### (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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Double.main([Ljava/lang/String;)V: Graph of 12 nodes with 0 SCCs.

Double.test(I)V: Graph of 20 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 18 rules for P and 2 rules for R.

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

Filtered ground terms:

154_0_test_Store(x1, x2, x3) → 154_0_test_Store(x2)
Cond_264_1_test_InvokeMethod(x1, x2, x3, x4, x5) → Cond_264_1_test_InvokeMethod(x1, x3, x4, x5)
254_0_test_Return(x1) → 254_0_test_Return
Cond_154_0_test_Store(x1, x2, x3, x4) → Cond_154_0_test_Store(x1, x3)

Filtered duplicate args:

264_1_test_InvokeMethod(x1, x2, x3, x4) → 264_1_test_InvokeMethod(x1, x2, x4)
Cond_264_1_test_InvokeMethod(x1, x2, x3, x4) → Cond_264_1_test_InvokeMethod(x1, x2, x4)

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

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

### (4) 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): 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(x0[0] > 0, x0[0])
(1): COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
(3): 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(x1[3] >= 0 && x0[3] > x1[3] + 1, 254_0_test_Return, x0[3], x1[3])
(4): COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(x1[4] + 1), x0[4], x1[4] + 1)
(5): COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(x1[5] + 1)

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

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

(1) -> (3), if ((154_0_test_Store(0) →* 254_0_test_Return)∧(x0[1]* x0[3])∧(0* x1[3]))

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

(3) -> (4), if ((x1[3] >= 0 && x0[3] > x1[3] + 1* TRUE)∧(x0[3]* x0[4])∧(x1[3]* x1[4]))

(3) -> (5), if ((x1[3] >= 0 && x0[3] > x1[3] + 1* TRUE)∧(x0[3]* x0[5])∧(x1[3]* x1[5]))

(4) -> (3), if ((154_0_test_Store(x1[4] + 1) →* 254_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.

### (5) IDPNonInfProof (SOUND transformation)

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 154_0_TEST_STORE(x0) → COND_154_0_TEST_STORE(>(x0, 0), x0) the following chains were created:
• We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_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)=TRUE154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_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_154_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_154_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_154_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_154_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 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_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)=TRUE154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_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_154_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_154_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_154_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_154_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_154_0_TEST_STORE(TRUE, x0) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0, 0) the following chains were created:
• We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0), 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]) which results in the following constraint:

(13)    (>(x0[0], 0)=TRUEx0[0]=x0[1]154_0_test_Store(0)=254_0_test_Returnx0[1]=x0[3]0=x1[3]COND_154_0_TEST_STORE(TRUE, x0[1])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[1])≥264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)∧(UIncreasing(264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)), ≥))

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

For Pair COND_154_0_TEST_STORE(TRUE, x0) → 154_0_TEST_STORE(0) the following chains were created:
• We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0), 154_0_TEST_STORE(x0[0]) → COND_154_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_154_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[2])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))

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

(15)    (>(x0[0], 0)=TRUECOND_154_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[0])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))

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

(16)    (0 ≥ 0 ⇒ (UIncreasing(154_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(154_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(154_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(154_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0, x1) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), 254_0_test_Return, x0, x1) the following chains were created:
• We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)) which results in the following constraint:

(20)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[4]x1[3]=x1[4]264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥))

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

(21)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1)) which results in the following constraint:

(26)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[5]x1[3]=x1[5]264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥))

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

(27)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1, 1)), x0, +(x1, 1)) the following chains were created:
• We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)), 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]) which results in the following constraint:

(32)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[4]x1[3]=x1[4]154_0_test_Store(+(x1[4], 1))=254_0_test_Returnx0[4]=x0[3]1+(x1[4], 1)=x1[3]1COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4])≥NonInfC∧COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4])≥264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))∧(UIncreasing(264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))), ≥))

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

For Pair COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 154_0_TEST_STORE(+(x1, 1)) the following chains were created:
• We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1)), 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

(33)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUEx0[3]=x0[5]x1[3]=x1[5]+(x1[5], 1)=x0[0]COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5])≥NonInfC∧COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5])≥154_0_TEST_STORE(+(x1[5], 1))∧(UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥))

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

(34)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUECOND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[3], x1[3])≥NonInfC∧COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[3], x1[3])≥154_0_TEST_STORE(+(x1[3], 1))∧(UIncreasing(154_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(154_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(154_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(154_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(154_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.
• 154_0_TEST_STORE(x0) → COND_154_0_TEST_STORE(>(x0, 0), x0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_154_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_154_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_154_0_TEST_STORE(TRUE, x0) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0, 0)

• COND_154_0_TEST_STORE(TRUE, x0) → 154_0_TEST_STORE(0)
• (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

• 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0, x1) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), 254_0_test_Return, x0, x1)
• (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_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_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1, 1)), x0, +(x1, 1))

• COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 154_0_TEST_STORE(+(x1, 1))
• (0 ≥ 0 ⇒ (UIncreasing(154_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(154_0_TEST_STORE(x1)) = [-1] + x1
POL(COND_154_0_TEST_STORE(x1, x2)) = [-1]
POL(>(x1, x2)) = 0
POL(0) = 0
POL(264_1_TEST_INVOKEMETHOD(x1, x2, x3)) = [1] + x3 + x2 + [-1]x1
POL(154_0_test_Store(x1)) = 0
POL(254_0_test_Return) = 0
POL(COND_264_1_TEST_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [-1]x2
POL(&&(x1, x2)) = 0
POL(>=(x1, x2)) = 0
POL(+(x1, x2)) = 0
POL(1) = 0

The following pairs are in P>:

COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)
264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))

The following pairs are in Pbound:

154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)
COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1))

The following pairs are in P:

154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1))

There are no usable 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): 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(x0[0] > 0, x0[0])
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
(5): COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_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* TRUE)∧(x0[0]* x0[2]))

The set Q is empty.

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
(0): 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(x0[0] > 0, x0[0])

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

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

The set Q is empty.

### (9) IDPNonInfProof (SOUND transformation)

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_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0) the following chains were created:
• We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0), 154_0_TEST_STORE(x0[0]) → COND_154_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_154_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[2])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))

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

(2)    (>(x0[0], 0)=TRUECOND_154_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[0])≥154_0_TEST_STORE(0)∧(UIncreasing(154_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(154_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(154_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(154_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(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]) the following chains were created:
• We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[2]154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_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)=TRUE154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_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_154_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_154_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_154_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_154_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_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)

• 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_154_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_154_0_TEST_STORE(x1, x2)) = [2]
POL(154_0_TEST_STORE(x1)) = [2] + x1
POL(0) = 0
POL(>(x1, x2)) = 0

The following pairs are in P>:

154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in Pbound:

COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in P:

COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)

There are no usable rules.

### (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:
none

R is empty.

The integer pair graph contains the following rules and edges:
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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