return
978_1_appE_InvokeMethod(1042_0_appE_Return, x0) → 1042_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))
(1): 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))
(2): COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_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 →* TRUE)∧(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:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0)
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0)
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, 0)
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

(6) IDPNonInfProof (SOUND transformation)

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 534_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 534_0_APPE_FIELDACCESS(x1, java.lang.Object(x0)) the following chains were created:
  • We consider the chain 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_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))) ⇒ 534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥NonInfC∧534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))∧(UIncreasing(534_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)    (534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(x0[0]2))))))≥NonInfC∧534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(java.lang.Object(x0[0]2))))))≥534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]2))))∧(UIncreasing(534_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)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[4 + (-1)bso_25] + [4]x0[0]2 ≥ 0)



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

    (4)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[4 + (-1)bso_25] + [4]x0[0]2 ≥ 0)



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

    (5)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[4 + (-1)bso_25] + [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)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧0 ≥ 0∧[4 + (-1)bso_25] ≥ 0∧[1] ≥ 0)



  • We consider the chain COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_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))) ⇒ 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥NonInfC∧534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))∧(UIncreasing(534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))), ≥))



    We solved constraint (7) using rules (I), (II).
  • We consider the chain 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_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)) ⇒ 534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥NonInfC∧534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(AppE(java.lang.Object(x0[0]1))))≥534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))∧(UIncreasing(534_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)    (534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(NULL)))))≥NonInfC∧534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(AppE(NULL)))))≥534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(NULL)))∧(UIncreasing(534_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)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[2 + (-1)bso_25] ≥ 0)



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

    (11)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[2 + (-1)bso_25] ≥ 0)



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

    (12)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧[2 + (-1)bso_25] ≥ 0)



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

    (13)    ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧0 ≥ 0∧[2 + (-1)bso_25] ≥ 0)



  • We consider the chain COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0])), 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_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)) ⇒ 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥NonInfC∧534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0]))))≥534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))∧(UIncreasing(534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))), ≥))



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




For Pair 534_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL))) the following chains were created:
  • We consider the chain 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_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] ⇒ 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(UIncreasing(COND_534_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 ⇒ 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(UIncreasing(COND_534_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 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[1] ≥ 0∧[(-1)bso_27] + x0[1] ≥ 0)



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

    (18)    (0 ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[1] ≥ 0∧[(-1)bso_27] + x0[1] ≥ 0)



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

    (19)    (0 ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[1] ≥ 0∧[(-1)bso_27] + x0[1] ≥ 0)



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

    (20)    (0 ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[bni_26] ≥ 0∧[(2)bni_26 + (-1)Bound*bni_26] ≥ 0∧[1] ≥ 0∧[(-1)bso_27] ≥ 0)







For Pair COND_534_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL))) the following chains were created:
  • We consider the chain 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_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_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥))



    We solved constraint (21) using rules (I), (II).
  • We consider the chain 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_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_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(UIncreasing(534_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_534_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_534_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥534_0_APPE_FIELDACCESS(+(x0[1], -1), java.lang.Object(AppE(NULL)))∧(UIncreasing(534_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 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_28 + (-1)Bound*bni_28] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (25)    (0 ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_28 + (-1)Bound*bni_28] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (26)    (0 ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(2)bni_28 + (-1)Bound*bni_28] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (27)    (0 ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧0 ≥ 0∧[(2)bni_28 + (-1)Bound*bni_28] ≥ 0∧0 ≥ 0∧[(-1)bso_29] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 534_0_APPE_FIELDACCESS(x1, java.lang.Object(AppE(java.lang.Object(x0)))) → 534_0_APPE_FIELDACCESS(x1, java.lang.Object(x0))
    • ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧0 ≥ 0∧[4 + (-1)bso_25] ≥ 0∧[1] ≥ 0)
    • ((UIncreasing(534_0_APPE_FIELDACCESS(x1[0]1, java.lang.Object(x0[0]1))), ≥)∧0 ≥ 0∧[2 + (-1)bso_25] ≥ 0)

  • 534_0_APPE_FIELDACCESS(x0, java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0, 0), x0, java.lang.Object(AppE(NULL)))
    • (0 ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[bni_26] ≥ 0∧[(2)bni_26 + (-1)Bound*bni_26] ≥ 0∧[1] ≥ 0∧[(-1)bso_27] ≥ 0)

  • COND_534_0_APPE_FIELDACCESS(TRUE, x0, java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0, -1), java.lang.Object(AppE(NULL)))
    • (0 ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧0 ≥ 0∧[(2)bni_28 + (-1)Bound*bni_28] ≥ 0∧0 ≥ 0∧[(-1)bso_29] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
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(707_1_appE_InvokeMethod(x1, x2, x3)) = 0   
POL(672_0_appE_Return) = 0   
POL(java.lang.Object(x1)) = [1] + [2]x1   
POL(AppE(x1)) = x1   
POL(NULL) = 0   
POL(0) = 0   
POL(759_0_appE_Return) = 0   
POL(1042_0_appE_Return) = 0   
POL(1000_0_appE_Return) = 0   
POL(978_1_appE_InvokeMethod(x1, x2)) = 0   
POL(534_0_APPE_FIELDACCESS(x1, x2)) = [1] + x2 + x1   
POL(COND_534_0_APPE_FIELDACCESS(x1, x2, x3)) = [2]   
POL(>(x1, x2)) = 0   
POL(+(x1, x2)) = 0   
POL(-1) = 0   

The following pairs are in P>:

534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The following pairs are in Pbound:

534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P≥:

534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

There are no usable rules.

(7) Complex Obligation (AND)

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


The ITRS R consists of the following rules:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 759_0_appE_Return
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
978_1_appE_InvokeMethod(672_0_appE_Return, 0) → 1000_0_appE_Return
978_1_appE_InvokeMethod(1000_0_appE_Return, x0) → 1042_0_appE_Return
978_1_appE_InvokeMethod(1042_0_appE_Return, x0) → 1042_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))
(2): COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_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 →* TRUE)∧(x0[1] →* x0[2]))



The set Q consists of the following terms:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0)
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0)
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, 0)
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

(9) IDPNonInfProof (SOUND transformation)

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 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))) the following chains were created:
  • We consider the chain 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_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] ⇒ 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(UIncreasing(COND_534_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 ⇒ 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL)))≥COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))∧(UIncreasing(COND_534_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 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[1] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (4)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[1] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (5)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[1] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (6)    (x0[1] ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_27 + bni_27] + [bni_27]x0[1] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))) the following chains were created:
  • We consider the chain 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL))), COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL))), 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_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_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL)))≥534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))∧(UIncreasing(534_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_534_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥NonInfC∧COND_534_0_APPE_FIELDACCESS(TRUE, x0[1], java.lang.Object(AppE(NULL)))≥534_0_APPE_FIELDACCESS(+(x0[1], -1), java.lang.Object(AppE(NULL)))∧(UIncreasing(534_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 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_29] + [bni_29]x0[1] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (10)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_29] + [bni_29]x0[1] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (11)    (x0[1] + [-1] ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_29] + [bni_29]x0[1] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (12)    (x0[1] ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]x0[1] ≥ 0∧[1 + (-1)bso_30] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
    • (x0[1] ≥ 0 ⇒ (UIncreasing(COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_27 + bni_27] + [bni_27]x0[1] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))
    • (x0[1] ≥ 0 ⇒ (UIncreasing(534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]x0[1] ≥ 0∧[1 + (-1)bso_30] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [3]   
POL(FALSE) = 0   
POL(707_1_appE_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1   
POL(672_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(759_0_appE_Return) = [-1]   
POL(1042_0_appE_Return) = [-1]   
POL(1000_0_appE_Return) = [-1]   
POL(978_1_appE_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(534_0_APPE_FIELDACCESS(x1, x2)) = [-1] + [-1]x2 + x1   
POL(COND_534_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_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in Pbound:

534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))
COND_534_0_APPE_FIELDACCESS(TRUE, x0[2], java.lang.Object(AppE(NULL))) → 534_0_APPE_FIELDACCESS(+(x0[2], -1), java.lang.Object(AppE(NULL)))

The following pairs are in P≥:

534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(>(x0[1], 0), x0[1], java.lang.Object(AppE(NULL)))

There are no usable rules.

(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:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 759_0_appE_Return
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
978_1_appE_InvokeMethod(672_0_appE_Return, 0) → 1000_0_appE_Return
978_1_appE_InvokeMethod(1000_0_appE_Return, x0) → 1042_0_appE_Return
978_1_appE_InvokeMethod(1042_0_appE_Return, x0) → 1042_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): 534_0_APPE_FIELDACCESS(x0[1], java.lang.Object(AppE(NULL))) → COND_534_0_APPE_FIELDACCESS(x0[1] > 0, x0[1], java.lang.Object(AppE(NULL)))


The set Q consists of the following terms:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0)
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0)
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, 0)
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE

(13) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:
none


The ITRS R consists of the following rules:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0) → 759_0_appE_Return
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
978_1_appE_InvokeMethod(672_0_appE_Return, 0) → 1000_0_appE_Return
978_1_appE_InvokeMethod(1000_0_appE_Return, x0) → 1042_0_appE_Return
978_1_appE_InvokeMethod(1042_0_appE_Return, x0) → 1042_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_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:
707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), 0)
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), 0)
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, 0)
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

(14) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(15) Obligation:

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

534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The TRS R consists of the following rules:

707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), pos(01)) → 759_0_appE_Return
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), pos(01)) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1) → 1042_0_appE_Return
978_1_appE_InvokeMethod(672_0_appE_Return, pos(01)) → 1000_0_appE_Return
978_1_appE_InvokeMethod(1000_0_appE_Return, x0) → 1042_0_appE_Return
978_1_appE_InvokeMethod(1042_0_appE_Return, x0) → 1042_0_appE_Return

The set Q consists of the following terms:

707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), pos(01))
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), pos(01))
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, pos(01))
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

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

(16) 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.

(17) Obligation:

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

534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

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

707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), pos(01))
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), pos(01))
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, pos(01))
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

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

(18) 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].

707_1_appE_InvokeMethod(672_0_appE_Return, java.lang.Object(AppE(NULL)), pos(01))
707_1_appE_InvokeMethod(759_0_appE_Return, java.lang.Object(AppE(java.lang.Object(AppE(NULL)))), pos(01))
707_1_appE_InvokeMethod(1000_0_appE_Return, java.lang.Object(x0), x1)
707_1_appE_InvokeMethod(1042_0_appE_Return, java.lang.Object(x0), x1)
978_1_appE_InvokeMethod(672_0_appE_Return, pos(01))
978_1_appE_InvokeMethod(1000_0_appE_Return, x0)
978_1_appE_InvokeMethod(1042_0_appE_Return, x0)

(19) Obligation:

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

534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_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.

(20) 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:

  • 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(AppE(java.lang.Object(x0[0])))) → 534_0_APPE_FIELDACCESS(x1[0], java.lang.Object(x0[0]))
    The graph contains the following edges 1 >= 1, 2 > 2

(21) YES

(22) 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:
352_0_createList_LE(0) → 378_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 352_0_CREATELIST_LE(x0[0]) → COND_352_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_352_0_CREATELIST_LE(TRUE, x0[1]) → 352_0_CREATELIST_LE(x0[1] + -1)

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


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



The set Q consists of the following terms:
352_0_createList_LE(0)

(23) IDPNonInfProof (SOUND transformation)

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 352_0_CREATELIST_LE(x0) → COND_352_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 352_0_CREATELIST_LE(x0[0]) → COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_352_0_CREATELIST_LE(TRUE, x0[1]) → 352_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 352_0_CREATELIST_LE(x0[0])≥NonInfC∧352_0_CREATELIST_LE(x0[0])≥COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_352_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 ⇒ 352_0_CREATELIST_LE(x0[0])≥NonInfC∧352_0_CREATELIST_LE(x0[0])≥COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_352_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 ⇒ (UIncreasing(COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)







For Pair COND_352_0_CREATELIST_LE(TRUE, x0) → 352_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
  • We consider the chain COND_352_0_CREATELIST_LE(TRUE, x0[1]) → 352_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_352_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_352_0_CREATELIST_LE(TRUE, x0[1])≥352_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(352_0_CREATELIST_LE(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(352_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)



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

    (9)    ((UIncreasing(352_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(352_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(352_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 352_0_CREATELIST_LE(x0) → COND_352_0_CREATELIST_LE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  • COND_352_0_CREATELIST_LE(TRUE, x0) → 352_0_CREATELIST_LE(+(x0, -1))
    • ((UIncreasing(352_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(352_0_createList_LE(x1)) = [-1]   
POL(0) = 0   
POL(378_0_createList_Return) = [-1]   
POL(352_0_CREATELIST_LE(x1)) = [2]x1   
POL(COND_352_0_CREATELIST_LE(x1, x2)) = [2]x2   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_352_0_CREATELIST_LE(TRUE, x0[1]) → 352_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

352_0_CREATELIST_LE(x0[0]) → COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

352_0_CREATELIST_LE(x0[0]) → COND_352_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(24) Complex Obligation (AND)

(25) 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:
352_0_createList_LE(0) → 378_0_createList_Return

The integer pair graph contains the following rules and edges:
(0): 352_0_CREATELIST_LE(x0[0]) → COND_352_0_CREATELIST_LE(x0[0] > 0, x0[0])


The set Q consists of the following terms:
352_0_createList_LE(0)

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(27) TRUE

(28) 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:
352_0_createList_LE(0) → 378_0_createList_Return

The integer pair graph contains the following rules and edges:
(1): COND_352_0_CREATELIST_LE(TRUE, x0[1]) → 352_0_CREATELIST_LE(x0[1] + -1)


The set Q consists of the following terms:
352_0_createList_LE(0)

(29) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) TRUE