Termination proof of /tmp/tmp9sG

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 630 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 AND

    ↳5 JBCTerminationSCC

        ↳6 SCCToIDPv1Proof (⇒, 120 ms)

        ↳7 IDP

        ↳8 IDPNonInfProof (⇒, 170 ms)

        ↳9 AND

            ↳10 IDP

                ↳11 IDependencyGraphProof (⇔, 0 ms)

                ↳12 TRUE

            ↳13 IDP

                ↳14 IDependencyGraphProof (⇔, 0 ms)

                ↳15 TRUE

    ↳16 JBCTerminationSCC

        ↳17 SCCToIDPv1Proof (⇒, 490 ms)

        ↳18 IDP

        ↳19 IDPNonInfProof (⇒, 2200 ms)

        ↳20 IDP

        ↳21 IDependencyGraphProof (⇔, 0 ms)

        ↳22 IDP

        ↳23 IDPNonInfProof (⇒, 430 ms)

        ↳24 IDP

        ↳25 IDependencyGraphProof (⇔, 0 ms)

        ↳26 TRUE

    ↳27 JBCTerminationSCC

        ↳28 SCCToIDPv1Proof (⇒, 190 ms)

        ↳29 IDP

        ↳30 IDPNonInfProof (⇒, 150 ms)

        ↳31 AND

            ↳32 IDP

                ↳33 IDependencyGraphProof (⇔, 0 ms)

                ↳34 TRUE

            ↳35 IDP

                ↳36 IDependencyGraphProof (⇔, 0 ms)

                ↳37 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: NestedLoop

public class NestedLoop {
	public static void main(String[] args) {
		int i, j, z, n;
		n = args.length;
		int[] a = new int[n];
		for(i = 0;i< n-1;i++){
			a[i] = args[i].length();
		}

		for (i = 0; i < n - 1; i++) {
			for (j = i + 1; j < n; j++) {
				if(a[i]< a[j]){
					z = a[i]; a[i] = a[j]; a[j] = z;
				}
			}
		}
		for(i = 0;i< n -1;i++){
		}
	}
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
NestedLoop.main([Ljava/lang/String;)V: Graph of 349 nodes with 3 SCCs.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 3 SCCss.

(4) Complex Obligation (AND)

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: NestedLoop.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 9 rules for P and 0 rules for R.

P rules:
1679_0_main_Load(EOS(STATIC_1679), i423, i223, i423) → 1696_0_main_ConstantStackPush(EOS(STATIC_1696), i423, i223, i423, i223)
1696_0_main_ConstantStackPush(EOS(STATIC_1696), i423, i223, i423, i223) → 1726_0_main_IntArithmetic(EOS(STATIC_1726), i423, i223, i423, i223, 1)
1726_0_main_IntArithmetic(EOS(STATIC_1726), i423, i223, i423, i223, matching1) → 1753_0_main_GE(EOS(STATIC_1753), i423, i223, i423, -(i223, 1)) | &&(>=(i223, 0), =(matching1, 1))
1753_0_main_GE(EOS(STATIC_1753), i423, i223, i423, i466) → 1779_0_main_GE(EOS(STATIC_1779), i423, i223, i423, i466)
1779_0_main_GE(EOS(STATIC_1779), i423, i223, i423, i466) → 1804_0_main_Inc(EOS(STATIC_1804), i423, i223) | <(i423, i466)
1804_0_main_Inc(EOS(STATIC_1804), i423, i223) → 1827_0_main_JMP(EOS(STATIC_1827), +(i423, 1), i223) | >=(i423, 0)
1827_0_main_JMP(EOS(STATIC_1827), i504, i223) → 1847_0_main_Load(EOS(STATIC_1847), i504, i223)
1847_0_main_Load(EOS(STATIC_1847), i504, i223) → 1648_0_main_Load(EOS(STATIC_1648), i504, i223)
1648_0_main_Load(EOS(STATIC_1648), i423, i223) → 1679_0_main_Load(EOS(STATIC_1679), i423, i223, i423)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
1679_0_main_Load(EOS(STATIC_1679), x0, x1, x0) → 1679_0_main_Load(EOS(STATIC_1679), +(x0, 1), x1, +(x0, 1)) | &&(&&(>(+(x1, 1), 0), >(+(x0, 1), 0)), <(x0, -(x1, 1)))
R rules:

Filtered ground terms:

1679_0_main_Load(x1, x2, x3, x4) → 1679_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_1679_0_main_Load(x1, x2, x3, x4, x5) → Cond_1679_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

1679_0_main_Load(x1, x2, x3) → 1679_0_main_Load(x2, x3)
Cond_1679_0_main_Load(x1, x2, x3, x4) → Cond_1679_0_main_Load(x1, x3, x4)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
1679_0_main_Load(x1, x0) → 1679_0_main_Load(x1, +(x0, 1)) | &&(&&(>(x1, -1), >(x0, -1)), <(x0, -(x1, 1)))
R rules:

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

P rules:
1679_0_MAIN_LOAD(x1, x0) → COND_1679_0_MAIN_LOAD(&&(&&(>(x1, -1), >(x0, -1)), <(x0, -(x1, 1))), x1, x0)
COND_1679_0_MAIN_LOAD(TRUE, x1, x0) → 1679_0_MAIN_LOAD(x1, +(x0, 1))
R rules:

(7) 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): 1679_0_MAIN_LOAD(x1[0], x0[0]) → COND_1679_0_MAIN_LOAD(x1[0] > -1 && x0[0] > -1 && x0[0] < x1[0] - 1, x1[0], x0[0])
(1): COND_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1679_0_MAIN_LOAD(x1[1], x0[1] + 1)

(0) -> (1), if (x1[0] > -1 && x0[0] > -1 && x0[0] < x1[0] - 1 ∧x1[0] →^* x1[1]∧x0[0] →^* x0[1])

(1) -> (0), if (x1[1] →^* x1[0]∧x0[1] + 1 →^* x0[0])

The set Q is empty.

(8) 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@79711174 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 1679_0_MAIN_LOAD(x1, x0) → COND_1679_0_MAIN_LOAD(&&(&&(>(x1, -1), >(x0, -1)), <(x0, -(x1, 1))), x1, x0) the following chains were created:

We consider the chain 1679_0_MAIN_LOAD(x1[0], x0[0]) → COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0]), COND_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1679_0_MAIN_LOAD(x1[1], +(x0[1], 1)) which results in the following constraint:
(1)    (&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1)))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 1679_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1679_0_MAIN_LOAD(x1[0], x0[0])≥COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])∧(U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<(x0[0], -(x1[0], 1))=TRUE∧>(x1[0], -1)=TRUE∧>(x0[0], -1)=TRUE ⇒ 1679_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1679_0_MAIN_LOAD(x1[0], x0[0])≥COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])∧(U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-2] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x0[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-2] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x0[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-2] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x0[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_1679_0_MAIN_LOAD(TRUE, x1, x0) → 1679_0_MAIN_LOAD(x1, +(x0, 1)) the following chains were created:

We consider the chain COND_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1679_0_MAIN_LOAD(x1[1], +(x0[1], 1)) which results in the following constraint:
(7)    (COND_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))∧(U^Increasing(1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1679_0_MAIN_LOAD(x1, x0) → COND_1679_0_MAIN_LOAD(&&(&&(>(x1, -1), >(x0, -1)), <(x0, -(x1, 1))), x1, x0)
- (x1[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_1679_0_MAIN_LOAD(TRUE, x1, x0) → 1679_0_MAIN_LOAD(x1, +(x0, 1))
- ((U^Increasing(1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(1679_0_MAIN_LOAD(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(COND_1679_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂

The following pairs are in P_>:

COND_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1679_0_MAIN_LOAD(x1[1], +(x0[1], 1))

The following pairs are in P_bound:

1679_0_MAIN_LOAD(x1[0], x0[0]) → COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])

The following pairs are in P_≥:

1679_0_MAIN_LOAD(x1[0], x0[0]) → COND_1679_0_MAIN_LOAD(&&(&&(>(x1[0], -1), >(x0[0], -1)), <(x0[0], -(x1[0], 1))), x1[0], x0[0])

There are no usable rules.

(9) Complex Obligation (AND)

(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:

Boolean, Integer

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE

(13) 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_1679_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1679_0_MAIN_LOAD(x1[1], x0[1] + 1)

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 53 rules for P and 0 rules for R.

P rules:
1292_0_main_Load(EOS(STATIC_1292), i296, i223, java.lang.Object(ARRAY(i223)), i296) → 1294_0_main_ConstantStackPush(EOS(STATIC_1294), i296, i223, java.lang.Object(ARRAY(i223)), i296, i223)
1294_0_main_ConstantStackPush(EOS(STATIC_1294), i296, i223, java.lang.Object(ARRAY(i223)), i296, i223) → 1296_0_main_IntArithmetic(EOS(STATIC_1296), i296, i223, java.lang.Object(ARRAY(i223)), i296, i223, 1)
1296_0_main_IntArithmetic(EOS(STATIC_1296), i296, i223, java.lang.Object(ARRAY(i223)), i296, i223, matching1) → 1298_0_main_GE(EOS(STATIC_1298), i296, i223, java.lang.Object(ARRAY(i223)), i296, -(i223, 1)) | &&(>=(i223, 0), =(matching1, 1))
1298_0_main_GE(EOS(STATIC_1298), i296, i223, java.lang.Object(ARRAY(i223)), i296, i301) → 1301_0_main_GE(EOS(STATIC_1301), i296, i223, java.lang.Object(ARRAY(i223)), i296, i301)
1301_0_main_GE(EOS(STATIC_1301), i296, i223, java.lang.Object(ARRAY(i223)), i296, i301) → 1305_0_main_Load(EOS(STATIC_1305), i296, i223, java.lang.Object(ARRAY(i223))) | <(i296, i301)
1305_0_main_Load(EOS(STATIC_1305), i296, i223, java.lang.Object(ARRAY(i223))) → 1309_0_main_ConstantStackPush(EOS(STATIC_1309), i296, i223, java.lang.Object(ARRAY(i223)), i296)
1309_0_main_ConstantStackPush(EOS(STATIC_1309), i296, i223, java.lang.Object(ARRAY(i223)), i296) → 1313_0_main_IntArithmetic(EOS(STATIC_1313), i296, i223, java.lang.Object(ARRAY(i223)), i296, 1)
1313_0_main_IntArithmetic(EOS(STATIC_1313), i296, i223, java.lang.Object(ARRAY(i223)), i296, matching1) → 1316_0_main_Store(EOS(STATIC_1316), i296, i223, java.lang.Object(ARRAY(i223)), +(i296, 1)) | &&(>=(i296, 0), =(matching1, 1))
1316_0_main_Store(EOS(STATIC_1316), i296, i223, java.lang.Object(ARRAY(i223)), i304) → 1320_0_main_Load(EOS(STATIC_1320), i296, i304, i223, java.lang.Object(ARRAY(i223)))
1320_0_main_Load(EOS(STATIC_1320), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1325_0_main_Load(EOS(STATIC_1325), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304)
1325_0_main_Load(EOS(STATIC_1325), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304) → 1330_0_main_GE(EOS(STATIC_1330), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223)
1330_0_main_GE(EOS(STATIC_1330), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223) → 1335_0_main_GE(EOS(STATIC_1335), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223)
1330_0_main_GE(EOS(STATIC_1330), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223) → 1336_0_main_GE(EOS(STATIC_1336), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223)
1335_0_main_GE(EOS(STATIC_1335), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223) → 1342_0_main_Inc(EOS(STATIC_1342), i296, i223, java.lang.Object(ARRAY(i223))) | >=(i304, i223)
1342_0_main_Inc(EOS(STATIC_1342), i296, i223, java.lang.Object(ARRAY(i223))) → 1350_0_main_JMP(EOS(STATIC_1350), +(i296, 1), i223, java.lang.Object(ARRAY(i223))) | >=(i296, 0)
1350_0_main_JMP(EOS(STATIC_1350), i310, i223, java.lang.Object(ARRAY(i223))) → 1363_0_main_Load(EOS(STATIC_1363), i310, i223, java.lang.Object(ARRAY(i223)))
1363_0_main_Load(EOS(STATIC_1363), i310, i223, java.lang.Object(ARRAY(i223))) → 1289_0_main_Load(EOS(STATIC_1289), i310, i223, java.lang.Object(ARRAY(i223)))
1289_0_main_Load(EOS(STATIC_1289), i296, i223, java.lang.Object(ARRAY(i223))) → 1292_0_main_Load(EOS(STATIC_1292), i296, i223, java.lang.Object(ARRAY(i223)), i296)
1336_0_main_GE(EOS(STATIC_1336), i296, i304, i223, java.lang.Object(ARRAY(i223)), i304, i223) → 1344_0_main_Load(EOS(STATIC_1344), i296, i304, i223, java.lang.Object(ARRAY(i223))) | <(i304, i223)
1344_0_main_Load(EOS(STATIC_1344), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1352_0_main_Load(EOS(STATIC_1352), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)))
1352_0_main_Load(EOS(STATIC_1352), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223))) → 1365_0_main_ArrayAccess(EOS(STATIC_1365), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296)
1365_0_main_ArrayAccess(EOS(STATIC_1365), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1369_0_main_ArrayAccess(EOS(STATIC_1369), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296)
1369_0_main_ArrayAccess(EOS(STATIC_1369), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1373_0_main_Load(EOS(STATIC_1373), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318) | <(i296, i223)
1373_0_main_Load(EOS(STATIC_1373), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318) → 1379_0_main_Load(EOS(STATIC_1379), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, java.lang.Object(ARRAY(i223)))
1379_0_main_Load(EOS(STATIC_1379), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, java.lang.Object(ARRAY(i223))) → 1384_0_main_ArrayAccess(EOS(STATIC_1384), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, java.lang.Object(ARRAY(i223)), i304)
1384_0_main_ArrayAccess(EOS(STATIC_1384), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, java.lang.Object(ARRAY(i223)), i304) → 1393_0_main_ArrayAccess(EOS(STATIC_1393), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, java.lang.Object(ARRAY(i223)), i304)
1393_0_main_ArrayAccess(EOS(STATIC_1393), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, java.lang.Object(ARRAY(i223)), i304) → 1402_0_main_GE(EOS(STATIC_1402), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322) | <(i304, i223)
1402_0_main_GE(EOS(STATIC_1402), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322) → 1410_0_main_GE(EOS(STATIC_1410), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322)
1402_0_main_GE(EOS(STATIC_1402), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322) → 1411_0_main_GE(EOS(STATIC_1411), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322)
1410_0_main_GE(EOS(STATIC_1410), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322) → 1424_0_main_Inc(EOS(STATIC_1424), i296, i304, i223, java.lang.Object(ARRAY(i223))) | >=(i318, i322)
1424_0_main_Inc(EOS(STATIC_1424), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1699_0_main_Inc(EOS(STATIC_1699), i296, i304, i223, java.lang.Object(ARRAY(i223)))
1699_0_main_Inc(EOS(STATIC_1699), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1728_0_main_JMP(EOS(STATIC_1728), i296, +(i304, 1), i223, java.lang.Object(ARRAY(i223))) | >(i304, 0)
1728_0_main_JMP(EOS(STATIC_1728), i296, i454, i223, java.lang.Object(ARRAY(i223))) → 1755_0_main_Load(EOS(STATIC_1755), i296, i454, i223, java.lang.Object(ARRAY(i223)))
1755_0_main_Load(EOS(STATIC_1755), i296, i454, i223, java.lang.Object(ARRAY(i223))) → 1320_0_main_Load(EOS(STATIC_1320), i296, i454, i223, java.lang.Object(ARRAY(i223)))
1411_0_main_GE(EOS(STATIC_1411), i296, i304, i223, java.lang.Object(ARRAY(i223)), i318, i322) → 1426_0_main_Load(EOS(STATIC_1426), i296, i304, i223, java.lang.Object(ARRAY(i223))) | <(i318, i322)
1426_0_main_Load(EOS(STATIC_1426), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1439_0_main_Load(EOS(STATIC_1439), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)))
1439_0_main_Load(EOS(STATIC_1439), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223))) → 1447_0_main_ArrayAccess(EOS(STATIC_1447), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296)
1447_0_main_ArrayAccess(EOS(STATIC_1447), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1454_0_main_ArrayAccess(EOS(STATIC_1454), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296)
1454_0_main_ArrayAccess(EOS(STATIC_1454), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1464_0_main_Store(EOS(STATIC_1464), i296, i304, i223, java.lang.Object(ARRAY(i223))) | <(i296, i223)
1464_0_main_Store(EOS(STATIC_1464), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1472_0_main_Load(EOS(STATIC_1472), i296, i304, i223, java.lang.Object(ARRAY(i223)))
1472_0_main_Load(EOS(STATIC_1472), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1481_0_main_Load(EOS(STATIC_1481), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)))
1481_0_main_Load(EOS(STATIC_1481), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223))) → 1496_0_main_Load(EOS(STATIC_1496), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296)
1496_0_main_Load(EOS(STATIC_1496), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1507_0_main_Load(EOS(STATIC_1507), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296, java.lang.Object(ARRAY(i223)))
1507_0_main_Load(EOS(STATIC_1507), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296, java.lang.Object(ARRAY(i223))) → 1515_0_main_ArrayAccess(EOS(STATIC_1515), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296, java.lang.Object(ARRAY(i223)), i304)
1515_0_main_ArrayAccess(EOS(STATIC_1515), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296, java.lang.Object(ARRAY(i223)), i304) → 1523_0_main_ArrayAccess(EOS(STATIC_1523), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296, java.lang.Object(ARRAY(i223)), i304)
1523_0_main_ArrayAccess(EOS(STATIC_1523), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296, java.lang.Object(ARRAY(i223)), i304) → 1533_0_main_ArrayAccess(EOS(STATIC_1533), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) | <(i304, i223)
1533_0_main_ArrayAccess(EOS(STATIC_1533), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1542_0_main_ArrayAccess(EOS(STATIC_1542), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296)
1542_0_main_ArrayAccess(EOS(STATIC_1542), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i296) → 1570_0_main_Load(EOS(STATIC_1570), i296, i304, i223, java.lang.Object(ARRAY(i223))) | <(i296, i223)
1570_0_main_Load(EOS(STATIC_1570), i296, i304, i223, java.lang.Object(ARRAY(i223))) → 1597_0_main_Load(EOS(STATIC_1597), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)))
1597_0_main_Load(EOS(STATIC_1597), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223))) → 1615_0_main_Load(EOS(STATIC_1615), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i304)
1615_0_main_Load(EOS(STATIC_1615), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i304) → 1651_0_main_ArrayAccess(EOS(STATIC_1651), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i304)
1651_0_main_ArrayAccess(EOS(STATIC_1651), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i304) → 1681_0_main_ArrayAccess(EOS(STATIC_1681), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i304)
1681_0_main_ArrayAccess(EOS(STATIC_1681), i296, i304, i223, java.lang.Object(ARRAY(i223)), java.lang.Object(ARRAY(i223)), i304) → 1699_0_main_Inc(EOS(STATIC_1699), i296, i304, i223, java.lang.Object(ARRAY(i223))) | <(i304, i223)
R rules:

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

P rules:
1330_0_main_GE(EOS(STATIC_1330), x0, x1, x2, java.lang.Object(ARRAY(x2)), x1, x2) → 1330_0_main_GE(EOS(STATIC_1330), +(x0, 1), +(x0, 2), x2, java.lang.Object(ARRAY(x2)), +(x0, 2), x2) | &&(&&(&&(>(+(x2, 1), 0), <=(x2, x1)), >(+(x0, 1), 0)), >(-(x2, 1), +(x0, 1)))
1330_0_main_GE(EOS(STATIC_1330), x0, x1, x2, java.lang.Object(ARRAY(x2)), x1, x2) → 1330_0_main_GE(EOS(STATIC_1330), x0, +(x1, 1), x2, java.lang.Object(ARRAY(x2)), +(x1, 1), x2) | &&(&&(>(x2, x1), >(x2, x0)), >(x1, 0))
R rules:

Filtered ground terms:

1330_0_main_GE(x1, x2, x3, x4, x5, x6, x7) → 1330_0_main_GE(x2, x3, x4, x5, x6, x7)
EOS(x1) → EOS
Cond_1330_0_main_GE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1330_0_main_GE1(x1, x3, x4, x5, x6, x7, x8)
Cond_1330_0_main_GE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1330_0_main_GE(x1, x3, x4, x5, x6, x7, x8)

Filtered duplicate args:

1330_0_main_GE(x1, x2, x3, x4, x5, x6) → 1330_0_main_GE(x1, x4, x5)
Cond_1330_0_main_GE(x1, x2, x3, x4, x5, x6, x7) → Cond_1330_0_main_GE(x1, x2, x5, x6)
Cond_1330_0_main_GE1(x1, x2, x3, x4, x5, x6, x7) → Cond_1330_0_main_GE1(x1, x2, x5, x6)

Filtered unneeded arguments:

Cond_1330_0_main_GE(x1, x2, x3, x4) → Cond_1330_0_main_GE(x1, x2, x3)

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

P rules:
1330_0_main_GE(x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_main_GE(+(x0, 1), java.lang.Object(ARRAY(x2)), +(x0, 2)) | &&(&&(&&(>(x2, -1), <=(x2, x1)), >(x0, -1)), >(-(x2, 1), +(x0, 1)))
1330_0_main_GE(x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_main_GE(x0, java.lang.Object(ARRAY(x2)), +(x1, 1)) | &&(&&(>(x2, x1), >(x2, x0)), >(x1, 0))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), x1) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2, -1), <=(x2, x1)), >(x0, -1)), >(-(x2, 1), +(x0, 1))), x0, java.lang.Object(ARRAY(x2)), x1)
COND_1330_0_MAIN_GE(TRUE, x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_MAIN_GE(+(x0, 1), java.lang.Object(ARRAY(x2)), +(x0, 2))
1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), x1) → COND_1330_0_MAIN_GE1(&&(&&(>(x2, x1), >(x2, x0)), >(x1, 0)), x0, java.lang.Object(ARRAY(x2)), x1)
COND_1330_0_MAIN_GE1(TRUE, x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), +(x1, 1))
R rules:

(18) 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): 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(x2[0] > -1 && x2[0] <= x1[0] && x0[0] > -1 && x2[0] - 1 > x0[0] + 1, x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])
(1): COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1]) → 1330_0_MAIN_GE(x0[1] + 1, java.lang.Object(ARRAY(x2[1])), x0[1] + 2)
(2): 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])
(3): COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), x1[3] + 1)

(0) -> (1), if (x2[0] > -1 && x2[0] <= x1[0] && x0[0] > -1 && x2[0] - 1 > x0[0] + 1 ∧x0[0] →^* x0[1]∧java.lang.Object(ARRAY(x2[0])) →^* java.lang.Object(ARRAY(x2[1]))∧x1[0] →^* x1[1])

(1) -> (0), if (x0[1] + 1 →^* x0[0]∧java.lang.Object(ARRAY(x2[1])) →^* java.lang.Object(ARRAY(x2[0]))∧x0[1] + 2 →^* x1[0])

(1) -> (2), if (x0[1] + 1 →^* x0[2]∧java.lang.Object(ARRAY(x2[1])) →^* java.lang.Object(ARRAY(x2[2]))∧x0[1] + 2 →^* x1[2])

(2) -> (3), if (x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0 ∧x0[2] →^* x0[3]∧java.lang.Object(ARRAY(x2[2])) →^* java.lang.Object(ARRAY(x2[3]))∧x1[2] →^* x1[3])

(3) -> (0), if (x0[3] →^* x0[0]∧java.lang.Object(ARRAY(x2[3])) →^* java.lang.Object(ARRAY(x2[0]))∧x1[3] + 1 →^* x1[0])

(3) -> (2), if (x0[3] →^* x0[2]∧java.lang.Object(ARRAY(x2[3])) →^* java.lang.Object(ARRAY(x2[2]))∧x1[3] + 1 →^* x1[2])

The set Q is empty.

(19) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5f01aa96 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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

For Pair 1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), x1) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2, -1), <=(x2, x1)), >(x0, -1)), >(-(x2, 1), +(x0, 1))), x0, java.lang.Object(ARRAY(x2)), x1) the following chains were created:

We consider the chain 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]), COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1]) → 1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2)) which results in the following constraint:
(1)    (&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1)))=TRUE∧x0[0]=x0[1]∧java.lang.Object(ARRAY(x2[0]))=java.lang.Object(ARRAY(x2[1]))∧x1[0]=x1[1] ⇒ 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥NonInfC∧1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])∧(U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(-(x2[0], 1), +(x0[0], 1))=TRUE∧>(x0[0], -1)=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x1[0])=TRUE ⇒ 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥NonInfC∧1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])∧(U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[0] + [(-1)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[0] + [(-1)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[0] + [(-1)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] + [-3] + [-1]x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[0] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_1330_0_MAIN_GE(TRUE, x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_MAIN_GE(+(x0, 1), java.lang.Object(ARRAY(x2)), +(x0, 2)) the following chains were created:

We consider the chain 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]), COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1]) → 1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2)), 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) which results in the following constraint:
(8)    (&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1)))=TRUE∧x0[0]=x0[1]∧java.lang.Object(ARRAY(x2[0]))=java.lang.Object(ARRAY(x2[1]))∧x1[0]=x1[1]∧+(x0[1], 1)=x0[0]1∧java.lang.Object(ARRAY(x2[1]))=java.lang.Object(ARRAY(x2[0]1))∧+(x0[1], 2)=x1[0]1 ⇒ COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1])≥NonInfC∧COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1])≥1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))∧(U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥))

We simplified constraint (8) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(-(x2[0], 1), +(x0[0], 1))=TRUE∧>(x0[0], -1)=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x1[0])=TRUE ⇒ COND_1330_0_MAIN_GE(TRUE, x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥NonInfC∧COND_1330_0_MAIN_GE(TRUE, x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥1330_0_MAIN_GE(+(x0[0], 1), java.lang.Object(ARRAY(x2[0])), +(x0[0], 2))∧(U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] + [(-1)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] + [(-1)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] + [(-1)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] + [-3] + [-1]x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
We consider the chain 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]), COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1]) → 1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2)), 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) which results in the following constraint:
(15)    (&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1)))=TRUE∧x0[0]=x0[1]∧java.lang.Object(ARRAY(x2[0]))=java.lang.Object(ARRAY(x2[1]))∧x1[0]=x1[1]∧+(x0[1], 1)=x0[2]∧java.lang.Object(ARRAY(x2[1]))=java.lang.Object(ARRAY(x2[2]))∧+(x0[1], 2)=x1[2] ⇒ COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1])≥NonInfC∧COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1])≥1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))∧(U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥))

We simplified constraint (15) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(16)    (>(-(x2[0], 1), +(x0[0], 1))=TRUE∧>(x0[0], -1)=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x1[0])=TRUE ⇒ COND_1330_0_MAIN_GE(TRUE, x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥NonInfC∧COND_1330_0_MAIN_GE(TRUE, x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])≥1330_0_MAIN_GE(+(x0[0], 1), java.lang.Object(ARRAY(x2[0])), +(x0[0], 2))∧(U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] + [(-1)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] + [(-1)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x2[0] + [-3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] + [(-1)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] + [-3] + [-1]x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

For Pair 1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), x1) → COND_1330_0_MAIN_GE1(&&(&&(>(x2, x1), >(x2, x0)), >(x1, 0)), x0, java.lang.Object(ARRAY(x2)), x1) the following chains were created:

We consider the chain 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]), COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1)) which results in the following constraint:
(22)    (&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0))=TRUE∧x0[2]=x0[3]∧java.lang.Object(ARRAY(x2[2]))=java.lang.Object(ARRAY(x2[3]))∧x1[2]=x1[3] ⇒ 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])∧(U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥))

We simplified constraint (22) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>(x1[2], 0)=TRUE∧>(x2[2], x1[2])=TRUE∧>(x2[2], x0[2])=TRUE ⇒ 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])∧(U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (x1[2] ≥ 0∧x2[2] + [-2] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[2] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (28) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(29)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[2] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

(30)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[2] + [bni_28]x2[2] + [bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)

For Pair COND_1330_0_MAIN_GE1(TRUE, x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), +(x1, 1)) the following chains were created:

We consider the chain 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]), COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1)), 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) which results in the following constraint:
(31)    (&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0))=TRUE∧x0[2]=x0[3]∧java.lang.Object(ARRAY(x2[2]))=java.lang.Object(ARRAY(x2[3]))∧x1[2]=x1[3]∧x0[3]=x0[0]∧java.lang.Object(ARRAY(x2[3]))=java.lang.Object(ARRAY(x2[0]))∧+(x1[3], 1)=x1[0] ⇒ COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3])≥NonInfC∧COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3])≥1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))∧(U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥))

We simplified constraint (31) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(32)    (>(x1[2], 0)=TRUE∧>(x2[2], x1[2])=TRUE∧>(x2[2], x0[2])=TRUE ⇒ COND_1330_0_MAIN_GE1(TRUE, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧COND_1330_0_MAIN_GE1(TRUE, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), +(x1[2], 1))∧(U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥))

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(36)    (x1[2] ≥ 0∧x2[2] + [-2] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(37)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(38)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

(39)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)
We consider the chain 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]), COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1)), 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) which results in the following constraint:
(40)    (&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0))=TRUE∧x0[2]=x0[3]∧java.lang.Object(ARRAY(x2[2]))=java.lang.Object(ARRAY(x2[3]))∧x1[2]=x1[3]∧x0[3]=x0[2]1∧java.lang.Object(ARRAY(x2[3]))=java.lang.Object(ARRAY(x2[2]1))∧+(x1[3], 1)=x1[2]1 ⇒ COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3])≥NonInfC∧COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3])≥1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))∧(U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥))

We simplified constraint (40) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(41)    (>(x1[2], 0)=TRUE∧>(x2[2], x1[2])=TRUE∧>(x2[2], x0[2])=TRUE ⇒ COND_1330_0_MAIN_GE1(TRUE, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧COND_1330_0_MAIN_GE1(TRUE, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), +(x1[2], 1))∧(U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(42)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(43)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(44)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(45)    (x1[2] ≥ 0∧x2[2] + [-2] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(46)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(47)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

(48)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), x1) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2, -1), <=(x2, x1)), >(x0, -1)), >(-(x2, 1), +(x0, 1))), x0, java.lang.Object(ARRAY(x2)), x1)
- (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])), ≥)∧[(2)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[0] ≥ 0∧[(-1)bso_25] ≥ 0)
COND_1330_0_MAIN_GE(TRUE, x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_MAIN_GE(+(x0, 1), java.lang.Object(ARRAY(x2)), +(x0, 2))
- (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
- (x2[0] ≥ 0∧x0[0] ≥ 0∧[3] + x0[0] + x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), x1) → COND_1330_0_MAIN_GE1(&&(&&(>(x2, x1), >(x2, x0)), >(x1, 0)), x0, java.lang.Object(ARRAY(x2)), x1)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[2] + [bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[2] + [bni_28]x2[2] + [bni_28]x0[2] ≥ 0∧[(-1)bso_29] ≥ 0)
COND_1330_0_MAIN_GE1(TRUE, x0, java.lang.Object(ARRAY(x2)), x1) → 1330_0_MAIN_GE(x0, java.lang.Object(ARRAY(x2)), +(x1, 1))
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] + [bni_30]x2[2] + [(-1)bni_30]x0[2] ≥ 0∧[(-1)bso_31] ≥ 0)

POL(TRUE) = [2]
POL(FALSE) = [3]
POL(1330_0_MAIN_GE(x₁, x₂, x₃)) = [-1] + [-1]x₂ + [-1]x₁
POL(java.lang.Object(x₁)) = [-1] + [-1]x₁
POL(ARRAY(x₁)) = [-1] + x₁
POL(COND_1330_0_MAIN_GE(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<=(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(2) = [2]
POL(COND_1330_0_MAIN_GE1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃ + [-1]x₂
POL(0) = 0

The following pairs are in P_>:

COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1]) → 1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))

The following pairs are in P_bound:

1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])
COND_1330_0_MAIN_GE(TRUE, x0[1], java.lang.Object(ARRAY(x2[1])), x1[1]) → 1330_0_MAIN_GE(+(x0[1], 1), java.lang.Object(ARRAY(x2[1])), +(x0[1], 2))
1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])
COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))

The following pairs are in P_≥:

1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(&&(&&(&&(>(x2[0], -1), <=(x2[0], x1[0])), >(x0[0], -1)), >(-(x2[0], 1), +(x0[0], 1))), x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])
1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])
COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹

(20) 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): 1330_0_MAIN_GE(x0[0], java.lang.Object(ARRAY(x2[0])), x1[0]) → COND_1330_0_MAIN_GE(x2[0] > -1 && x2[0] <= x1[0] && x0[0] > -1 && x2[0] - 1 > x0[0] + 1, x0[0], java.lang.Object(ARRAY(x2[0])), x1[0])
(2): 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])
(3): COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), x1[3] + 1)

(3) -> (0), if (x0[3] →^* x0[0]∧java.lang.Object(ARRAY(x2[3])) →^* java.lang.Object(ARRAY(x2[0]))∧x1[3] + 1 →^* x1[0])

(3) -> (2), if (x0[3] →^* x0[2]∧java.lang.Object(ARRAY(x2[3])) →^* java.lang.Object(ARRAY(x2[2]))∧x1[3] + 1 →^* x1[2])

(2) -> (3), if (x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0 ∧x0[2] →^* x0[3]∧java.lang.Object(ARRAY(x2[2])) →^* java.lang.Object(ARRAY(x2[3]))∧x1[2] →^* x1[3])

The set Q is empty.

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) 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:
(3): COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), x1[3] + 1)
(2): 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])

(3) -> (2), if (x0[3] →^* x0[2]∧java.lang.Object(ARRAY(x2[3])) →^* java.lang.Object(ARRAY(x2[2]))∧x1[3] + 1 →^* x1[2])

(2) -> (3), if (x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0 ∧x0[2] →^* x0[3]∧java.lang.Object(ARRAY(x2[2])) →^* java.lang.Object(ARRAY(x2[3]))∧x1[2] →^* x1[3])

The set Q is empty.

(23) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5f01aa96 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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

For Pair COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1)) the following chains were created:

We consider the chain 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]), COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1)), 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) which results in the following constraint:
(1)    (&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0))=TRUE∧x0[2]=x0[3]∧java.lang.Object(ARRAY(x2[2]))=java.lang.Object(ARRAY(x2[3]))∧x1[2]=x1[3]∧x0[3]=x0[2]1∧java.lang.Object(ARRAY(x2[3]))=java.lang.Object(ARRAY(x2[2]1))∧+(x1[3], 1)=x1[2]1 ⇒ COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3])≥NonInfC∧COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3])≥1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))∧(U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥))

We simplified constraint (1) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[2], 0)=TRUE∧>(x2[2], x1[2])=TRUE∧>(x2[2], x0[2])=TRUE ⇒ COND_1330_0_MAIN_GE1(TRUE, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧COND_1330_0_MAIN_GE1(TRUE, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), +(x1[2], 1))∧(U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[2] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[2] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[2] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[2] ≥ 0∧x2[2] + [-2] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[2] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(8)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

(9)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

For Pair 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) the following chains were created:

We consider the chain 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]), COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1)) which results in the following constraint:
(10)    (&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0))=TRUE∧x0[2]=x0[3]∧java.lang.Object(ARRAY(x2[2]))=java.lang.Object(ARRAY(x2[3]))∧x1[2]=x1[3] ⇒ 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])∧(U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥))

We simplified constraint (10) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(11)    (>(x1[2], 0)=TRUE∧>(x2[2], x1[2])=TRUE∧>(x2[2], x0[2])=TRUE ⇒ 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥NonInfC∧1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])≥COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])∧(U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥))

We simplified constraint (11) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(12)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[2] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (12) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(13)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[2] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (13) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(14)    (x1[2] + [-1] ≥ 0∧x2[2] + [-1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[2] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    (x1[2] ≥ 0∧x2[2] + [-2] + [-1]x1[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x1[2] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(17)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

(18)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x2[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)
- (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + x1[2] + x2[2] + [-1]x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x2[2] ≥ 0∧[(-1)bso_21] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = [3]
POL(COND_1330_0_MAIN_GE1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + [-1]x₃
POL(java.lang.Object(x₁)) = [-1] + [-1]x₁
POL(ARRAY(x₁)) = [-1] + x₁
POL(1330_0_MAIN_GE(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))

The following pairs are in P_bound:

COND_1330_0_MAIN_GE1(TRUE, x0[3], java.lang.Object(ARRAY(x2[3])), x1[3]) → 1330_0_MAIN_GE(x0[3], java.lang.Object(ARRAY(x2[3])), +(x1[3], 1))
1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])

The following pairs are in P_≥:

1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(&&(&&(>(x2[2], x1[2]), >(x2[2], x0[2])), >(x1[2], 0)), x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹

(24) 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:
(2): 1330_0_MAIN_GE(x0[2], java.lang.Object(ARRAY(x2[2])), x1[2]) → COND_1330_0_MAIN_GE1(x2[2] > x1[2] && x2[2] > x0[2] && x1[2] > 0, x0[2], java.lang.Object(ARRAY(x2[2])), x1[2])

The set Q is empty.

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE

(27) Obligation:

(28) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 23 rules for P and 0 rules for R.

P rules:
734_0_main_Load(EOS(STATIC_734), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181) → 736_0_main_ConstantStackPush(EOS(STATIC_736), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i50)
736_0_main_ConstantStackPush(EOS(STATIC_736), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i50) → 737_0_main_IntArithmetic(EOS(STATIC_737), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i50, 1)
737_0_main_IntArithmetic(EOS(STATIC_737), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i50, matching1) → 739_0_main_GE(EOS(STATIC_739), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, -(i50, 1)) | &&(>=(i50, 0), =(matching1, 1))
739_0_main_GE(EOS(STATIC_739), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i187) → 743_0_main_GE(EOS(STATIC_743), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i187)
743_0_main_GE(EOS(STATIC_743), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181, i187) → 746_0_main_Load(EOS(STATIC_746), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50))) | <(i181, i187)
746_0_main_Load(EOS(STATIC_746), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50))) → 750_0_main_Load(EOS(STATIC_750), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)))
750_0_main_Load(EOS(STATIC_750), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50))) → 753_0_main_Load(EOS(STATIC_753), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181)
753_0_main_Load(EOS(STATIC_753), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181) → 756_0_main_Load(EOS(STATIC_756), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(ARRAY(i50)))
756_0_main_Load(EOS(STATIC_756), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(ARRAY(i50))) → 761_0_main_ArrayAccess(EOS(STATIC_761), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(ARRAY(i50)), i181)
761_0_main_ArrayAccess(EOS(STATIC_761), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(ARRAY(i50)), i181) → 765_0_main_ArrayAccess(EOS(STATIC_765), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(ARRAY(i50)), i181)
765_0_main_ArrayAccess(EOS(STATIC_765), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(ARRAY(i50)), i181) → 771_0_main_InvokeMethod(EOS(STATIC_771), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, o65) | <(i181, i50)
771_0_main_InvokeMethod(EOS(STATIC_771), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(o68sub)) → 778_0_main_InvokeMethod(EOS(STATIC_778), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(o68sub))
778_0_main_InvokeMethod(EOS(STATIC_778), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(o68sub)) → 786_0_length_Load(EOS(STATIC_786), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(o68sub), java.lang.Object(o68sub))
786_0_length_Load(EOS(STATIC_786), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(o68sub), java.lang.Object(o68sub)) → 803_0_length_FieldAccess(EOS(STATIC_803), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(o68sub), java.lang.Object(o68sub))
803_0_length_FieldAccess(EOS(STATIC_803), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(java.lang.String(o72sub, i202)), java.lang.Object(java.lang.String(o72sub, i202))) → 810_0_length_FieldAccess(EOS(STATIC_810), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(java.lang.String(o72sub, i202)), java.lang.Object(java.lang.String(o72sub, i202))) | &&(>=(i202, 0), >=(i203, 0))
810_0_length_FieldAccess(EOS(STATIC_810), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(java.lang.String(o72sub, i202)), java.lang.Object(java.lang.String(o72sub, i202))) → 820_0_length_Return(EOS(STATIC_820), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(java.lang.String(o72sub, i202)))
820_0_length_Return(EOS(STATIC_820), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181, java.lang.Object(java.lang.String(o72sub, i202))) → 830_0_main_ArrayAccess(EOS(STATIC_830), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181)
830_0_main_ArrayAccess(EOS(STATIC_830), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181) → 838_0_main_ArrayAccess(EOS(STATIC_838), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181)
838_0_main_ArrayAccess(EOS(STATIC_838), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), java.lang.Object(ARRAY(i50)), i181) → 853_0_main_Inc(EOS(STATIC_853), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50))) | <(i181, i50)
853_0_main_Inc(EOS(STATIC_853), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50))) → 869_0_main_JMP(EOS(STATIC_869), java.lang.Object(ARRAY(i50)), +(i181, 1), i50, java.lang.Object(ARRAY(i50))) | >=(i181, 0)
869_0_main_JMP(EOS(STATIC_869), java.lang.Object(ARRAY(i50)), i218, i50, java.lang.Object(ARRAY(i50))) → 882_0_main_Load(EOS(STATIC_882), java.lang.Object(ARRAY(i50)), i218, i50, java.lang.Object(ARRAY(i50)))
882_0_main_Load(EOS(STATIC_882), java.lang.Object(ARRAY(i50)), i218, i50, java.lang.Object(ARRAY(i50))) → 728_0_main_Load(EOS(STATIC_728), java.lang.Object(ARRAY(i50)), i218, i50, java.lang.Object(ARRAY(i50)))
728_0_main_Load(EOS(STATIC_728), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50))) → 734_0_main_Load(EOS(STATIC_734), java.lang.Object(ARRAY(i50)), i181, i50, java.lang.Object(ARRAY(i50)), i181)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
734_0_main_Load(EOS(STATIC_734), java.lang.Object(ARRAY(x0)), x1, x0, java.lang.Object(ARRAY(x0)), x1) → 734_0_main_Load(EOS(STATIC_734), java.lang.Object(ARRAY(x0)), +(x1, 1), x0, java.lang.Object(ARRAY(x0)), +(x1, 1)) | &&(&&(&&(>(+(x1, 1), 0), <(x1, x0)), <(x1, -(x0, 1))), >(+(x0, 1), 0))
R rules:

Filtered ground terms:

734_0_main_Load(x1, x2, x3, x4, x5, x6) → 734_0_main_Load(x2, x3, x4, x5, x6)
EOS(x1) → EOS
Cond_734_0_main_Load(x1, x2, x3, x4, x5, x6, x7) → Cond_734_0_main_Load(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

734_0_main_Load(x1, x2, x3, x4, x5) → 734_0_main_Load(x4, x5)
Cond_734_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_734_0_main_Load(x1, x5, x6)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
734_0_main_Load(java.lang.Object(ARRAY(x0)), x1) → 734_0_main_Load(java.lang.Object(ARRAY(x0)), +(x1, 1)) | &&(&&(&&(>(x1, -1), <(x1, x0)), <(x1, -(x0, 1))), >(x0, -1))
R rules:

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

P rules:
734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1, -1), <(x1, x0)), <(x1, -(x0, 1))), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1)
COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1))
R rules:

(29) 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): 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_734_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0] && x1[0] < x0[0] - 1 && x0[0] > -1, java.lang.Object(ARRAY(x0[0])), x1[0])
(1): COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), x1[1] + 1)

(0) -> (1), if (x1[0] > -1 && x1[0] < x0[0] && x1[0] < x0[0] - 1 && x0[0] > -1 ∧java.lang.Object(ARRAY(x0[0])) →^* java.lang.Object(ARRAY(x0[1]))∧x1[0] →^* x1[1])

(1) -> (0), if (java.lang.Object(ARRAY(x0[1])) →^* java.lang.Object(ARRAY(x0[0]))∧x1[1] + 1 →^* x1[0])

The set Q is empty.

(30) 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@79711174 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 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1, -1), <(x1, x0)), <(x1, -(x0, 1))), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1) the following chains were created:

We consider the chain 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0]), COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:
(1)    (&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1))=TRUE∧java.lang.Object(ARRAY(x0[0]))=java.lang.Object(ARRAY(x0[1]))∧x1[0]=x1[1] ⇒ 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥NonInfC∧734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])∧(U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], -1)=TRUE∧<(x1[0], -(x0[0], 1))=TRUE∧>(x1[0], -1)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥NonInfC∧734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])∧(U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    ([2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(3)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1)) the following chains were created:

We consider the chain COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:
(7)    (COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1])≥NonInfC∧COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1])≥734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))∧(U^Increasing(734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1, -1), <(x1, x0)), <(x1, -(x0, 1))), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1)
- ([2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (U^Increasing(COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(3)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1))
- ((U^Increasing(734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(734_0_MAIN_LOAD(x₁, x₂)) = [-1] + [-1]x₂ + [2]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁)) = x₁
POL(COND_734_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂

The following pairs are in P_>:

COND_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))

The following pairs are in P_bound:

734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])

The following pairs are in P_≥:

734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_734_0_MAIN_LOAD(&&(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), <(x1[0], -(x0[0], 1))), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])

There are no usable rules.

(31) Complex Obligation (AND)

(32) 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

(33) IDependencyGraphProof (EQUIVALENT transformation)

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

(34) TRUE

(35) 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_734_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 734_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), x1[1] + 1)

The set Q is empty.

(36) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) SCCToIDPv1Proof (SOUND transformation)

(7) Obligation:

(8) IDPNonInfProof (SOUND transformation)

(9) Complex Obligation (AND)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) SCCToIDPv1Proof (SOUND transformation)

(18) Obligation:

(19) IDPNonInfProof (SOUND transformation)

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) Obligation:

(23) IDPNonInfProof (SOUND transformation)

(24) Obligation:

(25) IDependencyGraphProof (EQUIVALENT transformation)

(26) TRUE

(27) Obligation:

(28) SCCToIDPv1Proof (SOUND transformation)

(29) Obligation:

(30) IDPNonInfProof (SOUND transformation)

(31) Complex Obligation (AND)

(32) Obligation:

(33) IDependencyGraphProof (EQUIVALENT transformation)

(34) TRUE

(35) Obligation:

(36) IDependencyGraphProof (EQUIVALENT transformation)

(37) TRUE