(0) Obligation:

JBC Problem based on JBC Program:
public class LogIterative {
public static int log(int x, int y) {
int res = 0;
while (x >= y && y > 1) {
res++;
x = x/y;
}
return res;
}

public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();
log(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:
LogIterative.main([Ljava/lang/String;)V: Graph of 198 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: LogIterative.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses:
  • Used field analysis yielded the following read fields:
  • Marker field analysis yielded the following relations that could be markers:

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 15 rules for P and 0 rules for R.


P rules:
f815_0_log_Load(EOS(STATIC_815), i113, i112, i113, i112) → f817_0_log_LT(EOS(STATIC_817), i113, i112, i113, i112, i113)
f817_0_log_LT(EOS(STATIC_817), i113, i112, i113, i112, i113) → f820_0_log_LT(EOS(STATIC_820), i113, i112, i113, i112, i113)
f820_0_log_LT(EOS(STATIC_820), i113, i112, i113, i112, i113) → f825_0_log_Load(EOS(STATIC_825), i113, i112, i113) | >=(i112, i113)
f825_0_log_Load(EOS(STATIC_825), i113, i112, i113) → f830_0_log_ConstantStackPush(EOS(STATIC_830), i113, i112, i113, i113)
f830_0_log_ConstantStackPush(EOS(STATIC_830), i113, i112, i113, i113) → f835_0_log_LE(EOS(STATIC_835), i113, i112, i113, i113, 1)
f835_0_log_LE(EOS(STATIC_835), i125, i112, i125, i125, matching1) → f841_0_log_LE(EOS(STATIC_841), i125, i112, i125, i125, 1) | =(matching1, 1)
f841_0_log_LE(EOS(STATIC_841), i125, i112, i125, i125, matching1) → f849_0_log_Inc(EOS(STATIC_849), i125, i112, i125) | &&(>(i125, 1), =(matching1, 1))
f849_0_log_Inc(EOS(STATIC_849), i125, i112, i125) → f855_0_log_Load(EOS(STATIC_855), i125, i112, i125)
f855_0_log_Load(EOS(STATIC_855), i125, i112, i125) → f867_0_log_Load(EOS(STATIC_867), i125, i125, i112)
f867_0_log_Load(EOS(STATIC_867), i125, i125, i112) → f872_0_log_IntArithmetic(EOS(STATIC_872), i125, i125, i112, i125)
f872_0_log_IntArithmetic(EOS(STATIC_872), i125, i125, i112, i125) → f876_0_log_Store(EOS(STATIC_876), i125, i125, /(i112, i125)) | >(i125, 1)
f876_0_log_Store(EOS(STATIC_876), i125, i125, i129) → f878_0_log_JMP(EOS(STATIC_878), i125, i129, i125)
f878_0_log_JMP(EOS(STATIC_878), i125, i129, i125) → f907_0_log_Load(EOS(STATIC_907), i125, i129, i125)
f907_0_log_Load(EOS(STATIC_907), i125, i129, i125) → f811_0_log_Load(EOS(STATIC_811), i125, i129, i125)
f811_0_log_Load(EOS(STATIC_811), i113, i112, i113) → f815_0_log_Load(EOS(STATIC_815), i113, i112, i113, i112)
R rules:

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


P rules:
f815_0_log_Load(EOS(STATIC_815), x0, x1, x0, x1) → f815_0_log_Load(EOS(STATIC_815), x0, /(x1, x0), x0, /(x1, x0)) | &&(>=(x1, x0), >(x0, 1))
R rules:

Filtered ground terms:



f815_0_log_Load(x1, x2, x3, x4, x5) → f815_0_log_Load(x2, x3, x4, x5)
Cond_f815_0_log_Load(x1, x2, x3, x4, x5, x6) → Cond_f815_0_log_Load(x1, x3, x4, x5, x6)
EOS(x1) → EOS

Filtered duplicate args:



f815_0_log_Load(x1, x2, x3, x4) → f815_0_log_Load(x3, x4)
Cond_f815_0_log_Load(x1, x2, x3, x4, x5) → Cond_f815_0_log_Load(x1, x4, x5)

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


P rules:
F815_0_LOG_LOAD(x0, x1) → F815_0_LOG_LOAD(x0, /(x1, x0)) | &&(>(x0, 1), >=(x1, x0))
R rules:

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


P rules:
F815_0_LOG_LOAD'(x0, x1) → COND_F815_0_LOG_LOAD(&&(>(x0, 1), >=(x1, x0)), x0, x1)
COND_F815_0_LOG_LOAD(TRUE, x0, x1) → F815_0_LOG_LOAD'(x0, /(x1, x0))
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): F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(x0[0] > 1 && x1[0] >= x0[0], x0[0], x1[0])
(1): COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], x1[1] / x0[1])

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


(1) -> (0), if (x0[1]* x0[0]x1[1] / x0[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.IdpDefaultShapeHeuristic@4c7bd82c 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 F815_0_LOG_LOAD'(x0, x1) → COND_F815_0_LOG_LOAD(&&(>(x0, 1), >=(x1, x0)), x0, x1) the following chains were created:
  • We consider the chain F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0]), COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1])) which results in the following constraint:

    (1)    (&&(>(x0[0], 1), >=(x1[0], x0[0]))=TRUEx0[0]=x0[1]x1[0]=x1[1]F815_0_LOG_LOAD'(x0[0], x1[0])≥NonInfC∧F815_0_LOG_LOAD'(x0[0], x1[0])≥COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])∧(UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥))



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

    (2)    (>(x0[0], 1)=TRUE>=(x1[0], x0[0])=TRUEF815_0_LOG_LOAD'(x0[0], x1[0])≥NonInfC∧F815_0_LOG_LOAD'(x0[0], x1[0])≥COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])∧(UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥))



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

    (3)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (7)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_F815_0_LOG_LOAD(TRUE, x0, x1) → F815_0_LOG_LOAD'(x0, /(x1, x0)) the following chains were created:
  • We consider the chain F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0]), COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1])), F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0]) which results in the following constraint:

    (8)    (&&(>(x0[0], 1), >=(x1[0], x0[0]))=TRUEx0[0]=x0[1]x1[0]=x1[1]x0[1]=x0[0]1/(x1[1], x0[1])=x1[0]1COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1])≥NonInfC∧COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1])≥F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))∧(UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥))



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

    (9)    (>(x0[0], 1)=TRUE>=(x1[0], x0[0])=TRUECOND_F815_0_LOG_LOAD(TRUE, x0[0], x1[0])≥NonInfC∧COND_F815_0_LOG_LOAD(TRUE, x0[0], x1[0])≥F815_0_LOG_LOAD'(x0[0], /(x1[0], x0[0]))∧(UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥))



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

    (10)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] + [-1]max{x1[0], [-1]x1[0]} + min{max{x0[0], [-1]x0[0]} + [-1], max{x1[0], [-1]x1[0]}} ≥ 0)



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

    (11)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] + [-1]max{x1[0], [-1]x1[0]} + min{max{x0[0], [-1]x0[0]} + [-1], max{x1[0], [-1]x1[0]}} ≥ 0)



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

    (12)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧[2]x1[0] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[-1 + (-1)bso_19] + x0[0] ≥ 0)



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

    (13)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0∧[2]x1[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 0)



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

    (14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[4] + [2]x0[0] + [2]x1[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 0)



    We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:

    (15)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • F815_0_LOG_LOAD'(x0, x1) → COND_F815_0_LOG_LOAD(&&(>(x0, 1), >=(x1, x0)), x0, x1)
    • (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_F815_0_LOG_LOAD(TRUE, x0, x1) → F815_0_LOG_LOAD'(x0, /(x1, x0))
    • (x0[0] ≥ 0∧x1[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 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) = [3]   
POL(FALSE) = 0   
POL(F815_0_LOG_LOAD'(x1, x2)) = [2] + x2 + [-1]x1   
POL(COND_F815_0_LOG_LOAD(x1, x2, x3)) = [2] + x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(>=(x1, x2)) = [-1]   

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

POL(/(x1, x0[0])1 @ {F815_0_LOG_LOAD'_2/1}) = max{x1, [-1]x1} + [-1]min{max{x2, [-1]x2} + [-1], max{x1, [-1]x1}}   

The following pairs are in P>:

COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))

The following pairs are in Pbound:

F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])
COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))

The following pairs are in P:

F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])

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

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

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(x0[0] > 1 && x1[0] >= x0[0], x0[0], x1[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.

(10) TRUE