(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC11
`/** * Example taken from "A Term Rewriting Approach to the Automated Termination * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf) * and converted to Java. */public class PastaC11 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (true) {			if (x >= 0) {				x--;				y = Random.random();			} else if (y >= 0) {				y--;			} else {				break;			}        }    } }public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
PastaC11.main([Ljava/lang/String;)V: Graph of 230 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 33 rules for P and 48 rules for R.

Combined rules. Obtained 4 rules for P and 0 rules for R.

Filtered ground terms:

1617_0_main_LT(x1, x2, x3, x4) → 1617_0_main_LT(x2, x3, x4)
1694_0_random_IntArithmetic(x1, x2, x3, x4) → 1694_0_random_IntArithmetic(x2, x3)
1659_0_random_ArrayAccess(x1, x2, x3) → 1659_0_random_ArrayAccess(x2, x3)
Cond_1617_0_main_LT1(x1, x2, x3, x4, x5) → Cond_1617_0_main_LT1(x1, x3, x4, x5)
Cond_1617_0_main_LT(x1, x2, x3, x4, x5) → Cond_1617_0_main_LT(x1, x4)

Filtered duplicate args:

1617_0_main_LT(x1, x2, x3) → 1617_0_main_LT(x2, x3)
Cond_1617_0_main_LT1(x1, x2, x3, x4) → Cond_1617_0_main_LT1(x1, x3, x4)

Filtered unneeded arguments:

Cond_1617_0_main_LT1(x1, x2, x3) → Cond_1617_0_main_LT1(x1, x3)

Combined rules. Obtained 4 rules for P and 0 rules for R.

Finished conversion. Obtained 4 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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(x1[0] >= 0 && 0 > -1, x1[0], -1)
(1): COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(x1[1] + -1, -1)
(2): 1617_0_MAIN_LT(x1[2], x0[2]) → COND_1617_0_MAIN_LT1(x0[2] >= 0, x1[2], x0[2])
(3): COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3]) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), x0[3] + -1)
(4): 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4]) → COND_1659_1_MAIN_INVOKEMETHOD(x2[4] >= 2 && x2[4] < x0[4], 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])
(5): COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5]) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])
(6): 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6]) → COND_1694_1_MAIN_INVOKEMETHOD(x2[6] > 0, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])
(7): COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7]) → 1617_0_MAIN_LT(x1[7], x4[7])

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

(1) -> (0), if (x1[1] + -1* x1[0])

(1) -> (2), if ((x1[1] + -1* x1[2])∧(-1* x0[2]))

(2) -> (3), if ((x0[2] >= 0* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

(3) -> (4), if ((1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]) →* 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]))∧(x0[3] + -1* x3[4]))

(4) -> (5), if ((x2[4] >= 2 && x2[4] < x0[4]* TRUE)∧(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]) →* 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]))∧(x3[4]* x3[5]))

(5) -> (6), if ((1694_0_random_IntArithmetic(x4[5], x5[5]) →* 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]))∧(x3[5]* x4[6]))

(6) -> (7), if ((x2[6] > 0* TRUE)∧(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]) →* 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]))∧(x4[6]* x4[7]))

(7) -> (0), if ((x1[7]* x1[0])∧(x4[7]* -1))

(7) -> (2), if ((x1[7]* x1[2])∧(x4[7]* x0[2]))

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 1617_0_MAIN_LT(x1, -1) → COND_1617_0_MAIN_LT(&&(>=(x1, 0), >(0, -1)), x1, -1) the following chains were created:
• We consider the chain 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1), COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:

(1)    (&&(>=(x1[0], 0), >(0, -1))=TRUEx1[0]=x1[1]1617_0_MAIN_LT(x1[0], -1)≥NonInfC∧1617_0_MAIN_LT(x1[0], -1)≥COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)∧(UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥))

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

(2)    (>=(x1[0], 0)=TRUE1617_0_MAIN_LT(x1[0], -1)≥NonInfC∧1617_0_MAIN_LT(x1[0], -1)≥COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)∧(UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥))

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

(3)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(4)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(5)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧[(-1)bso_32] ≥ 0)

For Pair COND_1617_0_MAIN_LT(TRUE, x1, -1) → 1617_0_MAIN_LT(+(x1, -1), -1) the following chains were created:
• We consider the chain COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:

(6)    (COND_1617_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_1617_0_MAIN_LT(TRUE, x1[1], -1)≥1617_0_MAIN_LT(+(x1[1], -1), -1)∧(UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(7)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)bso_34] ≥ 0)

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

(8)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)bso_34] ≥ 0)

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

(9)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)bso_34] ≥ 0)

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

(10)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 1617_0_MAIN_LT(x1, x0) → COND_1617_0_MAIN_LT1(>=(x0, 0), x1, x0) the following chains were created:
• We consider the chain 1617_0_MAIN_LT(x1[2], x0[2]) → COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2]), COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3]) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1)) which results in the following constraint:

(11)    (>=(x0[2], 0)=TRUEx1[2]=x1[3]x0[2]=x0[3]1617_0_MAIN_LT(x1[2], x0[2])≥NonInfC∧1617_0_MAIN_LT(x1[2], x0[2])≥COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])∧(UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥))

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

(12)    (>=(x0[2], 0)=TRUE1617_0_MAIN_LT(x1[2], x0[2])≥NonInfC∧1617_0_MAIN_LT(x1[2], x0[2])≥COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])∧(UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥))

We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(13)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[2] ≥ 0∧[1 + (-1)bso_36] ≥ 0)

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

(14)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[2] ≥ 0∧[1 + (-1)bso_36] ≥ 0)

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

(15)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[2] ≥ 0∧[1 + (-1)bso_36] ≥ 0)

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

(16)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_35] + [bni_35]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_36] ≥ 0)

For Pair COND_1617_0_MAIN_LT1(TRUE, x1, x0) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4), +(x0, -1)) the following chains were created:
• We consider the chain COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3]) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1)) which results in the following constraint:

(17)    (COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3])≥1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))∧(UIncreasing(1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))), ≥))

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

(18)    ((UIncreasing(1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))), ≥)∧[(-1)bso_38] ≥ 0)

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

(19)    ((UIncreasing(1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))), ≥)∧[(-1)bso_38] ≥ 0)

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

(20)    ((UIncreasing(1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))), ≥)∧[(-1)bso_38] ≥ 0)

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

(21)    ((UIncreasing(1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2, 2), <(x2, x0)), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) the following chains were created:
• We consider the chain 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4]) → COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4]), COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5]) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5]) which results in the following constraint:

(22)    (&&(>=(x2[4], 2), <(x2[4], x0[4]))=TRUE1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4])=1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5])∧x3[4]=x3[5]1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])≥NonInfC∧1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])≥COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])∧(UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥))

We simplified constraint (22) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(23)    (>=(x2[4], 2)=TRUE<(x2[4], x0[4])=TRUE1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])≥NonInfC∧1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])≥COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])∧(UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥))

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

(24)    (x2[4] + [-2] ≥ 0∧x0[4] + [-1] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x3[4] ≥ 0∧[(-1)bso_40] ≥ 0)

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

(25)    (x2[4] + [-2] ≥ 0∧x0[4] + [-1] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x3[4] ≥ 0∧[(-1)bso_40] ≥ 0)

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

(26)    (x2[4] + [-2] ≥ 0∧x0[4] + [-1] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x3[4] ≥ 0∧[(-1)bso_40] ≥ 0)

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

(27)    (x2[4] + [-2] ≥ 0∧x0[4] + [-1] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)Bound*bni_39] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

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

(28)    (x2[4] ≥ 0∧x0[4] + [-3] + [-1]x2[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)Bound*bni_39] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

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

(29)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)Bound*bni_39] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

For Pair COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4, x5), x3) the following chains were created:
• We consider the chain COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5]) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5]) which results in the following constraint:

(30)    (COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5])≥NonInfC∧COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5])≥1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])∧(UIncreasing(1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])), ≥))

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

(31)    ((UIncreasing(1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])), ≥)∧[(-1)bso_42] ≥ 0)

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

(32)    ((UIncreasing(1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])), ≥)∧[(-1)bso_42] ≥ 0)

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

(33)    ((UIncreasing(1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])), ≥)∧[(-1)bso_42] ≥ 0)

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

(34)    ((UIncreasing(1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

For Pair 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_1694_1_MAIN_INVOKEMETHOD(>(x2, 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) the following chains were created:
• We consider the chain 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6]) → COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6]), COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7]) → 1617_0_MAIN_LT(x1[7], x4[7]) which results in the following constraint:

(35)    (>(x2[6], 0)=TRUE1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6])=1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7])∧x4[6]=x4[7]1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])≥NonInfC∧1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])≥COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])∧(UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥))

We simplified constraint (35) using rules (I), (II), (IV) which results in the following new constraint:

(36)    (>(x2[6], 0)=TRUE1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])≥NonInfC∧1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])≥COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])∧(UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥))

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

(37)    (x2[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x4[6] ≥ 0∧[(-1)bso_44] ≥ 0)

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

(38)    (x2[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x4[6] ≥ 0∧[(-1)bso_44] ≥ 0)

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

(39)    (x2[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x4[6] ≥ 0∧[(-1)bso_44] ≥ 0)

We simplified constraint (39) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(40)    (x2[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥)∧0 = 0∧0 = 0∧[bni_43] = 0∧[(-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

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

(41)    (x2[6] ≥ 0 ⇒ (UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥)∧0 = 0∧0 = 0∧[bni_43] = 0∧[(-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

For Pair COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 1617_0_MAIN_LT(x1, x4) the following chains were created:
• We consider the chain COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7]) → 1617_0_MAIN_LT(x1[7], x4[7]) which results in the following constraint:

(42)    (COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7])≥NonInfC∧COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7])≥1617_0_MAIN_LT(x1[7], x4[7])∧(UIncreasing(1617_0_MAIN_LT(x1[7], x4[7])), ≥))

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

(43)    ((UIncreasing(1617_0_MAIN_LT(x1[7], x4[7])), ≥)∧[(-1)bso_46] ≥ 0)

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

(44)    ((UIncreasing(1617_0_MAIN_LT(x1[7], x4[7])), ≥)∧[(-1)bso_46] ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(45)    ((UIncreasing(1617_0_MAIN_LT(x1[7], x4[7])), ≥)∧[(-1)bso_46] ≥ 0)

We simplified constraint (45) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(46)    ((UIncreasing(1617_0_MAIN_LT(x1[7], x4[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_46] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1617_0_MAIN_LT(x1, -1) → COND_1617_0_MAIN_LT(&&(>=(x1, 0), >(0, -1)), x1, -1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧[(-1)bso_32] ≥ 0)

• COND_1617_0_MAIN_LT(TRUE, x1, -1) → 1617_0_MAIN_LT(+(x1, -1), -1)
• ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧0 = 0∧[(-1)bso_34] ≥ 0)

• 1617_0_MAIN_LT(x1, x0) → COND_1617_0_MAIN_LT1(>=(x0, 0), x1, x0)
• (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_35] + [bni_35]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_36] ≥ 0)

• COND_1617_0_MAIN_LT1(TRUE, x1, x0) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2, x3)), x4), +(x0, -1))
• ((UIncreasing(1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2, 2), <(x2, x0)), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3)
• (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)Bound*bni_39] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

• COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0, x1)), x2), x3) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4, x5), x3)
• ((UIncreasing(1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)

• 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → COND_1694_1_MAIN_INVOKEMETHOD(>(x2, 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4)
• (x2[6] ≥ 0 ⇒ (UIncreasing(COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])), ≥)∧0 = 0∧0 = 0∧[bni_43] = 0∧[(-1)Bound*bni_43] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_44] ≥ 0)

• COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0, x1)), x2), x4) → 1617_0_MAIN_LT(x1, x4)
• ((UIncreasing(1617_0_MAIN_LT(x1[7], x4[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_46] ≥ 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(1617_0_MAIN_LT(x1, x2)) = x2
POL(-1) = [-1]
POL(COND_1617_0_MAIN_LT(x1, x2, x3)) = [-1]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(COND_1617_0_MAIN_LT1(x1, x2, x3)) = [-1] + x3
POL(1659_1_MAIN_INVOKEMETHOD(x1, x2)) = x2 + [-1]x1
POL(1659_0_random_ArrayAccess(x1, x2)) = [-1] + [-1]x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1]
POL(COND_1659_1_MAIN_INVOKEMETHOD(x1, x2, x3)) = x3 + [-1]x2
POL(2) = [2]
POL(<(x1, x2)) = [-1]
POL(1694_1_MAIN_INVOKEMETHOD(x1, x2)) = x2
POL(1694_0_random_IntArithmetic(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(java.lang.String(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(COND_1694_1_MAIN_INVOKEMETHOD(x1, x2, x3)) = x3

The following pairs are in P>:

1617_0_MAIN_LT(x1[2], x0[2]) → COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])

The following pairs are in Pbound:

1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)
1617_0_MAIN_LT(x1[2], x0[2]) → COND_1617_0_MAIN_LT1(>=(x0[2], 0), x1[2], x0[2])

The following pairs are in P:

1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)
COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1)
COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3]) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), +(x0[3], -1))
1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4]) → COND_1659_1_MAIN_INVOKEMETHOD(&&(>=(x2[4], 2), <(x2[4], x0[4])), 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])
COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5]) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])
1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6]) → COND_1694_1_MAIN_INVOKEMETHOD(>(x2[6], 0), 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])
COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7]) → 1617_0_MAIN_LT(x1[7], x4[7])

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(x1[0] >= 0 && 0 > -1, x1[0], -1)
(1): COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(x1[1] + -1, -1)
(3): COND_1617_0_MAIN_LT1(TRUE, x1[3], x0[3]) → 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]), x0[3] + -1)
(4): 1659_1_MAIN_INVOKEMETHOD(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4]) → COND_1659_1_MAIN_INVOKEMETHOD(x2[4] >= 2 && x2[4] < x0[4], 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]), x3[4])
(5): COND_1659_1_MAIN_INVOKEMETHOD(TRUE, 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]), x3[5]) → 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(x4[5], x5[5]), x3[5])
(6): 1694_1_MAIN_INVOKEMETHOD(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6]) → COND_1694_1_MAIN_INVOKEMETHOD(x2[6] > 0, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]), x4[6])
(7): COND_1694_1_MAIN_INVOKEMETHOD(TRUE, 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]), x4[7]) → 1617_0_MAIN_LT(x1[7], x4[7])

(1) -> (0), if (x1[1] + -1* x1[0])

(7) -> (0), if ((x1[7]* x1[0])∧(x4[7]* -1))

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

(3) -> (4), if ((1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x2[3], x3[3])), x4[3]) →* 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]))∧(x0[3] + -1* x3[4]))

(4) -> (5), if ((x2[4] >= 2 && x2[4] < x0[4]* TRUE)∧(1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[4], x1[4])), x2[4]) →* 1659_0_random_ArrayAccess(java.lang.Object(ARRAY(x0[5], x1[5])), x2[5]))∧(x3[4]* x3[5]))

(5) -> (6), if ((1694_0_random_IntArithmetic(x4[5], x5[5]) →* 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]))∧(x3[5]* x4[6]))

(6) -> (7), if ((x2[6] > 0* TRUE)∧(1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[6], x1[6])), x2[6]) →* 1694_0_random_IntArithmetic(java.lang.Object(java.lang.String(x0[7], x1[7])), x2[7]))∧(x4[6]* x4[7]))

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(x1[1] + -1, -1)
(0): 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(x1[0] >= 0 && 0 > -1, x1[0], -1)

(1) -> (0), if (x1[1] + -1* x1[0])

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

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_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1) the following chains were created:
• We consider the chain COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:

(1)    (COND_1617_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_1617_0_MAIN_LT(TRUE, x1[1], -1)≥1617_0_MAIN_LT(+(x1[1], -1), -1)∧(UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

(2)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1) the following chains were created:
• We consider the chain 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1), COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:

(6)    (&&(>=(x1[0], 0), >(0, -1))=TRUEx1[0]=x1[1]1617_0_MAIN_LT(x1[0], -1)≥NonInfC∧1617_0_MAIN_LT(x1[0], -1)≥COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)∧(UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥))

We simplified constraint (6) using rules (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (>=(x1[0], 0)=TRUE1617_0_MAIN_LT(x1[0], -1)≥NonInfC∧1617_0_MAIN_LT(x1[0], -1)≥COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)∧(UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥))

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

(8)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x1[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)

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

(9)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x1[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)

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

(10)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x1[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1)
• ((UIncreasing(1617_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• 1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]x1[0] ≥ 0∧[1 + (-1)bso_11] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_1617_0_MAIN_LT(x1, x2, x3)) = [1] + [2]x2
POL(-1) = [-1]
POL(1617_0_MAIN_LT(x1, x2)) = [2] + [2]x1
POL(+(x1, x2)) = x1 + x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [2]

The following pairs are in P>:

COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(+(x1[1], -1), -1)
1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)

The following pairs are in Pbound:

1617_0_MAIN_LT(x1[0], -1) → COND_1617_0_MAIN_LT(&&(>=(x1[0], 0), >(0, -1)), x1[0], -1)

The following pairs are in P:
none

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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1617_0_MAIN_LT(TRUE, x1[1], -1) → 1617_0_MAIN_LT(x1[1] + -1, -1)

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.