### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: CyclicList
`/** * This class represents a list. The function get(n) can be used to access * the n-th element.  * @author Marc Brockschmidt */public class CyclicList {	/**	 * A reference to the next list element.	 */	private CyclicList next;		public static void main(String[] args) {		CyclicList list = CyclicList.create(args.length);		list.get(args[0].length());	}		/**	 * Create a new list element.	 * @param n a reference to the next element.	 */	public CyclicList(final CyclicList n) {		this.next = n;	}		/**	 * Create a new cyclical list of a length l.	 * @param l some length	 * @return cyclical list of length max(1, l)	 */	public static CyclicList create(int x) {		CyclicList last, current;		last = current = new CyclicList(null);		while (--x > 0)			current = new CyclicList(current);		return last.next = current;	}	public CyclicList get(int n) {		CyclicList cur = this;		while (--n > 0) {			cur = cur.next;		}		return cur;	}	}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
CyclicList.main([Ljava/lang/String;)V: Graph of 246 nodes with 2 SCCs.

### (3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 2 SCCss.

### (5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: CyclicList.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 18 rules for P and 0 rules for R.

P rules:
680_0_get_Load(EOS(STATIC_680), java.lang.Object(o291sub0), i169, o2900) → 694_0_get_LE(EOS(STATIC_694), java.lang.Object(o291sub0), i169, o2900, i169)
694_0_get_LE(EOS(STATIC_694), java.lang.Object(o291sub0), i179, o2900, i179) → 710_0_get_LE(EOS(STATIC_710), java.lang.Object(o291sub0), i179, o2900, i179)
710_0_get_LE(EOS(STATIC_710), java.lang.Object(o291sub0), i179, o2900, i179) → 727_0_get_Load(EOS(STATIC_727), java.lang.Object(o291sub0), i179, o2900) | >(i179, 0)
727_0_get_Load(EOS(STATIC_727), java.lang.Object(o291sub0), i179, o2900) → 749_0_get_FieldAccess(EOS(STATIC_749), java.lang.Object(o291sub0), i179, o2900)
749_0_get_FieldAccess(EOS(STATIC_749), java.lang.Object(o291sub0), i179, java.lang.Object(o359sub0)) → 765_0_get_FieldAccess(EOS(STATIC_765), java.lang.Object(o291sub0), i179, java.lang.Object(o359sub0))
765_0_get_FieldAccess(EOS(STATIC_765), java.lang.Object(o291sub0), i179, java.lang.Object(o359sub0)) → 782_0_get_FieldAccess(EOS(STATIC_782), java.lang.Object(o291sub0), i179, java.lang.Object(o359sub0))
765_0_get_FieldAccess(EOS(STATIC_765), java.lang.Object(o291sub0), i179, java.lang.Object(o291sub0)) → 783_0_get_FieldAccess(EOS(STATIC_783), java.lang.Object(o291sub0), i179, java.lang.Object(o291sub0))
782_0_get_FieldAccess(EOS(STATIC_782), java.lang.Object(o291sub0), i179, java.lang.Object(CyclicList(EOC, o3742125973745))) → 792_0_get_FieldAccess(EOS(STATIC_792), java.lang.Object(o291sub0), i179, java.lang.Object(CyclicList(EOC, o3742125973745)))
792_0_get_FieldAccess(EOS(STATIC_792), java.lang.Object(o291sub0), i179, java.lang.Object(CyclicList(EOC, o3742125973745))) → 797_0_get_Store(EOS(STATIC_797), java.lang.Object(o291sub0), i179, o3740)
797_0_get_Store(EOS(STATIC_797), java.lang.Object(o291sub0), i179, o3740) → 802_0_get_JMP(EOS(STATIC_802), java.lang.Object(o291sub0), i179, o3740)
802_0_get_JMP(EOS(STATIC_802), java.lang.Object(o291sub0), i179, o3740) → 811_0_get_Inc(EOS(STATIC_811), java.lang.Object(o291sub0), i179, o3740)
811_0_get_Inc(EOS(STATIC_811), java.lang.Object(o291sub0), i179, o3740) → 665_0_get_Inc(EOS(STATIC_665), java.lang.Object(o291sub0), i179, o3740)
665_0_get_Inc(EOS(STATIC_665), java.lang.Object(o291sub0), i160, o2900) → 680_0_get_Load(EOS(STATIC_680), java.lang.Object(o291sub0), +(i160, -1), o2900) | >=(i160, 0)
783_0_get_FieldAccess(EOS(STATIC_783), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, java.lang.Object(CyclicList(EOC, o3762125973807))) → 793_0_get_FieldAccess(EOS(STATIC_793), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, java.lang.Object(CyclicList(EOC, o3762125973807)))
793_0_get_FieldAccess(EOS(STATIC_793), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, java.lang.Object(CyclicList(EOC, o3762125973807))) → 800_0_get_Store(EOS(STATIC_800), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760)
800_0_get_Store(EOS(STATIC_800), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760) → 804_0_get_JMP(EOS(STATIC_804), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760)
804_0_get_JMP(EOS(STATIC_804), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760) → 814_0_get_Inc(EOS(STATIC_814), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760)
814_0_get_Inc(EOS(STATIC_814), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760) → 665_0_get_Inc(EOS(STATIC_665), java.lang.Object(CyclicList(EOC, o3762125973807)), i179, o3760)
R rules:

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

P rules:
680_0_get_Load(EOS(STATIC_680), java.lang.Object(x0), x1, java.lang.Object(CyclicList(EOC, x2))) → 680_0_get_Load(EOS(STATIC_680), java.lang.Object(x0), +(x1, -1), x3) | >(x1, 0)
680_0_get_Load(EOS(STATIC_680), java.lang.Object(CyclicList(EOC, x0)), x1, java.lang.Object(CyclicList(EOC, x0))) → 680_0_get_Load(EOS(STATIC_680), java.lang.Object(CyclicList(EOC, x0)), +(x1, -1), x2) | >(x1, 0)
R rules:

Filtered ground terms:

CyclicList(x1, x2) → CyclicList(x2)
EOS(x1) → EOS
Cond_680_0_get_Load1(x1, x2, x3, x4, x5, x6) → Cond_680_0_get_Load1(x1, x3, x4, x5, x6)
Cond_680_0_get_Load(x1, x2, x3, x4, x5, x6) → Cond_680_0_get_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

Filtered unneeded arguments:

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

P rules:
R rules:

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

P rules:
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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(x1[0] > 0, java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])
(1): COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), x1[1] + -1, x3[1])
(2): 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(x1[2] > 0, java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])
(3): COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), x1[3] + -1, x2[3])

(0) -> (1), if (x1[0] > 0java.lang.Object(x0[0]) →* java.lang.Object(x0[1])∧x1[0]* x1[1]java.lang.Object(CyclicList(x2[0])) →* java.lang.Object(CyclicList(x2[1]))∧x3[0]* x3[1])

(1) -> (0), if (java.lang.Object(x0[1]) →* java.lang.Object(x0[0])∧x1[1] + -1* x1[0]x3[1]* java.lang.Object(CyclicList(x2[0])))

(1) -> (2), if (java.lang.Object(x0[1]) →* java.lang.Object(CyclicList(x0[2]))∧x1[1] + -1* x1[2]x3[1]* java.lang.Object(CyclicList(x0[2])))

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

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

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

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.IdpDefaultShapeHeuristic@c3e122 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 680_0_GET_LOAD(java.lang.Object(x0), x1, java.lang.Object(CyclicList(x2))) → COND_680_0_GET_LOAD(>(x1, 0), java.lang.Object(x0), x1, java.lang.Object(CyclicList(x2)), x3) the following chains were created:
• We consider the chain COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]), 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]), COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]) which results in the following constraint:

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

(2)    (>(+(x1[1], -1), 0)=TRUE680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), java.lang.Object(CyclicList(x2[0])))≥NonInfC∧680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), java.lang.Object(CyclicList(x2[0])))≥COND_680_0_GET_LOAD(>(+(x1[1], -1), 0), java.lang.Object(x0[1]), +(x1[1], -1), java.lang.Object(CyclicList(x2[0])), x3[0])∧(UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥))

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

(3)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x1[1] + [(-2)bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(4)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x1[1] + [(-2)bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(5)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x1[1] + [(-2)bni_24]x0[1] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(6)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧[(-2)bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x1[1] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

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

(7)    (x1[1] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧[(-2)bni_24] = 0∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[1] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

• We consider the chain COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3]), 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]), COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]) which results in the following constraint:

We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint:

(9)    (>(+(x1[3], -1), 0)=TRUE680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), java.lang.Object(CyclicList(x2[0])))≥NonInfC∧680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), java.lang.Object(CyclicList(x2[0])))≥COND_680_0_GET_LOAD(>(+(x1[3], -1), 0), java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), java.lang.Object(CyclicList(x2[0])), x3[0])∧(UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥))

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

(10)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[3] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(11)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[3] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(12)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[3] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(13)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧0 = 0∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[3] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

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

(14)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧0 = 0∧[(3)bni_24 + (-1)Bound*bni_24] + [bni_24]x1[3] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

For Pair COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0), x1, java.lang.Object(CyclicList(x2)), x3) → 680_0_GET_LOAD(java.lang.Object(x0), +(x1, -1), x3) the following chains were created:
• We consider the chain 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]), COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]), 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]) which results in the following constraint:

We simplified constraint (15) using rules (I), (II), (III), (IV) which results in the following new constraint:

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

(17)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[0] + [(-2)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)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[0] + [(-2)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)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[0] + [(-2)bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

(20)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧[(-2)bni_26] = 0∧[(-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

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

(21)    (x1[0] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧[(-2)bni_26] = 0∧[(-1)Bound*bni_26 + bni_26] + [bni_26]x1[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

• We consider the chain 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]), COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]), 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2]) which results in the following constraint:

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

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

(24)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

(25)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

(26)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧[1 + (-1)bso_27] ≥ 0)

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

(27)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧[(2)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

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

(28)    (x1[0] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

For Pair 680_0_GET_LOAD(java.lang.Object(CyclicList(x0)), x1, java.lang.Object(CyclicList(x0))) → COND_680_0_GET_LOAD1(>(x1, 0), java.lang.Object(CyclicList(x0)), x1, java.lang.Object(CyclicList(x0)), x2) the following chains were created:
• We consider the chain COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]), 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2]), COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3]) which results in the following constraint:

We simplified constraint (29) using rules (I), (II), (III), (IV) which results in the following new constraint:

(30)    (>(+(x1[1], -1), 0)=TRUE680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), +(x1[1], -1), java.lang.Object(CyclicList(x0[2])))≥NonInfC∧680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), +(x1[1], -1), java.lang.Object(CyclicList(x0[2])))≥COND_680_0_GET_LOAD1(>(+(x1[1], -1), 0), java.lang.Object(CyclicList(x0[2])), +(x1[1], -1), java.lang.Object(CyclicList(x0[2])), x2[2])∧(UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥))

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

(31)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[1] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(32)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[1] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(33)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[1] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(34)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧0 = 0∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(35)    (x1[1] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧0 = 0∧[(3)bni_28 + (-1)Bound*bni_28] + [bni_28]x1[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

• We consider the chain COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3]), 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2]), COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3]) which results in the following constraint:

We simplified constraint (36) using rules (I), (II), (III), (IV) which results in the following new constraint:

(37)    (>(+(x1[3], -1), 0)=TRUE680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), java.lang.Object(CyclicList(x0[3])))≥NonInfC∧680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), java.lang.Object(CyclicList(x0[3])))≥COND_680_0_GET_LOAD1(>(+(x1[3], -1), 0), java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), java.lang.Object(CyclicList(x0[3])), x2[2])∧(UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥))

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

(38)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[3] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(39)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[3] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(40)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[3] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(41)    (x1[3] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧0 = 0∧[bni_28 + (-1)Bound*bni_28] + [bni_28]x1[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

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

(42)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧0 = 0∧[(3)bni_28 + (-1)Bound*bni_28] + [bni_28]x1[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

For Pair COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0)), x1, java.lang.Object(CyclicList(x0)), x2) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0)), +(x1, -1), x2) the following chains were created:
• We consider the chain 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2]), COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3]), 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]) which results in the following constraint:

We simplified constraint (43) using rules (I), (II), (III), (IV) which results in the following new constraint:

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

(45)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧[1 + (-1)bso_31] ≥ 0)

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

(46)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(47)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (47) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(48)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧0 = 0∧0 = 0∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(49)    (x1[2] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

• We consider the chain 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2]), COND_680_0_GET_LOAD1(TRUE, java.lang.Object(CyclicList(x0[3])), x1[3], java.lang.Object(CyclicList(x0[3])), x2[3]) → 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3]), 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2]) which results in the following constraint:

We simplified constraint (50) using rules (I), (II), (III), (IV) which results in the following new constraint:

We simplified constraint (51) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(52)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(53)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(54)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (54) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(55)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧0 = 0∧[(2)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (55) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(56)    (x1[2] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (x1[1] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧[(-2)bni_24] = 0∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[1] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)
• (x1[3] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧0 = 0∧[(3)bni_24 + (-1)Bound*bni_24] + [bni_24]x1[3] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

• (x1[0] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧[(-2)bni_26] = 0∧[(-1)Bound*bni_26 + bni_26] + [bni_26]x1[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧[(3)bni_26 + (-1)Bound*bni_26] + [bni_26]x1[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

• (x1[1] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧0 = 0∧[(3)bni_28 + (-1)Bound*bni_28] + [bni_28]x1[1] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
• (x1[3] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD1(>(x1[2], 0), java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])), ≥)∧0 = 0∧[(3)bni_28 + (-1)Bound*bni_28] + [bni_28]x1[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

• (x1[2] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧0 = 0∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)
• (x1[2] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(CyclicList(x0[3])), +(x1[3], -1), x2[3])), ≥)∧0 = 0∧[(3)bni_30 + (-1)Bound*bni_30] + [bni_30]x1[2] ≥ 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

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

POL(TRUE) = [2]
POL(FALSE) = 0
POL(680_0_GET_LOAD(x1, x2, x3)) = [1] + [-1]x3 + x2 + [2]x1
POL(java.lang.Object(x1)) = [-1]x1
POL(CyclicList(x1)) = [-1]
POL(COND_680_0_GET_LOAD(x1, x2, x3, x4, x5)) = [-1] + x4 + x3 + [2]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_680_0_GET_LOAD1(x1, x2, x3, x4, x5)) = x4 + x3 + x2

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (10) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(x1[0] > 0, java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])
(2): 680_0_GET_LOAD(java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2]))) → COND_680_0_GET_LOAD1(x1[2] > 0, java.lang.Object(CyclicList(x0[2])), x1[2], java.lang.Object(CyclicList(x0[2])), x2[2])

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

### (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:
(0): 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(x1[0] > 0, java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])
(1): COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), x1[1] + -1, x3[1])

(1) -> (0), if (java.lang.Object(x0[1]) →* java.lang.Object(x0[0])∧x1[1] + -1* x1[0]x3[1]* java.lang.Object(CyclicList(x2[0])))

(0) -> (1), if (x1[0] > 0java.lang.Object(x0[0]) →* java.lang.Object(x0[1])∧x1[0]* x1[1]java.lang.Object(CyclicList(x2[0])) →* java.lang.Object(CyclicList(x2[1]))∧x3[0]* x3[1])

The set Q is empty.

### (14) 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@c3e122 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 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]) the following chains were created:
• We consider the chain COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]), 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]), COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]) which results in the following constraint:

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

(2)    (>(+(x1[1], -1), 0)=TRUE680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), java.lang.Object(CyclicList(x2[0])))≥NonInfC∧680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), java.lang.Object(CyclicList(x2[0])))≥COND_680_0_GET_LOAD(>(+(x1[1], -1), 0), java.lang.Object(x0[1]), +(x1[1], -1), java.lang.Object(CyclicList(x2[0])), x3[0])∧(UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥))

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

(3)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x1[1] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(4)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x1[1] ≥ 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[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x1[1] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(6)    (x1[1] + [-2] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧0 = 0∧[bni_18 + (-1)Bound*bni_18] + [bni_18]x1[1] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

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

(7)    (x1[1] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[1] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

For Pair COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]) the following chains were created:
• We consider the chain 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]), COND_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1]), 680_0_GET_LOAD(java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0]))) → COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0]) which results in the following constraint:

We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint:

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

(10)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(11)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(12)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(13)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

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

(14)    (x1[0] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (x1[1] ≥ 0 ⇒ (UIncreasing(COND_680_0_GET_LOAD(>(x1[0], 0), java.lang.Object(x0[0]), x1[0], java.lang.Object(CyclicList(x2[0])), x3[0])), ≥)∧0 = 0∧0 = 0∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[1] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

• (x1[0] ≥ 0 ⇒ (UIncreasing(680_0_GET_LOAD(java.lang.Object(x0[1]), +(x1[1], -1), x3[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(2)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(680_0_GET_LOAD(x1, x2, x3)) = [1] + x2 + [-1]x1
POL(java.lang.Object(x1)) = [-1]
POL(CyclicList(x1)) = [2] + [-1]x1
POL(COND_680_0_GET_LOAD(x1, x2, x3, x4, x5)) = [2] + x4 + x3
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (15) 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_680_0_GET_LOAD(TRUE, java.lang.Object(x0[1]), x1[1], java.lang.Object(CyclicList(x2[1])), x3[1]) → 680_0_GET_LOAD(java.lang.Object(x0[1]), x1[1] + -1, x3[1])

The set Q is empty.

### (16) IDependencyGraphProof (EQUIVALENT transformation)

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

### (18) Obligation:

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

### (19) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 17 rules for P and 0 rules for R.

P rules:
181_0_create_Load(EOS(STATIC_181), i23) → 183_0_create_LE(EOS(STATIC_183), i23, i23)
183_0_create_LE(EOS(STATIC_183), i27, i27) → 186_0_create_LE(EOS(STATIC_186), i27, i27)
186_0_create_LE(EOS(STATIC_186), i27, i27) → 190_0_create_New(EOS(STATIC_190), i27) | >(i27, 0)
190_0_create_New(EOS(STATIC_190), i27) → 194_0_create_Duplicate(EOS(STATIC_194), i27)
262_0_<init>_FieldAccess(EOS(STATIC_262), i27) → 275_0_<init>_Return(EOS(STATIC_275), i27)
275_0_<init>_Return(EOS(STATIC_275), i27) → 287_0_create_Store(EOS(STATIC_287), i27)
287_0_create_Store(EOS(STATIC_287), i27) → 315_0_create_JMP(EOS(STATIC_315), i27)
315_0_create_JMP(EOS(STATIC_315), i27) → 327_0_create_Inc(EOS(STATIC_327), i27)
327_0_create_Inc(EOS(STATIC_327), i27) → 176_0_create_Inc(EOS(STATIC_176), i27)
176_0_create_Inc(EOS(STATIC_176), i19) → 181_0_create_Load(EOS(STATIC_181), +(i19, -1)) | >=(i19, 0)
R rules:

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

P rules:
R rules:

Filtered ground terms:

EOS(x1) → EOS

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

P rules:
R rules:

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

P rules:
R rules:

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

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

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

The set Q is empty.

### (21) 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@4e9599e9 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 181_0_CREATE_LOAD(x0) → COND_181_0_CREATE_LOAD(>(x0, 0), x0) the following chains were created:

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

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_181_0_CREATE_LOAD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_181_0_CREATE_LOAD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_181_0_CREATE_LOAD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_181_0_CREATE_LOAD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_181_0_CREATE_LOAD(TRUE, x0) → 181_0_CREATE_LOAD(+(x0, -1)) the following chains were created:
• We consider the chain COND_181_0_CREATE_LOAD(TRUE, x0[1]) → 181_0_CREATE_LOAD(+(x0[1], -1)) which results in the following constraint:

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

(8)    ((UIncreasing(181_0_CREATE_LOAD(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(9)    ((UIncreasing(181_0_CREATE_LOAD(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(10)    ((UIncreasing(181_0_CREATE_LOAD(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(11)    ((UIncreasing(181_0_CREATE_LOAD(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_181_0_CREATE_LOAD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• ((UIncreasing(181_0_CREATE_LOAD(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (23) 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:

The set Q is empty.

### (24) IDependencyGraphProof (EQUIVALENT transformation)

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

### (26) 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: