### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Double2
`/** * A recursive loop. * * All calls terminate. * * Julia + BinTerm prove that all calls terminate. * * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A> */public class Double2 {    private static void test(int n) {	for (int i = n - 1; i >= 0; 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:
Double2.main([Ljava/lang/String;)V: Graph of 31 nodes with 0 SCCs.

Double2.test(I)V: Graph of 18 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: Double2.test(I)V
SCC calls the following helper methods: Double2.test(I)V
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 16 rules for P and 2 rules for R.

P rules:
205_0_test_ConstantStackPush(EOS(STATIC_205), i14) → 206_0_test_IntArithmetic(EOS(STATIC_206), i14, 1)
206_0_test_IntArithmetic(EOS(STATIC_206), i14, matching1) → 208_0_test_Store(EOS(STATIC_208), -(i14, 1)) | =(matching1, 1)
210_0_test_Load(EOS(STATIC_210), i15) → 211_0_test_LT(EOS(STATIC_211), i15, i15)
211_0_test_LT(EOS(STATIC_211), i19, i19) → 214_0_test_LT(EOS(STATIC_214), i19, i19)
214_0_test_LT(EOS(STATIC_214), i19, i19) → 217_0_test_Load(EOS(STATIC_217), i19) | >=(i19, 0)
217_0_test_Load(EOS(STATIC_217), i19) → 221_0_test_InvokeMethod(EOS(STATIC_221), i19, i19)
221_0_test_InvokeMethod(EOS(STATIC_221), i19, i19) → 226_1_test_InvokeMethod(226_0_test_Load(EOS(STATIC_226), i19), i19, i19)
226_1_test_InvokeMethod(216_0_test_Return(EOS(STATIC_216)), i22, i22) → 238_0_test_Return(EOS(STATIC_238), i22, i22)
238_0_test_Return(EOS(STATIC_238), i22, i22) → 240_0_test_Inc(EOS(STATIC_240), i22)
240_0_test_Inc(EOS(STATIC_240), i22) → 242_0_test_JMP(EOS(STATIC_242), +(i22, -1)) | >=(i22, 0)
R rules:
211_0_test_LT(EOS(STATIC_211), i18, i18) → 213_0_test_LT(EOS(STATIC_213), i18, i18)
213_0_test_LT(EOS(STATIC_213), i18, i18) → 216_0_test_Return(EOS(STATIC_216)) | <(i18, 0)

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

P rules:
205_0_test_ConstantStackPush(EOS(STATIC_205), x0) → 226_1_test_InvokeMethod(205_0_test_ConstantStackPush(EOS(STATIC_205), -(x0, 1)), -(x0, 1), -(x0, 1)) | >(+(x0, 1), 1)
226_1_test_InvokeMethod(216_0_test_Return(EOS(STATIC_216)), x0, x0) → 226_1_test_InvokeMethod(205_0_test_ConstantStackPush(EOS(STATIC_205), +(x0, -1)), +(x0, -1), +(x0, -1)) | >(+(x0, 1), 1)
R rules:

Filtered ground terms:

205_0_test_ConstantStackPush(x1, x2) → 205_0_test_ConstantStackPush(x2)
Cond_226_1_test_InvokeMethod(x1, x2, x3, x4) → Cond_226_1_test_InvokeMethod(x1, x3, x4)
216_0_test_Return(x1) → 216_0_test_Return
Cond_205_0_test_ConstantStackPush(x1, x2, x3) → Cond_205_0_test_ConstantStackPush(x1, x3)

Filtered duplicate args:

226_1_test_InvokeMethod(x1, x2, x3) → 226_1_test_InvokeMethod(x1, x3)
Cond_226_1_test_InvokeMethod(x1, x2, x3) → Cond_226_1_test_InvokeMethod(x1, x3)

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

P rules:
205_0_test_ConstantStackPush(x0) → 226_1_test_InvokeMethod(205_0_test_ConstantStackPush(-(x0, 1)), -(x0, 1)) | >(x0, 0)
226_1_test_InvokeMethod(216_0_test_Return, x0) → 226_1_test_InvokeMethod(205_0_test_ConstantStackPush(+(x0, -1)), +(x0, -1)) | >(x0, 0)
R rules:

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

P rules:
205_0_TEST_CONSTANTSTACKPUSH(x0) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0, 0), x0)
COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0, 1)), -(x0, 1))
COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0) → 205_0_TEST_CONSTANTSTACKPUSH(-(x0, 1))
226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0) → COND_226_1_TEST_INVOKEMETHOD(>(x0, 0), 216_0_test_Return, x0)
COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0, -1)), +(x0, -1))
COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0) → 205_0_TEST_CONSTANTSTACKPUSH(+(x0, -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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 205_0_TEST_CONSTANTSTACKPUSH(x0[0]) → COND_205_0_TEST_CONSTANTSTACKPUSH(x0[0] > 0, x0[0])
(1): COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(x0[1] - 1), x0[1] - 1)
(2): COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2]) → 205_0_TEST_CONSTANTSTACKPUSH(x0[2] - 1)
(3): 226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3]) → COND_226_1_TEST_INVOKEMETHOD(x0[3] > 0, 216_0_test_Return, x0[3])
(4): COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(x0[4] + -1), x0[4] + -1)
(5): COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5]) → 205_0_TEST_CONSTANTSTACKPUSH(x0[5] + -1)

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

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

(1) -> (3), if (205_0_test_ConstantStackPush(x0[1] - 1) →* 216_0_test_Returnx0[1] - 1* x0[3])

(2) -> (0), if (x0[2] - 1* x0[0])

(3) -> (4), if (x0[3] > 0x0[3]* x0[4])

(3) -> (5), if (x0[3] > 0x0[3]* x0[5])

(4) -> (3), if (205_0_test_ConstantStackPush(x0[4] + -1) →* 216_0_test_Returnx0[4] + -1* x0[3])

(5) -> (0), if (x0[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: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@79bf3519 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 205_0_TEST_CONSTANTSTACKPUSH(x0) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0, 0), x0) the following chains were created:
• We consider the chain 205_0_TEST_CONSTANTSTACKPUSH(x0[0]) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0]), COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])∧(UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])∧(UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(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_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 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_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• We consider the chain 205_0_TEST_CONSTANTSTACKPUSH(x0[0]) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0]), COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2]) → 205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1)) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])∧(UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥))

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

(8)    (>(x0[0], 0)=TRUE205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧205_0_TEST_CONSTANTSTACKPUSH(x0[0])≥COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])∧(UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(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_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 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_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0, 1)), -(x0, 1)) the following chains were created:
• We consider the chain COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:

(13)    (COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1])≥226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(14)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_16] = 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(15)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_16] = 0∧[(-1)bso_17] ≥ 0)

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

(16)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_16] = 0∧[(-1)bso_17] ≥ 0)

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

(17)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_16] = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0) → 205_0_TEST_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
• We consider the chain COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2]) → 205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1)) which results in the following constraint:

(18)    (COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2])≥NonInfC∧COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2])≥205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))∧(UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥))

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

(19)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(20)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(21)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(22)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_18] = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

For Pair 226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0) → COND_226_1_TEST_INVOKEMETHOD(>(x0, 0), 216_0_test_Return, x0) the following chains were created:
• We consider the chain 226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3]) → COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3]), COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1)) which results in the following constraint:

(23)    (>(x0[3], 0)=TRUEx0[3]=x0[4]226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥NonInfC∧226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])∧(UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥))

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

(24)    (>(x0[3], 0)=TRUE226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥NonInfC∧226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])∧(UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥))

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

(25)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(26)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(27)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(28)    (x0[3] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

• We consider the chain 226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3]) → COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3]), COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5]) → 205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1)) which results in the following constraint:

(29)    (>(x0[3], 0)=TRUEx0[3]=x0[5]226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥NonInfC∧226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])∧(UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥))

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

(30)    (>(x0[3], 0)=TRUE226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥NonInfC∧226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3])≥COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])∧(UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥))

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

(31)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(32)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(33)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(34)    (x0[3] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

For Pair COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0, -1)), +(x0, -1)) the following chains were created:
• We consider the chain COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1)) which results in the following constraint:

(35)    (COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4])≥NonInfC∧COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4])≥226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))∧(UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))), ≥))

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

(36)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))), ≥)∧[bni_22] = 0∧[(-1)bso_23] ≥ 0)

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

(37)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))), ≥)∧[bni_22] = 0∧[(-1)bso_23] ≥ 0)

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

(38)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))), ≥)∧[bni_22] = 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(39)    ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0) → 205_0_TEST_CONSTANTSTACKPUSH(+(x0, -1)) the following chains were created:
• We consider the chain COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5]) → 205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1)) which results in the following constraint:

(40)    (COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5])≥NonInfC∧COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5])≥205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))∧(UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))), ≥))

We simplified constraint (40) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(41)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (41) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(42)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(43)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (43) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(44)    ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))), ≥)∧[bni_24] = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 205_0_TEST_CONSTANTSTACKPUSH(x0) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0, 1)), -(x0, 1))
• ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[bni_16] = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

• COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0) → 205_0_TEST_CONSTANTSTACKPUSH(-(x0, 1))
• ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[bni_18] = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

• 226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0) → COND_226_1_TEST_INVOKEMETHOD(>(x0, 0), 216_0_test_Return, x0)
• (x0[3] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)
• (x0[3] ≥ 0 ⇒ (UIncreasing(COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])), ≥)∧[(3)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

• COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0, -1)), +(x0, -1))
• ((UIncreasing(226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

• COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0) → 205_0_TEST_CONSTANTSTACKPUSH(+(x0, -1))
• ((UIncreasing(205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))), ≥)∧[bni_24] = 0∧0 = 0∧[1 + (-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[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(205_0_TEST_CONSTANTSTACKPUSH(x1)) = [1] + x1
POL(COND_205_0_TEST_CONSTANTSTACKPUSH(x1, x2)) = [1] + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(226_1_TEST_INVOKEMETHOD(x1, x2)) = [2] + x2
POL(205_0_test_ConstantStackPush(x1)) = x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(216_0_test_Return) = [-1]
POL(COND_226_1_TEST_INVOKEMETHOD(x1, x2, x3)) = [1] + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2]) → 205_0_TEST_CONSTANTSTACKPUSH(-(x0[2], 1))
226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3]) → COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])
COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5]) → 205_0_TEST_CONSTANTSTACKPUSH(+(x0[5], -1))

The following pairs are in Pbound:

205_0_TEST_CONSTANTSTACKPUSH(x0[0]) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])
226_1_TEST_INVOKEMETHOD(216_0_test_Return, x0[3]) → COND_226_1_TEST_INVOKEMETHOD(>(x0[3], 0), 216_0_test_Return, x0[3])

The following pairs are in P:

205_0_TEST_CONSTANTSTACKPUSH(x0[0]) → COND_205_0_TEST_CONSTANTSTACKPUSH(>(x0[0], 0), x0[0])
COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(-(x0[1], 1)), -(x0[1], 1))
COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(+(x0[4], -1)), +(x0[4], -1))

There are no usable rules.

### (9) 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): 205_0_TEST_CONSTANTSTACKPUSH(x0[0]) → COND_205_0_TEST_CONSTANTSTACKPUSH(x0[0] > 0, x0[0])
(1): COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(x0[1] - 1), x0[1] - 1)
(4): COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(x0[4] + -1), x0[4] + -1)

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

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[1]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(x0[1] - 1), x0[1] - 1)
(2): COND_205_0_TEST_CONSTANTSTACKPUSH(TRUE, x0[2]) → 205_0_TEST_CONSTANTSTACKPUSH(x0[2] - 1)
(4): COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[4]) → 226_1_TEST_INVOKEMETHOD(205_0_test_ConstantStackPush(x0[4] + -1), x0[4] + -1)
(5): COND_226_1_TEST_INVOKEMETHOD(TRUE, 216_0_test_Return, x0[5]) → 205_0_TEST_CONSTANTSTACKPUSH(x0[5] + -1)

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.