### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: AProVEMath
`/** * Abstract class to provide some additional mathematical functions * which are not provided by java.lang.Math. * * @author fuhs */public abstract class AProVEMath {  /**   * Returns <code>base<sup>exponent</sup></code>.   * Works considerably faster than java.lang.Math.pow(double, double).   *   * @param base base of the power   * @param exponent non-negative exponent of the power   * @return base<sup>exponent</sup>   */  public static int power (int base, int exponent) {    if (exponent == 0) {      return 1;    }    else if (exponent == 1) {      return base;    }    else if (base == 2) {      return base << (exponent-1);    }    else {      int result = 1;      while (exponent > 0) {        if (exponent % 2 == 1) {          result *= base;        }        base *= base;        exponent /= 2;      }      return result;    }  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    power(x, y);  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
AProVEMath.main([Ljava/lang/String;)V: Graph of 235 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: AProVEMath.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:
665_0_power_LE(EOS(STATIC_665), i145, i145) → 668_0_power_LE(EOS(STATIC_668), i145, i145)
668_0_power_LE(EOS(STATIC_668), i145, i145) → 672_0_power_Load(EOS(STATIC_672), i145) | >(i145, 0)
672_0_power_Load(EOS(STATIC_672), i145) → 676_0_power_ConstantStackPush(EOS(STATIC_676), i145, i145)
676_0_power_ConstantStackPush(EOS(STATIC_676), i145, i145) → 681_0_power_IntArithmetic(EOS(STATIC_681), i145, i145, 2)
681_0_power_IntArithmetic(EOS(STATIC_681), i145, i145, matching1) → 686_0_power_ConstantStackPush(EOS(STATIC_686), i145) | =(matching1, 2)
686_0_power_ConstantStackPush(EOS(STATIC_686), i145) → 690_0_power_NE(EOS(STATIC_690), i145)
690_0_power_NE(EOS(STATIC_690), i145) → 692_0_power_NE(EOS(STATIC_692), i145)
690_0_power_NE(EOS(STATIC_690), i145) → 693_0_power_NE(EOS(STATIC_693), i145)
719_0_power_IntArithmetic(EOS(STATIC_719), i145) → 721_0_power_Store(EOS(STATIC_721), i145)
725_0_power_ConstantStackPush(EOS(STATIC_725), i145) → 727_0_power_IntArithmetic(EOS(STATIC_727), i145, 2)
727_0_power_IntArithmetic(EOS(STATIC_727), i145, matching1) → 729_0_power_Store(EOS(STATIC_729), /(i145, 2)) | &&(>=(i145, 1), =(matching1, 2))
729_0_power_Store(EOS(STATIC_729), i159) → 731_0_power_JMP(EOS(STATIC_731), i159)
661_0_power_Load(EOS(STATIC_661), i138) → 665_0_power_LE(EOS(STATIC_665), i138, i138)
705_0_power_IntArithmetic(EOS(STATIC_705), i145) → 709_0_power_Store(EOS(STATIC_709), i145)
R rules:

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

P rules:
665_0_power_LE(EOS(STATIC_665), x0, x0) → 665_0_power_LE(EOS(STATIC_665), /(x0, 2), /(x0, 2)) | >(+(x0, 1), 1)
R rules:

Filtered ground terms:

665_0_power_LE(x1, x2, x3) → 665_0_power_LE(x2, x3)
EOS(x1) → EOS
Cond_665_0_power_LE(x1, x2, x3, x4) → Cond_665_0_power_LE(x1, x3, x4)

Filtered duplicate args:

665_0_power_LE(x1, x2) → 665_0_power_LE(x2)
Cond_665_0_power_LE(x1, x2, x3) → Cond_665_0_power_LE(x1, x3)

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

P rules:
665_0_power_LE(x0) → 665_0_power_LE(/(x0, 2)) | >(x0, 0)
R rules:

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

P rules:
665_0_POWER_LE(x0) → COND_665_0_POWER_LE(>(x0, 0), x0)
COND_665_0_POWER_LE(TRUE, x0) → 665_0_POWER_LE(/(x0, 2))
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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(x0[0] > 0, x0[0])
(1): COND_665_0_POWER_LE(TRUE, x0[1]) → 665_0_POWER_LE(x0[1] / 2)

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

(1) -> (0), if (x0[1] / 2* x0[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.IdpDefaultShapeHeuristic@225274a3 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 665_0_POWER_LE(x0) → COND_665_0_POWER_LE(>(x0, 0), x0) the following chains were created:
• We consider the chain 665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(>(x0[0], 0), x0[0]), COND_665_0_POWER_LE(TRUE, x0[1]) → 665_0_POWER_LE(/(x0[1], 2)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]665_0_POWER_LE(x0[0])≥NonInfC∧665_0_POWER_LE(x0[0])≥COND_665_0_POWER_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE665_0_POWER_LE(x0[0])≥NonInfC∧665_0_POWER_LE(x0[0])≥COND_665_0_POWER_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_9 + (-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_665_0_POWER_LE(TRUE, x0) → 665_0_POWER_LE(/(x0, 2)) the following chains were created:
• We consider the chain 665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(>(x0[0], 0), x0[0]), COND_665_0_POWER_LE(TRUE, x0[1]) → 665_0_POWER_LE(/(x0[1], 2)), 665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(>(x0[0], 0), x0[0]) which results in the following constraint:

(7)    (>(x0[0], 0)=TRUEx0[0]=x0[1]/(x0[1], 2)=x0[0]1COND_665_0_POWER_LE(TRUE, x0[1])≥NonInfC∧COND_665_0_POWER_LE(TRUE, x0[1])≥665_0_POWER_LE(/(x0[1], 2))∧(UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥))

We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:

(8)    (>(x0[0], 0)=TRUECOND_665_0_POWER_LE(TRUE, x0[0])≥NonInfC∧COND_665_0_POWER_LE(TRUE, x0[0])≥665_0_POWER_LE(/(x0[0], 2))∧(UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥))

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

(9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

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

(10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(12)    (x0[0] ≥ 0∧[2] + [2]x0[0] ≥ 0 ⇒ (UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(13)    (x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 665_0_POWER_LE(x0) → COND_665_0_POWER_LE(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_665_0_POWER_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_9] + [bni_9]x0[0] ≥ 0∧[(-1)bso_10] ≥ 0)

• COND_665_0_POWER_LE(TRUE, x0) → 665_0_POWER_LE(/(x0, 2))
• (x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(665_0_POWER_LE(/(x0[1], 2))), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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(665_0_POWER_LE(x1)) = [-1] + x1
POL(COND_665_0_POWER_LE(x1, x2)) = [-1] + x2
POL(>(x1, x2)) = 0
POL(0) = 0
POL(2) = [2]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {665_0_POWER_LE_1/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

COND_665_0_POWER_LE(TRUE, x0[1]) → 665_0_POWER_LE(/(x0[1], 2))

The following pairs are in Pbound:

665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(>(x0[0], 0), x0[0])
COND_665_0_POWER_LE(TRUE, x0[1]) → 665_0_POWER_LE(/(x0[1], 2))

The following pairs are in P:

665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(>(x0[0], 0), x0[0])

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

/1

### (8) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 665_0_POWER_LE(x0[0]) → COND_665_0_POWER_LE(x0[0] > 0, x0[0])

The set Q is empty.

### (9) IDependencyGraphProof (EQUIVALENT transformation)

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