### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Take
`/** * Java can do infinite data objects, too. * Here we take the first n elements from an * ascending infinite list of integer numbers. * * @author Carsten Fuhs */public class Take {    public static int[] take(int n, MyIterator f) {        int[] result = new int[n];        for (int i = 0; i < n; ++i) {            if (f.hasNext()) {                result[i] = f.next();            }            else {                break;            }        }        return result;    }    public static void main(String args[]) {        int start = args[0].length();        int howMany = args[1].length();        From f = new From(start);        int[] firstHowMany = take(howMany, f);    }}interface MyIterator {    boolean hasNext();    int next();}class From implements MyIterator {    private int current;    public From(int start) {        this.current = start;    }    public boolean hasNext() {        return true;    }    public int next() {        return current++;    }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
Take.main([Ljava/lang/String;)V: Graph of 218 nodes with 1 SCC.

### (3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

### (4) Obligation:

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

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 26 rules for P and 0 rules for R.

P rules:
651_0_take_Load(EOS(STATIC_651), i47, i47, java.lang.Object(ARRAY(i47)), i80, i80) → 653_0_take_GE(EOS(STATIC_653), i47, i47, java.lang.Object(ARRAY(i47)), i80, i80, i47)
653_0_take_GE(EOS(STATIC_653), i47, i47, java.lang.Object(ARRAY(i47)), i80, i80, i47) → 656_0_take_GE(EOS(STATIC_656), i47, i47, java.lang.Object(ARRAY(i47)), i80, i80, i47)
656_0_take_GE(EOS(STATIC_656), i47, i47, java.lang.Object(ARRAY(i47)), i80, i80, i47) → 659_0_take_Load(EOS(STATIC_659), i47, i47, java.lang.Object(ARRAY(i47)), i80) | <(i80, i47)
659_0_take_Load(EOS(STATIC_659), i47, i47, java.lang.Object(ARRAY(i47)), i80) → 662_0_take_InvokeMethod(EOS(STATIC_662), i47, i47, java.lang.Object(ARRAY(i47)), i80)
662_0_take_InvokeMethod(EOS(STATIC_662), i47, i47, java.lang.Object(ARRAY(i47)), i80) → 665_0_hasNext_ConstantStackPush(EOS(STATIC_665), i47, i47, java.lang.Object(ARRAY(i47)), i80)
665_0_hasNext_ConstantStackPush(EOS(STATIC_665), i47, i47, java.lang.Object(ARRAY(i47)), i80) → 670_0_hasNext_Return(EOS(STATIC_670), i47, i47, java.lang.Object(ARRAY(i47)), i80, 1)
670_0_hasNext_Return(EOS(STATIC_670), i47, i47, java.lang.Object(ARRAY(i47)), i80, matching1) → 671_0_take_EQ(EOS(STATIC_671), i47, i47, java.lang.Object(ARRAY(i47)), i80, 1) | =(matching1, 1)
671_0_take_EQ(EOS(STATIC_671), i47, i47, java.lang.Object(ARRAY(i47)), i80, matching1) → 673_0_take_Load(EOS(STATIC_673), i47, i47, java.lang.Object(ARRAY(i47)), i80) | &&(>(1, 0), =(matching1, 1))
673_0_take_Load(EOS(STATIC_673), i47, i47, java.lang.Object(ARRAY(i47)), i80) → 675_0_take_Load(EOS(STATIC_675), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)))
675_0_take_Load(EOS(STATIC_675), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47))) → 676_0_take_Load(EOS(STATIC_676), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
676_0_take_Load(EOS(STATIC_676), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 678_0_take_InvokeMethod(EOS(STATIC_678), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
678_0_take_InvokeMethod(EOS(STATIC_678), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 680_0_next_Load(EOS(STATIC_680), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
680_0_next_Load(EOS(STATIC_680), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 681_0_next_Duplicate(EOS(STATIC_681), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
681_0_next_Duplicate(EOS(STATIC_681), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 683_0_next_FieldAccess(EOS(STATIC_683), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
683_0_next_FieldAccess(EOS(STATIC_683), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 685_0_next_Duplicate(EOS(STATIC_685), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
685_0_next_Duplicate(EOS(STATIC_685), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 687_0_next_ConstantStackPush(EOS(STATIC_687), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
687_0_next_ConstantStackPush(EOS(STATIC_687), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 689_0_next_IntArithmetic(EOS(STATIC_689), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80, 1)
689_0_next_IntArithmetic(EOS(STATIC_689), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80, matching1) → 692_0_next_FieldAccess(EOS(STATIC_692), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) | =(matching1, 1)
692_0_next_FieldAccess(EOS(STATIC_692), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 695_0_next_Return(EOS(STATIC_695), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
695_0_next_Return(EOS(STATIC_695), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 698_0_take_ArrayAccess(EOS(STATIC_698), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
698_0_take_ArrayAccess(EOS(STATIC_698), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 700_0_take_ArrayAccess(EOS(STATIC_700), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80)
700_0_take_ArrayAccess(EOS(STATIC_700), i47, i47, java.lang.Object(ARRAY(i47)), i80, java.lang.Object(ARRAY(i47)), i80) → 704_0_take_Inc(EOS(STATIC_704), i47, i47, java.lang.Object(ARRAY(i47)), i80) | <(i80, i47)
704_0_take_Inc(EOS(STATIC_704), i47, i47, java.lang.Object(ARRAY(i47)), i80) → 709_0_take_JMP(EOS(STATIC_709), i47, i47, java.lang.Object(ARRAY(i47)), +(i80, 1)) | >=(i80, 0)
709_0_take_JMP(EOS(STATIC_709), i47, i47, java.lang.Object(ARRAY(i47)), i86) → 726_0_take_Load(EOS(STATIC_726), i47, i47, java.lang.Object(ARRAY(i47)), i86)
649_0_take_Load(EOS(STATIC_649), i47, i47, java.lang.Object(ARRAY(i47)), i80) → 651_0_take_Load(EOS(STATIC_651), i47, i47, java.lang.Object(ARRAY(i47)), i80, i80)
R rules:

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

P rules:
651_0_take_Load(EOS(STATIC_651), x0, x0, java.lang.Object(ARRAY(x0)), x1, x1) → 651_0_take_Load(EOS(STATIC_651), x0, x0, java.lang.Object(ARRAY(x0)), +(x1, 1), +(x1, 1)) | &&(>(+(x1, 1), 0), <(x1, x0))
R rules:

Filtered ground terms:

651_0_take_Load(x1, x2, x3, x4, x5, x6) → 651_0_take_Load(x2, x3, x4, x5, x6)
EOS(x1) → EOS
Cond_651_0_take_Load(x1, x2, x3, x4, x5, x6, x7) → Cond_651_0_take_Load(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

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:

### (6) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_651_0_TAKE_LOAD(x1[0] > -1 && x1[0] < x0[0], java.lang.Object(ARRAY(x0[0])), x1[0])

(0) -> (1), if (x1[0] > -1 && x1[0] < x0[0]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.

### (7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@588a137c 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 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_651_0_TAKE_LOAD(&&(>(x1, -1), <(x1, x0)), java.lang.Object(ARRAY(x0)), x1) the following chains were created:
• We consider the chain 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_651_0_TAKE_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), java.lang.Object(ARRAY(x0[0])), x1[0]), COND_651_0_TAKE_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:

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

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

(3)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_651_0_TAKE_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), 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)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_651_0_TAKE_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), 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)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_651_0_TAKE_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), 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)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_651_0_TAKE_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_651_0_TAKE_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1)) the following chains were created:
• We consider the chain COND_651_0_TAKE_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[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(651_0_TAKE_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)    ((UIncreasing(651_0_TAKE_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)    ((UIncreasing(651_0_TAKE_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)    ((UIncreasing(651_0_TAKE_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.
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_651_0_TAKE_LOAD(&&(>(x1[0], -1), <(x1[0], x0[0])), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• ((UIncreasing(651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 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(651_0_TAKE_LOAD(x1, x2)) = [-1] + [-1]x2 + [2]x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1)) = x1
POL(COND_651_0_TAKE_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 + 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.

### (9) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 651_0_TAKE_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_651_0_TAKE_LOAD(x1[0] > -1 && x1[0] < x0[0], java.lang.Object(ARRAY(x0[0])), x1[0])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (12) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges: