### (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:
302_0_test_Store(EOS(STATIC_302), i23, matching1) → 303_0_test_Load(EOS(STATIC_303), i23, 0) | =(matching1, 0)
469_0_test_Load(EOS(STATIC_469), i29, i50, i50) → 471_0_test_GE(EOS(STATIC_471), i29, i50, i50, i29)
471_0_test_GE(EOS(STATIC_471), i29, i50, i50, i29) → 474_0_test_GE(EOS(STATIC_474), i29, i50, i50, i29)
474_0_test_GE(EOS(STATIC_474), i29, i50, i50, i29) → 478_0_test_Load(EOS(STATIC_478), i29, i50) | <(i50, i29)
478_0_test_Load(EOS(STATIC_478), i29, i50) → 481_0_test_InvokeMethod(EOS(STATIC_481), i29, i50, i50)
481_0_test_InvokeMethod(EOS(STATIC_481), i29, i50, i50) → 486_1_test_InvokeMethod(486_0_test_ConstantStackPush(EOS(STATIC_486), i50), i29, i50, i50)
486_0_test_ConstantStackPush(EOS(STATIC_486), i50) → 491_0_test_ConstantStackPush(EOS(STATIC_491), i50)
486_1_test_InvokeMethod(476_0_test_Return(EOS(STATIC_476)), i29, i54, i54) → 497_0_test_Return(EOS(STATIC_497), i29, i54, i54)
491_0_test_ConstantStackPush(EOS(STATIC_491), i50) → 300_0_test_ConstantStackPush(EOS(STATIC_300), i50)
300_0_test_ConstantStackPush(EOS(STATIC_300), i23) → 302_0_test_Store(EOS(STATIC_302), i23, 0)
497_0_test_Return(EOS(STATIC_497), i29, i54, i54) → 499_0_test_Inc(EOS(STATIC_499), i29, i54)
499_0_test_Inc(EOS(STATIC_499), i29, i54) → 501_0_test_JMP(EOS(STATIC_501), i29, +(i54, 1)) | >=(i54, 0)
501_0_test_JMP(EOS(STATIC_501), i29, i55) → 504_0_test_Load(EOS(STATIC_504), i29, i55)
R rules:
471_0_test_GE(EOS(STATIC_471), i29, i50, i50, i29) → 473_0_test_GE(EOS(STATIC_473), i29, i50, i50, i29)
473_0_test_GE(EOS(STATIC_473), i29, i50, i50, i29) → 476_0_test_Return(EOS(STATIC_476)) | >=(i50, i29)

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

P rules:
302_0_test_Store(EOS(STATIC_302), x0, 0) → 486_1_test_InvokeMethod(302_0_test_Store(EOS(STATIC_302), 0, 0), x0, 0, 0) | >(x0, 0)
486_1_test_InvokeMethod(476_0_test_Return(EOS(STATIC_476)), x0, x1, x1) → 486_1_test_InvokeMethod(302_0_test_Store(EOS(STATIC_302), +(x1, 1), 0), x0, +(x1, 1), +(x1, 1)) | &&(>(+(x1, 1), 0), >(x0, +(x1, 1)))
R rules:

Filtered ground terms:

302_0_test_Store(x1, x2, x3) → 302_0_test_Store(x2)
Cond_486_1_test_InvokeMethod(x1, x2, x3, x4, x5) → Cond_486_1_test_InvokeMethod(x1, x3, x4, x5)
476_0_test_Return(x1) → 476_0_test_Return
Cond_302_0_test_Store(x1, x2, x3, x4) → Cond_302_0_test_Store(x1, x3)

Filtered duplicate args:

486_1_test_InvokeMethod(x1, x2, x3, x4) → 486_1_test_InvokeMethod(x1, x2, x4)
Cond_486_1_test_InvokeMethod(x1, x2, x3, x4) → Cond_486_1_test_InvokeMethod(x1, x2, x4)

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

P rules:
302_0_test_Store(x0) → 486_1_test_InvokeMethod(302_0_test_Store(0), x0, 0) | >(x0, 0)
486_1_test_InvokeMethod(476_0_test_Return, x0, x1) → 486_1_test_InvokeMethod(302_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:
302_0_TEST_STORE(x0) → COND_302_0_TEST_STORE(>(x0, 0), x0)
COND_302_0_TEST_STORE(TRUE, x0) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0, 0)
COND_302_0_TEST_STORE(TRUE, x0) → 302_0_TEST_STORE(0)
486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0, x1) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1, -1), >(x0, +(x1, 1))), 476_0_test_Return, x0, x1)
COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0, x1) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1, 1)), x0, +(x1, 1))
COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0, x1) → 302_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): 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(x0[0] > 0, x0[0])
(1): COND_302_0_TEST_STORE(TRUE, x0[1]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0)
(2): COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
(3): 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(x1[3] > -1 && x0[3] > x1[3] + 1, 476_0_test_Return, x0[3], x1[3])
(4): COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(x1[4] + 1), x0[4], x1[4] + 1)
(5): COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5]) → 302_0_TEST_STORE(x1[5] + 1)

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

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

(1) -> (3), if (302_0_test_Store(0) →* 476_0_test_Returnx0[1]* x0[3]0* x1[3])

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

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

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

(4) -> (3), if (302_0_test_Store(x1[4] + 1) →* 476_0_test_Returnx0[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@634ae081 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 302_0_TEST_STORE(x0) → COND_302_0_TEST_STORE(>(x0, 0), x0) the following chains were created:
• We consider the chain 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_302_0_TEST_STORE(TRUE, x0[1]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]302_0_TEST_STORE(x0[0])≥NonInfC∧302_0_TEST_STORE(x0[0])≥COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_302_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)=TRUE302_0_TEST_STORE(x0[0])≥NonInfC∧302_0_TEST_STORE(x0[0])≥COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_302_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_302_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_302_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_302_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_302_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 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]302_0_TEST_STORE(x0[0])≥NonInfC∧302_0_TEST_STORE(x0[0])≥COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_302_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)=TRUE302_0_TEST_STORE(x0[0])≥NonInfC∧302_0_TEST_STORE(x0[0])≥COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_302_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_302_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_302_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_302_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_302_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_302_0_TEST_STORE(TRUE, x0) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0, 0) the following chains were created:
• We consider the chain 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_302_0_TEST_STORE(TRUE, x0[1]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0), 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3]) which results in the following constraint:

(13)    (>(x0[0], 0)=TRUEx0[0]=x0[1]302_0_test_Store(0)=476_0_test_Returnx0[1]=x0[3]0=x1[3]COND_302_0_TEST_STORE(TRUE, x0[1])≥NonInfC∧COND_302_0_TEST_STORE(TRUE, x0[1])≥486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0)∧(UIncreasing(486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0)), ≥))

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

For Pair COND_302_0_TEST_STORE(TRUE, x0) → 302_0_TEST_STORE(0) the following chains were created:
• We consider the chain 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0), 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

(14)    (>(x0[0], 0)=TRUEx0[0]=x0[2]0=x0[0]1COND_302_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_302_0_TEST_STORE(TRUE, x0[2])≥302_0_TEST_STORE(0)∧(UIncreasing(302_0_TEST_STORE(0)), ≥))

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

(15)    (>(x0[0], 0)=TRUECOND_302_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_302_0_TEST_STORE(TRUE, x0[0])≥302_0_TEST_STORE(0)∧(UIncreasing(302_0_TEST_STORE(0)), ≥))

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

(16)    (0 ≥ 0 ⇒ (UIncreasing(302_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(302_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(302_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(302_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0, x1) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1, -1), >(x0, +(x1, 1))), 476_0_test_Return, x0, x1) the following chains were created:
• We consider the chain 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3]), COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4]) → 486_1_TEST_INVOKEMETHOD(302_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)))=TRUEx0[3]=x0[4]x1[3]=x1[4]486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥NonInfC∧486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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)))=TRUE486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥NonInfC∧486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3]), COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5]) → 302_0_TEST_STORE(+(x1[5], 1)) which results in the following constraint:

(26)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[5]x1[3]=x1[5]486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥NonInfC∧486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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)))=TRUE486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥NonInfC∧486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3])≥COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0, x1) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1, 1)), x0, +(x1, 1)) the following chains were created:
• We consider the chain 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3]), COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)), 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3]) which results in the following constraint:

(32)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[4]x1[3]=x1[4]302_0_test_Store(+(x1[4], 1))=476_0_test_Returnx0[4]=x0[3]1+(x1[4], 1)=x1[3]1COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4])≥NonInfC∧COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4])≥486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))∧(UIncreasing(486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))), ≥))

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

For Pair COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0, x1) → 302_0_TEST_STORE(+(x1, 1)) the following chains were created:
• We consider the chain 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3]), COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5]) → 302_0_TEST_STORE(+(x1[5], 1)), 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

(33)    (&&(>(x1[3], -1), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[5]x1[3]=x1[5]+(x1[5], 1)=x0[0]COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5])≥NonInfC∧COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5])≥302_0_TEST_STORE(+(x1[5], 1))∧(UIncreasing(302_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)))=TRUECOND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[3], x1[3])≥NonInfC∧COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[3], x1[3])≥302_0_TEST_STORE(+(x1[3], 1))∧(UIncreasing(302_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(302_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(302_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(302_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(302_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.
• 302_0_TEST_STORE(x0) → COND_302_0_TEST_STORE(>(x0, 0), x0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_302_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_302_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_302_0_TEST_STORE(TRUE, x0) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0, 0)

• COND_302_0_TEST_STORE(TRUE, x0) → 302_0_TEST_STORE(0)
• (0 ≥ 0 ⇒ (UIncreasing(302_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

• 486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0, x1) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1, -1), >(x0, +(x1, 1))), 476_0_test_Return, x0, x1)
• (0 ≥ 0 ⇒ (UIncreasing(COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_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_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0, x1) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1, 1)), x0, +(x1, 1))

• COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0, x1) → 302_0_TEST_STORE(+(x1, 1))
• (0 ≥ 0 ⇒ (UIncreasing(302_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(302_0_TEST_STORE(x1)) = [-1] + x1
POL(COND_302_0_TEST_STORE(x1, x2)) = [-1]
POL(>(x1, x2)) = 0
POL(0) = 0
POL(486_1_TEST_INVOKEMETHOD(x1, x2, x3)) = [1] + x3 + x2 + [-1]x1
POL(302_0_test_Store(x1)) = 0
POL(476_0_test_Return) = 0
POL(COND_486_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_302_0_TEST_STORE(TRUE, x0[1]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0)
486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3])
COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))

The following pairs are in Pbound:

302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_302_0_TEST_STORE(TRUE, x0[1]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(0), x0[1], 0)
COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
486_1_TEST_INVOKEMETHOD(476_0_test_Return, x0[3], x1[3]) → COND_486_1_TEST_INVOKEMETHOD(&&(>(x1[3], -1), >(x0[3], +(x1[3], 1))), 476_0_test_Return, x0[3], x1[3])
COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[4], x1[4]) → 486_1_TEST_INVOKEMETHOD(302_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))
COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5]) → 302_0_TEST_STORE(+(x1[5], 1))

The following pairs are in P:

302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5]) → 302_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): 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(x0[0] > 0, x0[0])
(2): COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
(5): COND_486_1_TEST_INVOKEMETHOD(TRUE, 476_0_test_Return, x0[5], x1[5]) → 302_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] > 0x0[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_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
(0): 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(x0[0] > 0, x0[0])

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

(0) -> (2), if (x0[0] > 0x0[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@39401e69 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_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0) the following chains were created:
• We consider the chain 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0), 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[2]0=x0[0]1COND_302_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_302_0_TEST_STORE(TRUE, x0[2])≥302_0_TEST_STORE(0)∧(UIncreasing(302_0_TEST_STORE(0)), ≥))

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

(2)    (>(x0[0], 0)=TRUECOND_302_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_302_0_TEST_STORE(TRUE, x0[0])≥302_0_TEST_STORE(0)∧(UIncreasing(302_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(302_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(302_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(302_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(302_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]) the following chains were created:
• We consider the chain 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]302_0_TEST_STORE(x0[0])≥NonInfC∧302_0_TEST_STORE(x0[0])≥COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_302_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)=TRUE302_0_TEST_STORE(x0[0])≥NonInfC∧302_0_TEST_STORE(x0[0])≥COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_302_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_302_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_302_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_302_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_302_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_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(302_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)

• 302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_302_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_302_0_TEST_STORE(x1, x2)) = [2]
POL(302_0_TEST_STORE(x1)) = [2] + x1
POL(0) = 0
POL(>(x1, x2)) = 0

The following pairs are in P>:

302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in Pbound:

COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_0_TEST_STORE(0)
302_0_TEST_STORE(x0[0]) → COND_302_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in P:

COND_302_0_TEST_STORE(TRUE, x0[2]) → 302_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_302_0_TEST_STORE(TRUE, x0[2]) → 302_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.