Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 130 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 AND
    ↳5 JBCTerminationSCC
        ↳6 SCCToIDPv1Proof (⇒, 50 ms)
        ↳7 IDP
        ↳8 IDPNonInfProof (⇒, 110 ms)
        ↳9 AND
            ↳10 IDP
                ↳11 IDependencyGraphProof (⇔, 0 ms)
                ↳12 TRUE
            ↳13 IDP
                ↳14 IDependencyGraphProof (⇔, 0 ms)
                ↳15 TRUE
    ↳16 JBCTerminationSCC
        ↳17 SCCToIDPv1Proof (⇒, 270 ms)
        ↳18 IDP
        ↳19 IDPtoQDPProof (⇒, 20 ms)
        ↳20 QDP
        ↳21 QDPOrderProof (⇔, 30 ms)
        ↳22 QDP
        ↳23 DependencyGraphProof (⇔, 0 ms)
        ↳24 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_25 (Sun Microsystems Inc.) Main-Class: AlternatingGrowReduce2/AlternatingGrowReduce2 
package AlternatingGrowReduce2;

public class AlternatingGrowReduce2 {
	AlternatingGrowReduce2 next;

	public static void main(String[] argv) {
		Random.args = argv;
		AlternatingGrowReduce2 list = createList(Random.random());

		int mode = 0;
		while (list != null) {
			if (mode == 0) {
				list = list.next;
			} else if (mode == 1) {
				list = new AlternatingGrowReduce2(list);
			} else if (mode > 1) {
				list = list.next;
			}

			mode++;
			if (mode > 2) {
				mode = 0;
			}
		}
	}

	public AlternatingGrowReduce2(AlternatingGrowReduce2 old) {
		this.next = old;
	}

	public static AlternatingGrowReduce2 createList(int length) {
		AlternatingGrowReduce2 res = new AlternatingGrowReduce2(null);
		while (length > 0) {
			res = new AlternatingGrowReduce2(res);
			length--;
		}
		return res;
	}
}


package AlternatingGrowReduce2;

public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}


(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
AlternatingGrowReduce2.AlternatingGrowReduce2.main([Ljava/lang/String;)V: Graph of 163 nodes with 1 SCC.

AlternatingGrowReduce2.AlternatingGrowReduce2.createList(I)LAlternatingGrowReduce2/AlternatingGrowReduce2;: Graph of 33 nodes with 1 SCC.


(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: AlternatingGrowReduce2.AlternatingGrowReduce2.createList(I)LAlternatingGrowReduce2/AlternatingGrowReduce2;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 17 rules for P and 0 rules for R.


P rules:
323_0_createList_LE(EOS(STATIC_323), i36, i36) → 328_0_createList_LE(EOS(STATIC_328), i36, i36)
328_0_createList_LE(EOS(STATIC_328), i36, i36) → 335_0_createList_New(EOS(STATIC_335), i36) | >(i36, 0)
335_0_createList_New(EOS(STATIC_335), i36) → 341_0_createList_Duplicate(EOS(STATIC_341), i36)
341_0_createList_Duplicate(EOS(STATIC_341), i36) → 347_0_createList_Load(EOS(STATIC_347), i36)
347_0_createList_Load(EOS(STATIC_347), i36) → 365_0_createList_InvokeMethod(EOS(STATIC_365), i36)
365_0_createList_InvokeMethod(EOS(STATIC_365), i36) → 379_0_<init>_Load(EOS(STATIC_379), i36)
379_0_<init>_Load(EOS(STATIC_379), i36) → 384_0_<init>_InvokeMethod(EOS(STATIC_384), i36)
384_0_<init>_InvokeMethod(EOS(STATIC_384), i36) → 389_0_<init>_Load(EOS(STATIC_389), i36)
389_0_<init>_Load(EOS(STATIC_389), i36) → 393_0_<init>_Load(EOS(STATIC_393), i36)
393_0_<init>_Load(EOS(STATIC_393), i36) → 397_0_<init>_FieldAccess(EOS(STATIC_397), i36)
397_0_<init>_FieldAccess(EOS(STATIC_397), i36) → 403_0_<init>_Return(EOS(STATIC_403), i36)
403_0_<init>_Return(EOS(STATIC_403), i36) → 407_0_createList_Store(EOS(STATIC_407), i36)
407_0_createList_Store(EOS(STATIC_407), i36) → 411_0_createList_Inc(EOS(STATIC_411), i36)
411_0_createList_Inc(EOS(STATIC_411), i36) → 415_0_createList_JMP(EOS(STATIC_415), +(i36, -1)) | >(i36, 0)
415_0_createList_JMP(EOS(STATIC_415), i53) → 419_0_createList_Load(EOS(STATIC_419), i53)
419_0_createList_Load(EOS(STATIC_419), i53) → 315_0_createList_Load(EOS(STATIC_315), i53)
315_0_createList_Load(EOS(STATIC_315), i29) → 323_0_createList_LE(EOS(STATIC_323), i29, i29)
R rules:

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


P rules:
323_0_createList_LE(EOS(STATIC_323), x0, x0) → 323_0_createList_LE(EOS(STATIC_323), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:



323_0_createList_LE(x1, x2, x3) → 323_0_createList_LE(x2, x3)
EOS(x1) → EOS
Cond_323_0_createList_LE(x1, x2, x3, x4) → Cond_323_0_createList_LE(x1, x3, x4)

Filtered duplicate args:



323_0_createList_LE(x1, x2) → 323_0_createList_LE(x2)
Cond_323_0_createList_LE(x1, x2, x3) → Cond_323_0_createList_LE(x1, x3)

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


P rules:
323_0_createList_LE(x0) → 323_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:
323_0_CREATELIST_LE(x0) → COND_323_0_CREATELIST_LE(>(x0, 0), x0)
COND_323_0_CREATELIST_LE(TRUE, x0) → 323_0_CREATELIST_LE(+(x0, -1))
R rules:

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 323_0_CREATELIST_LE(x0[0]) → COND_323_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_323_0_CREATELIST_LE(TRUE, x0[1]) → 323_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.

(8) 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@54c90c7 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 323_0_CREATELIST_LE(x0) → COND_323_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 323_0_CREATELIST_LE(x0[0]) → COND_323_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_323_0_CREATELIST_LE(TRUE, x0[1]) → 323_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 323_0_CREATELIST_LE(x0[0])≥NonInfC∧323_0_CREATELIST_LE(x0[0])≥COND_323_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_323_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 ⇒ 323_0_CREATELIST_LE(x0[0])≥NonInfC∧323_0_CREATELIST_LE(x0[0])≥COND_323_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_323_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_323_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 ⇒ (UIncreasing(COND_323_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 ⇒ (UIncreasing(COND_323_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 ⇒ (UIncreasing(COND_323_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_323_0_CREATELIST_LE(TRUE, x0) → 323_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
  • We consider the chain COND_323_0_CREATELIST_LE(TRUE, x0[1]) → 323_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_323_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_323_0_CREATELIST_LE(TRUE, x0[1])≥323_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(323_0_CREATELIST_LE(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(323_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)    ((UIncreasing(323_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)    ((UIncreasing(323_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)    ((UIncreasing(323_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.
  • 323_0_CREATELIST_LE(x0) → COND_323_0_CREATELIST_LE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_323_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_323_0_CREATELIST_LE(TRUE, x0) → 323_0_CREATELIST_LE(+(x0, -1))
    • ((UIncreasing(323_0_CREATELIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 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(323_0_CREATELIST_LE(x1)) = [2]x1   
POL(COND_323_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_323_0_CREATELIST_LE(TRUE, x0[1]) → 323_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

323_0_CREATELIST_LE(x0[0]) → COND_323_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

323_0_CREATELIST_LE(x0[0]) → COND_323_0_CREATELIST_LE(>(x0[0], 0), x0[0])

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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 323_0_CREATELIST_LE(x0[0]) → COND_323_0_CREATELIST_LE(x0[0] > 0, x0[0])


The set Q is empty.

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_323_0_CREATELIST_LE(TRUE, x0[1]) → 323_0_CREATELIST_LE(x0[1] + -1)


The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: AlternatingGrowReduce2.AlternatingGrowReduce2.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 56 rules for P and 0 rules for R.


P rules:
530_0_main_NULL(EOS(STATIC_530), java.lang.Object(o98sub), i59, java.lang.Object(o98sub)) → 533_0_main_NULL(EOS(STATIC_533), java.lang.Object(o98sub), i59, java.lang.Object(o98sub))
533_0_main_NULL(EOS(STATIC_533), java.lang.Object(o98sub), i59, java.lang.Object(o98sub)) → 536_0_main_Load(EOS(STATIC_536), java.lang.Object(o98sub), i59)
536_0_main_Load(EOS(STATIC_536), java.lang.Object(o98sub), i59) → 538_0_main_NE(EOS(STATIC_538), java.lang.Object(o98sub), i59, i59)
538_0_main_NE(EOS(STATIC_538), java.lang.Object(o98sub), i63, i63) → 542_0_main_NE(EOS(STATIC_542), java.lang.Object(o98sub), i63, i63)
538_0_main_NE(EOS(STATIC_538), java.lang.Object(o98sub), matching1, matching2) → 543_0_main_NE(EOS(STATIC_543), java.lang.Object(o98sub), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
542_0_main_NE(EOS(STATIC_542), java.lang.Object(o98sub), i63, i63) → 545_0_main_Load(EOS(STATIC_545), java.lang.Object(o98sub), i63) | >(i63, 0)
545_0_main_Load(EOS(STATIC_545), java.lang.Object(o98sub), i63) → 549_0_main_ConstantStackPush(EOS(STATIC_549), java.lang.Object(o98sub), i63, i63)
549_0_main_ConstantStackPush(EOS(STATIC_549), java.lang.Object(o98sub), i63, i63) → 554_0_main_NE(EOS(STATIC_554), java.lang.Object(o98sub), i63, i63, 1)
554_0_main_NE(EOS(STATIC_554), java.lang.Object(o98sub), matching1, matching2, matching3) → 558_0_main_NE(EOS(STATIC_558), java.lang.Object(o98sub), 1, 1, 1) | &&(&&(=(matching1, 1), =(matching2, 1)), =(matching3, 1))
554_0_main_NE(EOS(STATIC_554), java.lang.Object(o98sub), matching1, matching2, matching3) → 559_0_main_NE(EOS(STATIC_559), java.lang.Object(o98sub), 2, 2, 1) | &&(&&(=(matching1, 2), =(matching2, 2)), =(matching3, 1))
558_0_main_NE(EOS(STATIC_558), java.lang.Object(o98sub), matching1, matching2, matching3) → 563_0_main_New(EOS(STATIC_563), java.lang.Object(o98sub), 1) | &&(&&(=(matching1, 1), =(matching2, 1)), =(matching3, 1))
563_0_main_New(EOS(STATIC_563), java.lang.Object(o98sub), matching1) → 569_0_main_Duplicate(EOS(STATIC_569), java.lang.Object(o98sub), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) | =(matching1, 1)
569_0_main_Duplicate(EOS(STATIC_569), java.lang.Object(o98sub), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) → 575_0_main_Load(EOS(STATIC_575), java.lang.Object(o98sub), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) | =(matching1, 1)
575_0_main_Load(EOS(STATIC_575), java.lang.Object(o98sub), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) → 581_0_main_InvokeMethod(EOS(STATIC_581), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) | =(matching1, 1)
581_0_main_InvokeMethod(EOS(STATIC_581), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) → 587_0_<init>_Load(EOS(STATIC_587), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) | =(matching1, 1)
587_0_<init>_Load(EOS(STATIC_587), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) → 600_0_<init>_InvokeMethod(EOS(STATIC_600), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) | =(matching1, 1)
600_0_<init>_InvokeMethod(EOS(STATIC_600), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) → 605_0_<init>_Load(EOS(STATIC_605), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) | =(matching1, 1)
605_0_<init>_Load(EOS(STATIC_605), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) → 609_0_<init>_Load(EOS(STATIC_609), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) | =(matching1, 1)
609_0_<init>_Load(EOS(STATIC_609), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL))) → 614_0_<init>_FieldAccess(EOS(STATIC_614), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) | =(matching1, 1)
614_0_<init>_FieldAccess(EOS(STATIC_614), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, NULL)), java.lang.Object(o98sub)) → 620_0_<init>_Return(EOS(STATIC_620), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), java.lang.Object(o98sub)) | =(matching1, 1)
620_0_<init>_Return(EOS(STATIC_620), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), java.lang.Object(o98sub)) → 625_0_main_Store(EOS(STATIC_625), 1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub)))) | =(matching1, 1)
625_0_main_Store(EOS(STATIC_625), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub)))) → 629_0_main_JMP(EOS(STATIC_629), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 1) | =(matching1, 1)
629_0_main_JMP(EOS(STATIC_629), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), matching1) → 634_0_main_Inc(EOS(STATIC_634), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 1) | =(matching1, 1)
634_0_main_Inc(EOS(STATIC_634), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), matching1) → 636_0_main_Load(EOS(STATIC_636), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 2) | =(matching1, 1)
636_0_main_Load(EOS(STATIC_636), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), matching1) → 644_0_main_ConstantStackPush(EOS(STATIC_644), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 2, 2) | =(matching1, 2)
644_0_main_ConstantStackPush(EOS(STATIC_644), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), matching1, matching2) → 647_0_main_LE(EOS(STATIC_647), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 2, 2, 2) | &&(=(matching1, 2), =(matching2, 2))
647_0_main_LE(EOS(STATIC_647), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), matching1, matching2, matching3) → 651_0_main_Load(EOS(STATIC_651), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 2) | &&(&&(=(matching1, 2), =(matching2, 2)), =(matching3, 2))
651_0_main_Load(EOS(STATIC_651), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), matching1) → 526_0_main_Load(EOS(STATIC_526), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(o98sub))), 2) | =(matching1, 2)
526_0_main_Load(EOS(STATIC_526), o92, i59) → 530_0_main_NULL(EOS(STATIC_530), o92, i59, o92)
559_0_main_NE(EOS(STATIC_559), java.lang.Object(o98sub), matching1, matching2, matching3) → 565_0_main_Load(EOS(STATIC_565), java.lang.Object(o98sub), 2) | &&(&&(=(matching1, 2), =(matching2, 2)), =(matching3, 1))
565_0_main_Load(EOS(STATIC_565), java.lang.Object(o98sub), matching1) → 571_0_main_ConstantStackPush(EOS(STATIC_571), java.lang.Object(o98sub), 2, 2) | =(matching1, 2)
571_0_main_ConstantStackPush(EOS(STATIC_571), java.lang.Object(o98sub), matching1, matching2) → 577_0_main_LE(EOS(STATIC_577), java.lang.Object(o98sub), 2, 2, 1) | &&(=(matching1, 2), =(matching2, 2))
577_0_main_LE(EOS(STATIC_577), java.lang.Object(o98sub), matching1, matching2, matching3) → 583_0_main_Load(EOS(STATIC_583), java.lang.Object(o98sub), 2) | &&(&&(=(matching1, 2), =(matching2, 2)), =(matching3, 1))
583_0_main_Load(EOS(STATIC_583), java.lang.Object(o98sub), matching1) → 590_0_main_FieldAccess(EOS(STATIC_590), 2, java.lang.Object(o98sub)) | =(matching1, 2)
590_0_main_FieldAccess(EOS(STATIC_590), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, o108))) → 594_0_main_FieldAccess(EOS(STATIC_594), 2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, o108))) | =(matching1, 2)
594_0_main_FieldAccess(EOS(STATIC_594), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, o108))) → 602_0_main_Store(EOS(STATIC_602), 2, o108) | =(matching1, 2)
602_0_main_Store(EOS(STATIC_602), matching1, o108) → 607_0_main_Inc(EOS(STATIC_607), o108, 2) | =(matching1, 2)
607_0_main_Inc(EOS(STATIC_607), o108, matching1) → 612_0_main_Load(EOS(STATIC_612), o108) | =(matching1, 2)
612_0_main_Load(EOS(STATIC_612), o108) → 616_0_main_ConstantStackPush(EOS(STATIC_616), o108)
616_0_main_ConstantStackPush(EOS(STATIC_616), o108) → 623_0_main_LE(EOS(STATIC_623), o108, 2)
623_0_main_LE(EOS(STATIC_623), o108, matching1) → 627_0_main_ConstantStackPush(EOS(STATIC_627), o108) | =(matching1, 2)
627_0_main_ConstantStackPush(EOS(STATIC_627), o108) → 631_0_main_Store(EOS(STATIC_631), o108, 0)
631_0_main_Store(EOS(STATIC_631), o108, matching1) → 635_0_main_JMP(EOS(STATIC_635), o108, 0) | =(matching1, 0)
635_0_main_JMP(EOS(STATIC_635), o108, matching1) → 640_0_main_Load(EOS(STATIC_640), o108, 0) | =(matching1, 0)
640_0_main_Load(EOS(STATIC_640), o108, matching1) → 526_0_main_Load(EOS(STATIC_526), o108, 0) | =(matching1, 0)
543_0_main_NE(EOS(STATIC_543), java.lang.Object(o98sub), matching1, matching2) → 547_0_main_Load(EOS(STATIC_547), java.lang.Object(o98sub), 0) | &&(=(matching1, 0), =(matching2, 0))
547_0_main_Load(EOS(STATIC_547), java.lang.Object(o98sub), matching1) → 552_0_main_FieldAccess(EOS(STATIC_552), 0, java.lang.Object(o98sub)) | =(matching1, 0)
552_0_main_FieldAccess(EOS(STATIC_552), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, o103))) → 556_0_main_FieldAccess(EOS(STATIC_556), 0, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, o103))) | =(matching1, 0)
556_0_main_FieldAccess(EOS(STATIC_556), matching1, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, o103))) → 561_0_main_Store(EOS(STATIC_561), 0, o103) | =(matching1, 0)
561_0_main_Store(EOS(STATIC_561), matching1, o103) → 567_0_main_JMP(EOS(STATIC_567), o103, 0) | =(matching1, 0)
567_0_main_JMP(EOS(STATIC_567), o103, matching1) → 573_0_main_Inc(EOS(STATIC_573), o103, 0) | =(matching1, 0)
573_0_main_Inc(EOS(STATIC_573), o103, matching1) → 579_0_main_Load(EOS(STATIC_579), o103, 1) | =(matching1, 0)
579_0_main_Load(EOS(STATIC_579), o103, matching1) → 585_0_main_ConstantStackPush(EOS(STATIC_585), o103, 1, 1) | =(matching1, 1)
585_0_main_ConstantStackPush(EOS(STATIC_585), o103, matching1, matching2) → 592_0_main_LE(EOS(STATIC_592), o103, 1, 1) | &&(=(matching1, 1), =(matching2, 1))
592_0_main_LE(EOS(STATIC_592), o103, matching1, matching2) → 598_0_main_Load(EOS(STATIC_598), o103, 1) | &&(=(matching1, 1), =(matching2, 1))
598_0_main_Load(EOS(STATIC_598), o103, matching1) → 526_0_main_Load(EOS(STATIC_526), o103, 1) | =(matching1, 1)
R rules:

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.


P rules:
530_0_main_NULL(EOS(STATIC_530), java.lang.Object(x0), 1, java.lang.Object(x0)) → 530_0_main_NULL(EOS(STATIC_530), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(x0))), 2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, java.lang.Object(x0))))
530_0_main_NULL(EOS(STATIC_530), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, x0)), 2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, x0))) → 530_0_main_NULL(EOS(STATIC_530), x0, 0, x0)
530_0_main_NULL(EOS(STATIC_530), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, x0)), 0, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(EOC, x0))) → 530_0_main_NULL(EOS(STATIC_530), x0, 1, x0)
R rules:

Filtered ground terms:



530_0_main_NULL(x1, x2, x3, x4) → 530_0_main_NULL(x2, x3, x4)
EOS(x1) → EOS
AlternatingGrowReduce2.AlternatingGrowReduce2(x1, x2) → AlternatingGrowReduce2.AlternatingGrowReduce2(x2)

Filtered duplicate args:



530_0_main_NULL(x1, x2, x3) → 530_0_main_NULL(x2, x3)

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.


P rules:
530_0_main_NULL(1, java.lang.Object(x0)) → 530_0_main_NULL(2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0))))
530_0_main_NULL(2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0))) → 530_0_main_NULL(0, x0)
530_0_main_NULL(0, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0))) → 530_0_main_NULL(1, x0)
R rules:

Finished conversion. Obtained 3 rules for P and 0 rules for R. System has no predefined symbols.


P rules:
530_0_MAIN_NULL(1, java.lang.Object(x0)) → 530_0_MAIN_NULL(2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0))))
530_0_MAIN_NULL(2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0))) → 530_0_MAIN_NULL(0, x0)
530_0_MAIN_NULL(0, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0))) → 530_0_MAIN_NULL(1, x0)
R rules:

(18) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph contains the following rules and edges:
(0): 530_0_MAIN_NULL(1, java.lang.Object(x0[0])) → 530_0_MAIN_NULL(2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0[0]))))
(1): 530_0_MAIN_NULL(2, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[1]))) → 530_0_MAIN_NULL(0, x0[1])
(2): 530_0_MAIN_NULL(0, java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[2]))) → 530_0_MAIN_NULL(1, x0[2])

(0) -> (0), if (2 →* 1∧java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0[0]))) →* java.lang.Object(x0[0]'))


(0) -> (1), if java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0[0]))) →* java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[1]))


(0) -> (2), if (2 →* 0∧java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0[0]))) →* java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[2])))


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


(1) -> (1), if (0 →* 2∧x0[1] →* java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[1]')))


(1) -> (2), if x0[1] →* java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[2]))


(2) -> (0), if x0[2] →* java.lang.Object(x0[0])


(2) -> (1), if (1 →* 2∧x0[2] →* java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[1])))


(2) -> (2), if (1 →* 0∧x0[2] →* java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[2]')))



The set Q is empty.

(19) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(20) Obligation:

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

530_0_MAIN_NULL(pos(s(01)), java.lang.Object(x0[0])) → 530_0_MAIN_NULL(pos(s(s(01))), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0[0]))))
530_0_MAIN_NULL(pos(s(s(01))), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[1]))) → 530_0_MAIN_NULL(pos(01), x0[1])
530_0_MAIN_NULL(pos(01), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[2]))) → 530_0_MAIN_NULL(pos(s(01)), x0[2])

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


530_0_MAIN_NULL(pos(s(s(01))), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[1]))) → 530_0_MAIN_NULL(pos(01), x0[1])
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

POL(530_0_MAIN_NULL(x1, x2)) = 0 +
[1,1]
·x1 +
[1,0]
·x2

POL(pos(x1)) =
/1\
\1/
 +
/01\
\00/
·x1

POL(s(x1)) =
/0\
\1/
 +
/00\
\10/
·x1

POL(01) =
/1\
\1/

POL(java.lang.Object(x1)) =
/0\
\0/
 +
/01\
\00/
·x1

POL(AlternatingGrowReduce2.AlternatingGrowReduce2(x1)) =
/0\
\1/
 +
/00\
\10/
·x1

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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

530_0_MAIN_NULL(pos(s(01)), java.lang.Object(x0[0])) → 530_0_MAIN_NULL(pos(s(s(01))), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(java.lang.Object(x0[0]))))
530_0_MAIN_NULL(pos(01), java.lang.Object(AlternatingGrowReduce2.AlternatingGrowReduce2(x0[2]))) → 530_0_MAIN_NULL(pos(s(01)), x0[2])

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

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(24) TRUE