(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: CAppE
public class CAppE {
CAppE next;

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

public static void cappE(int j) {
CAppE a = new CAppE();
if (j > 0) {
a.appE(j);
while (a.next == null) {}
}
}

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

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

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

public CAppE(CAppE 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:
CAppE.main([Ljava/lang/String;)V: Graph of 138 nodes with 0 SCCs.

CAppE.createList()LCAppE;: Graph of 91 nodes with 1 SCC.

CAppE.appE(I)V: Graph of 35 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: CAppE.appE(I)V
SCC calls the following helper methods: CAppE.appE(I)V
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 24 rules for P and 9 rules for R.


P rules:
668_0_appE_FieldAccess(EOS(STATIC_668), i76) → 670_0_appE_NONNULL(EOS(STATIC_670), i76)
670_0_appE_NONNULL(EOS(STATIC_670), i76) → 671_0_appE_Load(EOS(STATIC_671), i76)
671_0_appE_Load(EOS(STATIC_671), i76) → 672_0_appE_GT(EOS(STATIC_672), i76, i76)
672_0_appE_GT(EOS(STATIC_672), i83, i83) → 675_0_appE_GT(EOS(STATIC_675), i83, i83)
675_0_appE_GT(EOS(STATIC_675), i83, i83) → 678_0_appE_Load(EOS(STATIC_678), i83) | >(i83, 0)
678_0_appE_Load(EOS(STATIC_678), i83) → 682_0_appE_New(EOS(STATIC_682), i83)
682_0_appE_New(EOS(STATIC_682), i83) → 687_0_appE_Duplicate(EOS(STATIC_687), i83)
687_0_appE_Duplicate(EOS(STATIC_687), i83) → 691_0_appE_InvokeMethod(EOS(STATIC_691), i83)
691_0_appE_InvokeMethod(EOS(STATIC_691), i83) → 692_0_<init>_Load(EOS(STATIC_692), i83)
692_0_<init>_Load(EOS(STATIC_692), i83) → 695_0_<init>_InvokeMethod(EOS(STATIC_695), i83)
695_0_<init>_InvokeMethod(EOS(STATIC_695), i83) → 697_0_<init>_Load(EOS(STATIC_697), i83)
697_0_<init>_Load(EOS(STATIC_697), i83) → 699_0_<init>_ConstantStackPush(EOS(STATIC_699), i83)
699_0_<init>_ConstantStackPush(EOS(STATIC_699), i83) → 701_0_<init>_FieldAccess(EOS(STATIC_701), i83)
701_0_<init>_FieldAccess(EOS(STATIC_701), i83) → 702_0_<init>_Return(EOS(STATIC_702), i83)
702_0_<init>_Return(EOS(STATIC_702), i83) → 705_0_appE_FieldAccess(EOS(STATIC_705), i83)
705_0_appE_FieldAccess(EOS(STATIC_705), i83) → 708_0_appE_Inc(EOS(STATIC_708), i83)
708_0_appE_Inc(EOS(STATIC_708), i83) → 710_0_appE_Load(EOS(STATIC_710), +(i83, -1)) | >(i83, 0)
710_0_appE_Load(EOS(STATIC_710), i86) → 712_0_appE_FieldAccess(EOS(STATIC_712), i86)
712_0_appE_FieldAccess(EOS(STATIC_712), i86) → 714_0_appE_Load(EOS(STATIC_714), i86)
714_0_appE_Load(EOS(STATIC_714), i86) → 716_0_appE_InvokeMethod(EOS(STATIC_716), i86)
716_0_appE_InvokeMethod(EOS(STATIC_716), i86) → 718_1_appE_InvokeMethod(718_0_appE_Load(EOS(STATIC_718), i86), i86)
718_0_appE_Load(EOS(STATIC_718), i86) → 721_0_appE_Load(EOS(STATIC_721), i86)
721_0_appE_Load(EOS(STATIC_721), i86) → 665_0_appE_Load(EOS(STATIC_665), i86)
665_0_appE_Load(EOS(STATIC_665), i76) → 668_0_appE_FieldAccess(EOS(STATIC_668), i76)
R rules:
672_0_appE_GT(EOS(STATIC_672), matching1, matching2) → 674_0_appE_GT(EOS(STATIC_674), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
674_0_appE_GT(EOS(STATIC_674), matching1, matching2) → 677_0_appE_Return(EOS(STATIC_677), 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
718_1_appE_InvokeMethod(677_0_appE_Return(EOS(STATIC_677), matching1), matching2) → 732_0_appE_Return(EOS(STATIC_732), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
718_1_appE_InvokeMethod(734_0_appE_Return(EOS(STATIC_734)), i94) → 749_0_appE_Return(EOS(STATIC_749), i94)
718_1_appE_InvokeMethod(787_0_appE_Return(EOS(STATIC_787)), i113) → 815_0_appE_Return(EOS(STATIC_815), i113)
732_0_appE_Return(EOS(STATIC_732), matching1, matching2) → 734_0_appE_Return(EOS(STATIC_734)) | &&(=(matching1, 0), =(matching2, 0))
749_0_appE_Return(EOS(STATIC_749), i94) → 782_0_appE_Return(EOS(STATIC_782), i94)
782_0_appE_Return(EOS(STATIC_782), i107) → 787_0_appE_Return(EOS(STATIC_787))
815_0_appE_Return(EOS(STATIC_815), i113) → 782_0_appE_Return(EOS(STATIC_782), i113)

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.


P rules:
668_0_appE_FieldAccess(EOS(STATIC_668), x0) → 718_1_appE_InvokeMethod(668_0_appE_FieldAccess(EOS(STATIC_668), +(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:
718_1_appE_InvokeMethod(677_0_appE_Return(EOS(STATIC_677), 0), 0) → 734_0_appE_Return(EOS(STATIC_734))
718_1_appE_InvokeMethod(734_0_appE_Return(EOS(STATIC_734)), x0) → 787_0_appE_Return(EOS(STATIC_787))
718_1_appE_InvokeMethod(787_0_appE_Return(EOS(STATIC_787)), x0) → 787_0_appE_Return(EOS(STATIC_787))

Filtered ground terms:



668_0_appE_FieldAccess(x1, x2) → 668_0_appE_FieldAccess(x2)
Cond_668_0_appE_FieldAccess(x1, x2, x3) → Cond_668_0_appE_FieldAccess(x1, x3)
787_0_appE_Return(x1) → 787_0_appE_Return
734_0_appE_Return(x1) → 734_0_appE_Return
677_0_appE_Return(x1, x2) → 677_0_appE_Return

Combined rules. Obtained 1 conditional rules for P and 3 conditional rules for R.


P rules:
668_0_appE_FieldAccess(x0) → 718_1_appE_InvokeMethod(668_0_appE_FieldAccess(+(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 734_0_appE_Return
718_1_appE_InvokeMethod(734_0_appE_Return, x0) → 787_0_appE_Return
718_1_appE_InvokeMethod(787_0_appE_Return, x0) → 787_0_appE_Return

Performed bisimulation on rules. Used the following equivalence classes: {[677_0_appE_Return, 734_0_appE_Return, 787_0_appE_Return]=677_0_appE_Return}


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


P rules:
668_0_APPE_FIELDACCESS(x0) → COND_668_0_APPE_FIELDACCESS(>(x0, 0), x0)
COND_668_0_APPE_FIELDACCESS(TRUE, x0) → 668_0_APPE_FIELDACCESS(+(x0, -1))
R rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_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:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])
(1): COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(x0[1] + -1)

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


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



The set Q consists of the following terms:
718_1_appE_InvokeMethod(677_0_appE_Return, x0)

(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@42fcb4f 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 668_0_APPE_FIELDACCESS(x0) → COND_668_0_APPE_FIELDACCESS(>(x0, 0), x0) the following chains were created:
  • We consider the chain 668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0]), COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]668_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧668_0_APPE_FIELDACCESS(x0[0])≥COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE668_0_APPE_FIELDACCESS(x0[0])≥NonInfC∧668_0_APPE_FIELDACCESS(x0[0])≥COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])∧(UIncreasing(COND_668_0_APPE_FIELDACCESS(>(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_668_0_APPE_FIELDACCESS(>(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_668_0_APPE_FIELDACCESS(>(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_668_0_APPE_FIELDACCESS(>(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_668_0_APPE_FIELDACCESS(>(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_668_0_APPE_FIELDACCESS(TRUE, x0) → 668_0_APPE_FIELDACCESS(+(x0, -1)) the following chains were created:
  • We consider the chain COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_668_0_APPE_FIELDACCESS(TRUE, x0[1])≥NonInfC∧COND_668_0_APPE_FIELDACCESS(TRUE, x0[1])≥668_0_APPE_FIELDACCESS(+(x0[1], -1))∧(UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)



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

    (9)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 668_0_APPE_FIELDACCESS(x0) → COND_668_0_APPE_FIELDACCESS(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  • COND_668_0_APPE_FIELDACCESS(TRUE, x0) → 668_0_APPE_FIELDACCESS(+(x0, -1))
    • ((UIncreasing(668_0_APPE_FIELDACCESS(+(x0[1], -1))), ≥)∧[bni_12] = 0∧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(718_1_appE_InvokeMethod(x1, x2)) = [-1]   
POL(677_0_appE_Return) = [-1]   
POL(0) = 0   
POL(668_0_APPE_FIELDACCESS(x1)) = [2]x1   
POL(COND_668_0_APPE_FIELDACCESS(x1, x2)) = [2]x2   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(+(x0[1], -1))

The following pairs are in Pbound:

668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(>(x0[0], 0), x0[0])

The following pairs are in P:

668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(>(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


The ITRS R consists of the following rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

The integer pair graph contains the following rules and edges:
(0): 668_0_APPE_FIELDACCESS(x0[0]) → COND_668_0_APPE_FIELDACCESS(x0[0] > 0, x0[0])


The set Q consists of the following terms:
718_1_appE_InvokeMethod(677_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:

Integer


The ITRS R consists of the following rules:
718_1_appE_InvokeMethod(677_0_appE_Return, 0) → 677_0_appE_Return
718_1_appE_InvokeMethod(677_0_appE_Return, x0) → 677_0_appE_Return

The integer pair graph contains the following rules and edges:
(1): COND_668_0_APPE_FIELDACCESS(TRUE, x0[1]) → 668_0_APPE_FIELDACCESS(x0[1] + -1)


The set Q consists of the following terms:
718_1_appE_InvokeMethod(677_0_appE_Return, x0)

(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: CAppE.createList()LCAppE;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 17 rules for P and 0 rules for R.


P rules:
331_0_createList_LE(EOS(STATIC_331), i39, i39) → 335_0_createList_LE(EOS(STATIC_335), i39, i39)
335_0_createList_LE(EOS(STATIC_335), i39, i39) → 346_0_createList_New(EOS(STATIC_346), i39) | >(i39, 0)
346_0_createList_New(EOS(STATIC_346), i39) → 366_0_createList_Duplicate(EOS(STATIC_366), i39)
366_0_createList_Duplicate(EOS(STATIC_366), i39) → 377_0_createList_Load(EOS(STATIC_377), i39)
377_0_createList_Load(EOS(STATIC_377), i39) → 389_0_createList_InvokeMethod(EOS(STATIC_389), i39)
389_0_createList_InvokeMethod(EOS(STATIC_389), i39) → 403_0_<init>_Load(EOS(STATIC_403), i39)
403_0_<init>_Load(EOS(STATIC_403), i39) → 408_0_<init>_InvokeMethod(EOS(STATIC_408), i39)
408_0_<init>_InvokeMethod(EOS(STATIC_408), i39) → 412_0_<init>_Load(EOS(STATIC_412), i39)
412_0_<init>_Load(EOS(STATIC_412), i39) → 414_0_<init>_Load(EOS(STATIC_414), i39)
414_0_<init>_Load(EOS(STATIC_414), i39) → 417_0_<init>_FieldAccess(EOS(STATIC_417), i39)
417_0_<init>_FieldAccess(EOS(STATIC_417), i39) → 423_0_<init>_Return(EOS(STATIC_423), i39)
423_0_<init>_Return(EOS(STATIC_423), i39) → 428_0_createList_Store(EOS(STATIC_428), i39)
428_0_createList_Store(EOS(STATIC_428), i39) → 434_0_createList_Inc(EOS(STATIC_434), i39)
434_0_createList_Inc(EOS(STATIC_434), i39) → 438_0_createList_JMP(EOS(STATIC_438), +(i39, -1)) | >(i39, 0)
438_0_createList_JMP(EOS(STATIC_438), i52) → 445_0_createList_Load(EOS(STATIC_445), i52)
445_0_createList_Load(EOS(STATIC_445), i52) → 321_0_createList_Load(EOS(STATIC_321), i52)
321_0_createList_Load(EOS(STATIC_321), i35) → 331_0_createList_LE(EOS(STATIC_331), i35, i35)
R rules:

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


P rules:
331_0_createList_LE(EOS(STATIC_331), x0, x0) → 331_0_createList_LE(EOS(STATIC_331), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:



331_0_createList_LE(x1, x2, x3) → 331_0_createList_LE(x2, x3)
EOS(x1) → EOS
Cond_331_0_createList_LE(x1, x2, x3, x4) → Cond_331_0_createList_LE(x1, x3, x4)

Filtered duplicate args:



331_0_createList_LE(x1, x2) → 331_0_createList_LE(x2)
Cond_331_0_createList_LE(x1, x2, x3) → Cond_331_0_createList_LE(x1, x3)

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


P rules:
331_0_createList_LE(x0) → 331_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:
331_0_CREATELIST_LE(x0) → COND_331_0_CREATELIST_LE(>(x0, 0), x0)
COND_331_0_CREATELIST_LE(TRUE, x0) → 331_0_CREATELIST_LE(+(x0, -1))
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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(x0[0] > 0, x0[0])
(1): COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(x0[1] + -1)

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


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



The set Q is empty.

(19) 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@42fcb4f 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 331_0_CREATELIST_LE(x0) → COND_331_0_CREATELIST_LE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0]), COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]331_0_CREATELIST_LE(x0[0])≥NonInfC∧331_0_CREATELIST_LE(x0[0])≥COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_331_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)=TRUE331_0_CREATELIST_LE(x0[0])≥NonInfC∧331_0_CREATELIST_LE(x0[0])≥COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_331_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_331_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_331_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_331_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_331_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_331_0_CREATELIST_LE(TRUE, x0) → 331_0_CREATELIST_LE(+(x0, -1)) the following chains were created:
  • We consider the chain COND_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(+(x0[1], -1)) which results in the following constraint:

    (7)    (COND_331_0_CREATELIST_LE(TRUE, x0[1])≥NonInfC∧COND_331_0_CREATELIST_LE(TRUE, x0[1])≥331_0_CREATELIST_LE(+(x0[1], -1))∧(UIncreasing(331_0_CREATELIST_LE(+(x0[1], -1))), ≥))



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

    (8)    ((UIncreasing(331_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(331_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(331_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(331_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.
  • 331_0_CREATELIST_LE(x0) → COND_331_0_CREATELIST_LE(>(x0, 0), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_331_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_331_0_CREATELIST_LE(TRUE, x0) → 331_0_CREATELIST_LE(+(x0, -1))
    • ((UIncreasing(331_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(331_0_CREATELIST_LE(x1)) = [2]x1   
POL(COND_331_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_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P:

331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(20) Complex Obligation (AND)

(21) 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): 331_0_CREATELIST_LE(x0[0]) → COND_331_0_CREATELIST_LE(x0[0] > 0, x0[0])


The set Q is empty.

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(23) TRUE

(24) 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_331_0_CREATELIST_LE(TRUE, x0[1]) → 331_0_CREATELIST_LE(x0[1] + -1)


The set Q is empty.

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE