Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 100 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 120 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 180 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 IDP
↳11 IDPNonInfProof (⇒, 50 ms)
↳12 IDP
↳13 IDependencyGraphProof (⇔, 0 ms)
↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Double 
/**
 * A recursive loop.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that the call to <tt>test()</tt> terminates.
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Double {

    private static void test(int n) {
	for (int i = 0; i < n; i++)
	    test(i);
    }

    public static void main(String[] args) {
	test(10);
    }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Double.main([Ljava/lang/String;)V: Graph of 31 nodes with 0 SCCs.

Double.test(I)V: Graph of 21 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

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

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 19 rules for P and 2 rules for R.


P rules:
265_0_test_Store(EOS(STATIC_265), i23, matching1) → 267_0_test_Load(EOS(STATIC_267), i23, 0) | =(matching1, 0)
267_0_test_Load(EOS(STATIC_267), i23, matching1) → 307_0_test_Load(EOS(STATIC_307), i23, 0) | =(matching1, 0)
307_0_test_Load(EOS(STATIC_307), i29, i30) → 348_0_test_Load(EOS(STATIC_348), i29, i30)
348_0_test_Load(EOS(STATIC_348), i29, i38) → 391_0_test_Load(EOS(STATIC_391), i29, i38)
391_0_test_Load(EOS(STATIC_391), i29, i44) → 430_0_test_Load(EOS(STATIC_430), i29, i44)
430_0_test_Load(EOS(STATIC_430), i29, i50) → 432_0_test_Load(EOS(STATIC_432), i29, i50, i50)
432_0_test_Load(EOS(STATIC_432), i29, i50, i50) → 434_0_test_GE(EOS(STATIC_434), i29, i50, i50, i29)
434_0_test_GE(EOS(STATIC_434), i29, i50, i50, i29) → 436_0_test_GE(EOS(STATIC_436), i29, i50, i50, i29)
436_0_test_GE(EOS(STATIC_436), i29, i50, i50, i29) → 439_0_test_Load(EOS(STATIC_439), i29, i50) | <(i50, i29)
439_0_test_Load(EOS(STATIC_439), i29, i50) → 442_0_test_InvokeMethod(EOS(STATIC_442), i29, i50, i50)
442_0_test_InvokeMethod(EOS(STATIC_442), i29, i50, i50) → 446_1_test_InvokeMethod(446_0_test_ConstantStackPush(EOS(STATIC_446), i50), i29, i50, i50)
446_0_test_ConstantStackPush(EOS(STATIC_446), i50) → 450_0_test_ConstantStackPush(EOS(STATIC_450), i50)
446_1_test_InvokeMethod(438_0_test_Return(EOS(STATIC_438)), i29, i54, i54) → 455_0_test_Return(EOS(STATIC_455), i29, i54, i54)
450_0_test_ConstantStackPush(EOS(STATIC_450), i50) → 263_0_test_ConstantStackPush(EOS(STATIC_263), i50)
263_0_test_ConstantStackPush(EOS(STATIC_263), i23) → 265_0_test_Store(EOS(STATIC_265), i23, 0)
455_0_test_Return(EOS(STATIC_455), i29, i54, i54) → 456_0_test_Inc(EOS(STATIC_456), i29, i54)
456_0_test_Inc(EOS(STATIC_456), i29, i54) → 458_0_test_JMP(EOS(STATIC_458), i29, +(i54, 1)) | >=(i54, 0)
458_0_test_JMP(EOS(STATIC_458), i29, i55) → 460_0_test_Load(EOS(STATIC_460), i29, i55)
460_0_test_Load(EOS(STATIC_460), i29, i55) → 430_0_test_Load(EOS(STATIC_430), i29, i55)
R rules:
434_0_test_GE(EOS(STATIC_434), i29, i50, i50, i29) → 435_0_test_GE(EOS(STATIC_435), i29, i50, i50, i29)
435_0_test_GE(EOS(STATIC_435), i29, i50, i50, i29) → 438_0_test_Return(EOS(STATIC_438)) | >=(i50, i29)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.


P rules:
265_0_test_Store(EOS(STATIC_265), x0, 0) → 446_1_test_InvokeMethod(265_0_test_Store(EOS(STATIC_265), 0, 0), x0, 0, 0) | >(x0, 0)
446_1_test_InvokeMethod(438_0_test_Return(EOS(STATIC_438)), x0, x1, x1) → 446_1_test_InvokeMethod(265_0_test_Store(EOS(STATIC_265), +(x1, 1), 0), x0, +(x1, 1), +(x1, 1)) | &&(>(+(x1, 1), 0), >(x0, +(x1, 1)))
R rules:

Filtered ground terms:



265_0_test_Store(x1, x2, x3) → 265_0_test_Store(x2)
Cond_446_1_test_InvokeMethod(x1, x2, x3, x4, x5) → Cond_446_1_test_InvokeMethod(x1, x3, x4, x5)
438_0_test_Return(x1) → 438_0_test_Return
Cond_265_0_test_Store(x1, x2, x3, x4) → Cond_265_0_test_Store(x1, x3)

Filtered duplicate args:



446_1_test_InvokeMethod(x1, x2, x3, x4) → 446_1_test_InvokeMethod(x1, x2, x4)
Cond_446_1_test_InvokeMethod(x1, x2, x3, x4) → Cond_446_1_test_InvokeMethod(x1, x2, x4)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.


P rules:
265_0_test_Store(x0) → 446_1_test_InvokeMethod(265_0_test_Store(0), x0, 0) | >(x0, 0)
446_1_test_InvokeMethod(438_0_test_Return, x0, x1) → 446_1_test_InvokeMethod(265_0_test_Store(+(x1, 1)), x0, +(x1, 1)) | &&(>(x1, -1), >(x0, +(x1, 1)))
R rules:

Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.


P rules:
265_0_TEST_STORE(x0) → COND_265_0_TEST_STORE(>(x0, 0), x0)
COND_265_0_TEST_STORE(TRUE, x0) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0, 0)
COND_265_0_TEST_STORE(TRUE, x0) → 265_0_TEST_STORE(0)
446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0, x1) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1, -1), >(x0, +(x1, 1))), 438_0_test_Return, x0, x1)
COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0, x1) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1, 1)), x0, +(x1, 1))
COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0, x1) → 265_0_TEST_STORE(+(x1, 1))
R rules:

(6) 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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(x0[0] > 0, x0[0])
(1): COND_265_0_TEST_STORE(TRUE, x0[1]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0)
(2): COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
(3): 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(x1[3] > -1 && x0[3] > x1[3] + 1, 438_0_test_Return, x0[3], x1[3])
(4): COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(x1[4] + 1), x0[4], x1[4] + 1)
(5): COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5]) → 265_0_TEST_STORE(x1[5] + 1)

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


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


(1) -> (3), if (265_0_test_Store(0) →* 438_0_test_Return∧x0[1] →* x0[3]∧0 →* x1[3])


(2) -> (0), if (0 →* x0[0])


(3) -> (4), if (x1[3] > -1 && x0[3] > x1[3] + 1 ∧x0[3] →* x0[4]∧x1[3] →* x1[4])


(3) -> (5), if (x1[3] > -1 && x0[3] > x1[3] + 1 ∧x0[3] →* x0[5]∧x1[3] →* x1[5])


(4) -> (3), if (265_0_test_Store(x1[4] + 1) →* 438_0_test_Return∧x0[4] →* x0[3]∧x1[4] + 1 →* x1[3])


(5) -> (0), if (x1[5] + 1 →* x0[0])



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@2c5d7ace Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair 265_0_TEST_STORE(x0) → COND_265_0_TEST_STORE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_265_0_TEST_STORE(TRUE, x0[1]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 265_0_TEST_STORE(x0[0])≥NonInfC∧265_0_TEST_STORE(x0[0])≥COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ 265_0_TEST_STORE(x0[0])≥NonInfC∧265_0_TEST_STORE(x0[0])≥COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)



  • We consider the chain 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 265_0_TEST_STORE(x0[0])≥NonInfC∧265_0_TEST_STORE(x0[0])≥COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (8)    (>(x0[0], 0)=TRUE ⇒ 265_0_TEST_STORE(x0[0])≥NonInfC∧265_0_TEST_STORE(x0[0])≥COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (9)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (10)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (11)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (12)    (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_265_0_TEST_STORE(TRUE, x0) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0, 0) the following chains were created:
  • We consider the chain 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_265_0_TEST_STORE(TRUE, x0[1]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0), 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3]) which results in the following constraint:

    (13)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧265_0_test_Store(0)=438_0_test_Return∧x0[1]=x0[3]∧0=x1[3] ⇒ COND_265_0_TEST_STORE(TRUE, x0[1])≥NonInfC∧COND_265_0_TEST_STORE(TRUE, x0[1])≥446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0)∧(UIncreasing(446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0)), ≥))



    We solved constraint (13) using rules (I), (II).




For Pair COND_265_0_TEST_STORE(TRUE, x0) → 265_0_TEST_STORE(0) the following chains were created:
  • We consider the chain 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0), 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (14)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2]∧0=x0[0]1 ⇒ COND_265_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_265_0_TEST_STORE(TRUE, x0[2])≥265_0_TEST_STORE(0)∧(UIncreasing(265_0_TEST_STORE(0)), ≥))



    We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

    (15)    (>(x0[0], 0)=TRUE ⇒ COND_265_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_265_0_TEST_STORE(TRUE, x0[0])≥265_0_TEST_STORE(0)∧(UIncreasing(265_0_TEST_STORE(0)), ≥))



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

    (16)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (17)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (18)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (19)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0, x1) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1, -1), >(x0, +(x1, 1))), 438_0_test_Return, x0, x1) the following chains were created:
  • We consider the chain 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3]), COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)) which results in the following constraint:

    (20)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥NonInfC∧446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥))



    We simplified constraint (20) using rule (IV) which results in the following new constraint:

    (21)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE ⇒ 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥NonInfC∧446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥))



    We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (22)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



    We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (23)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (24)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (25)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)



  • We consider the chain 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3]), COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5]) → 265_0_TEST_STORE(+(x1[5], 1)) which results in the following constraint:

    (26)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5] ⇒ 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥NonInfC∧446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥))



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

    (27)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE ⇒ 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥NonInfC∧446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3])≥COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥))



    We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (28)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



    We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (29)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



    We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (30)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



    We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (31)    (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)







For Pair COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0, x1) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1, 1)), x0, +(x1, 1)) the following chains were created:
  • We consider the chain 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3]), COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)), 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3]) which results in the following constraint:

    (32)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧265_0_test_Store(+(x1[4], 1))=438_0_test_Return∧x0[4]=x0[3]1∧+(x1[4], 1)=x1[3]1 ⇒ COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4])≥NonInfC∧COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4])≥446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))∧(UIncreasing(446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))), ≥))



    We solved constraint (32) using rules (I), (II).




For Pair COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0, x1) → 265_0_TEST_STORE(+(x1, 1)) the following chains were created:
  • We consider the chain 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3]), COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5]) → 265_0_TEST_STORE(+(x1[5], 1)), 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (33)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5]∧+(x1[5], 1)=x0[0] ⇒ COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5])≥NonInfC∧COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5])≥265_0_TEST_STORE(+(x1[5], 1))∧(UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥))



    We simplified constraint (33) using rules (III), (IV) which results in the following new constraint:

    (34)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUE ⇒ COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[3], x1[3])≥NonInfC∧COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[3], x1[3])≥265_0_TEST_STORE(+(x1[3], 1))∧(UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥))



    We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (35)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (36)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (37)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (37) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (38)    (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 265_0_TEST_STORE(x0) → COND_265_0_TEST_STORE(>(x0, 0), x0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_265_0_TEST_STORE(TRUE, x0) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0, 0)

  • COND_265_0_TEST_STORE(TRUE, x0) → 265_0_TEST_STORE(0)
    • (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

  • 446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0, x1) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1, -1), >(x0, +(x1, 1))), 438_0_test_Return, x0, x1)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)

  • COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0, x1) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1, 1)), x0, +(x1, 1))

  • COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0, x1) → 265_0_TEST_STORE(+(x1, 1))
    • (0 ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(+(x1[5], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 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(265_0_TEST_STORE(x1)) = [-1] + x1   
POL(COND_265_0_TEST_STORE(x1, x2)) = [-1]   
POL(>(x1, x2)) = 0   
POL(0) = 0   
POL(446_1_TEST_INVOKEMETHOD(x1, x2, x3)) = [1] + x3 + x2 + [-1]x1   
POL(265_0_test_Store(x1)) = 0   
POL(438_0_test_Return) = 0   
POL(COND_446_1_TEST_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [-1]x2   
POL(&&(x1, x2)) = 0   
POL(-1) = 0   
POL(+(x1, x2)) = 0   
POL(1) = 0   

The following pairs are in P>:

COND_265_0_TEST_STORE(TRUE, x0[1]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0)
446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])
COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))

The following pairs are in Pbound:

265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_265_0_TEST_STORE(TRUE, x0[1]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(0), x0[1], 0)
COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
446_1_TEST_INVOKEMETHOD(438_0_test_Return, x0[3], x1[3]) → COND_446_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 438_0_test_Return, x0[3], x1[3])
COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[4], x1[4]) → 446_1_TEST_INVOKEMETHOD(265_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))
COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5]) → 265_0_TEST_STORE(+(x1[5], 1))

The following pairs are in P≥:

265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5]) → 265_0_TEST_STORE(+(x1[5], 1))

There are no usable rules.

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(x0[0] > 0, x0[0])
(2): COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
(5): COND_446_1_TEST_INVOKEMETHOD(TRUE, 438_0_test_Return, x0[5], x1[5]) → 265_0_TEST_STORE(x1[5] + 1)

(2) -> (0), if (0 →* x0[0])


(5) -> (0), if (x1[5] + 1 →* x0[0])


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



The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(2): COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
(0): 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(x0[0] > 0, x0[0])

(2) -> (0), if (0 →* x0[0])


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



The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@2d298123 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0) the following chains were created:
  • We consider the chain 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0), 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2]∧0=x0[0]1 ⇒ COND_265_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_265_0_TEST_STORE(TRUE, x0[2])≥265_0_TEST_STORE(0)∧(UIncreasing(265_0_TEST_STORE(0)), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ COND_265_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_265_0_TEST_STORE(TRUE, x0[0])≥265_0_TEST_STORE(0)∧(UIncreasing(265_0_TEST_STORE(0)), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)







For Pair 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]) the following chains were created:
  • We consider the chain 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 265_0_TEST_STORE(x0[0])≥NonInfC∧265_0_TEST_STORE(x0[0])≥COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (8)    (>(x0[0], 0)=TRUE ⇒ 265_0_TEST_STORE(x0[0])≥NonInfC∧265_0_TEST_STORE(x0[0])≥COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] + x0[0] ≥ 0)



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

    (10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] + x0[0] ≥ 0)



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

    (11)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] + x0[0] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] + x0[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(265_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)

  • 265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] + x0[0] ≥ 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(COND_265_0_TEST_STORE(x1, x2)) = [2]   
POL(265_0_TEST_STORE(x1)) = [2] + x1   
POL(0) = 0   
POL(>(x1, x2)) = 0   

The following pairs are in P>:

265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in Pbound:

COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)
265_0_TEST_STORE(x0[0]) → COND_265_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)

There are no usable rules.

(12) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph contains the following rules and edges:
(2): COND_265_0_TEST_STORE(TRUE, x0[2]) → 265_0_TEST_STORE(0)


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE