(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogMult
`public class LogMult{  public static int log(int x, int y) {    int res = 1;    if (x < 0 || y < 1) return 0;    else {      while (x > y) {         y = y*y;        res = 2*res;      }    }    return res;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    log(x,2);  }}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:
LogMult.main([Ljava/lang/String;)V: Graph of 131 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: LogMult.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 14 rules for P and 0 rules for R.

P rules:
491_0_log_Load(EOS(STATIC_491), i32, i32, i65, i32) → 494_0_log_LE(EOS(STATIC_494), i32, i32, i65, i32, i65)
494_0_log_LE(EOS(STATIC_494), i32, i32, i65, i32, i65) → 498_0_log_LE(EOS(STATIC_498), i32, i32, i65, i32, i65)
498_0_log_LE(EOS(STATIC_498), i32, i32, i65, i32, i65) → 502_0_log_Load(EOS(STATIC_502), i32, i32, i65) | >(i32, i65)
505_0_log_Load(EOS(STATIC_505), i32, i32, i65, i65) → 507_0_log_IntArithmetic(EOS(STATIC_507), i32, i32, i65, i65)
507_0_log_IntArithmetic(EOS(STATIC_507), i32, i32, i65, i65) → 509_0_log_Store(EOS(STATIC_509), i32, i32, *(i65, i65)) | &&(>(i65, 1), >(i65, 1))
509_0_log_Store(EOS(STATIC_509), i32, i32, i71) → 513_0_log_ConstantStackPush(EOS(STATIC_513), i32, i32, i71)
513_0_log_ConstantStackPush(EOS(STATIC_513), i32, i32, i71) → 515_0_log_Load(EOS(STATIC_515), i32, i32, i71)
515_0_log_Load(EOS(STATIC_515), i32, i32, i71) → 517_0_log_IntArithmetic(EOS(STATIC_517), i32, i32, i71)
517_0_log_IntArithmetic(EOS(STATIC_517), i32, i32, i71) → 519_0_log_Store(EOS(STATIC_519), i32, i32, i71)
519_0_log_Store(EOS(STATIC_519), i32, i32, i71) → 521_0_log_JMP(EOS(STATIC_521), i32, i32, i71)
521_0_log_JMP(EOS(STATIC_521), i32, i32, i71) → 526_0_log_Load(EOS(STATIC_526), i32, i32, i71)
R rules:

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

P rules:
491_0_log_Load(EOS(STATIC_491), x0, x0, x1, x0) → 491_0_log_Load(EOS(STATIC_491), x0, x0, *(x1, x1), x0) | &&(>(x1, 1), <(x1, x0))
R rules:

Filtered ground terms:

EOS(x1) → EOS
Cond_491_0_log_Load(x1, x2, x3, x4, x5, x6) → Cond_491_0_log_Load(x1, x3, x4, x5, x6)

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): 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(x1[0] > 1 && x1[0] < x0[0], x1[0], x0[0])

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

(1) -> (0), if (x1[1] * x1[1]* x1[0]x0[1]* 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@318430a5 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 491_0_LOG_LOAD(x1, x0) → COND_491_0_LOG_LOAD(&&(>(x1, 1), <(x1, x0)), x1, x0) the following chains were created:
• We consider the chain 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0]), COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1]) which results in the following constraint:

We simplified constraint (1) using rules (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] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(4)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(5)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] + [-3] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

For Pair COND_491_0_LOG_LOAD(TRUE, x1, x0) → 491_0_LOG_LOAD(*(x1, x1), x0) the following chains were created:
• We consider the chain 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0]), COND_491_0_LOG_LOAD(TRUE, x1[1], x0[1]) → 491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1]), 491_0_LOG_LOAD(x1[0], x0[0]) → COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0]) which results in the following constraint:

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) 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] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_16] + [-1]x1[0] + x1[0]2 ≥ 0)

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

(11)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_16] + [-1]x1[0] + x1[0]2 ≥ 0)

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

(12)    (x1[0] + [-2] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[-2 + (-1)bso_16] + [-1]x1[0] + x1[0]2 ≥ 0)

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

(13)    (x1[0] ≥ 0∧x0[0] + [-3] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] + [3]x1[0] + x1[0]2 ≥ 0)

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

(14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [3]x1[0] + x1[0]2 ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_491_0_LOG_LOAD(&&(>(x1[0], 1), <(x1[0], x0[0])), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(491_0_LOG_LOAD(*(x1[1], x1[1]), x0[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [3]x1[0] + x1[0]2 ≥ 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(491_0_LOG_LOAD(x1, x2)) = [1] + x2 + [-1]x1
POL(COND_491_0_LOG_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(<(x1, x2)) = [-1]
POL(*(x1, x2)) = x1·x2

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

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

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

(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: