(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogBuiltIn
`public class LogBuiltIn{  public static int log(int x) {    int res = 0;    while (x > 1) {      x = x/2;      res++;    }    return res;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    log(x);  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 138 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load653(1, i42, i38) → Cond_Load653(i42 > 1 && i38 + 1 > 0, 1, i42, i38)
Cond_Load653(TRUE, 1, i42, i38) → Load653(1, i42 / 2, i38 + 1)
The set Q consists of the following terms:

(5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
• 1

We removed arguments according to the following replacements:

(6) Obligation:

ITRS 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 TRS R consists of the following rules:
Load653(i42, i38) → Cond_Load653(i42 > 1 && i38 + 1 > 0, i42, i38)
The set Q consists of the following terms:

(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

The ITRS R consists of the following rules:
Load653(i42, i38) → Cond_Load653(i42 > 1 && i38 + 1 > 0, i42, i38)

The integer pair graph contains the following rules and edges:
(0): LOAD653(i42[0], i38[0]) → COND_LOAD653(i42[0] > 1 && i38[0] + 1 > 0, i42[0], i38[0])

(0) -> (1), if ((i42[0]* i42[1])∧(i38[0]* i38[1])∧(i42[0] > 1 && i38[0] + 1 > 0* TRUE))

(1) -> (0), if ((i42[1] / 2* i42[0])∧(i38[1] + 1* i38[0]))

The set Q consists of the following terms:

(9) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD653(i42[0], i38[0]) → COND_LOAD653(i42[0] > 1 && i38[0] + 1 > 0, i42[0], i38[0])

(0) -> (1), if ((i42[0]* i42[1])∧(i38[0]* i38[1])∧(i42[0] > 1 && i38[0] + 1 > 0* TRUE))

(1) -> (0), if ((i42[1] / 2* i42[0])∧(i38[1] + 1* i38[0]))

The set Q consists of the following terms:

(11) IDPNonInfProof (SOUND transformation)

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 LOAD653(i42, i38) → COND_LOAD653(&&(>(i42, 1), >(+(i38, 1), 0)), i42, i38) the following chains were created:
• We consider the chain LOAD653(i42[0], i38[0]) → COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0]), COND_LOAD653(TRUE, i42[1], i38[1]) → LOAD653(/(i42[1], 2), +(i38[1], 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)    (i42[0] + [-2] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i42[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (i42[0] + [-2] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i42[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(5)    (i42[0] + [-2] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i42[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (i42[0] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]i42[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_LOAD653(TRUE, i42, i38) → LOAD653(/(i42, 2), +(i38, 1)) the following chains were created:
• We consider the chain LOAD653(i42[0], i38[0]) → COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0]), COND_LOAD653(TRUE, i42[1], i38[1]) → LOAD653(/(i42[1], 2), +(i38[1], 1)), LOAD653(i42[0], i38[0]) → COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0]) which results in the following constraint:

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

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

(9)    (i42[0] + [-2] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(LOAD653(/(i42[1], 2), +(i38[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i42[0] ≥ 0∧[1 + (-1)bso_18] + i42[0] + [-1]max{i42[0], [-1]i42[0]} ≥ 0)

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

(10)    (i42[0] + [-2] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(LOAD653(/(i42[1], 2), +(i38[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i42[0] ≥ 0∧[1 + (-1)bso_18] + i42[0] + [-1]max{i42[0], [-1]i42[0]} ≥ 0)

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

(11)    (i42[0] + [-2] ≥ 0∧i38[0] ≥ 0∧[2]i42[0] ≥ 0 ⇒ (UIncreasing(LOAD653(/(i42[1], 2), +(i38[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i42[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(12)    (i42[0] ≥ 0∧i38[0] ≥ 0∧[4] + [2]i42[0] ≥ 0 ⇒ (UIncreasing(LOAD653(/(i42[1], 2), +(i38[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i42[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(13)    (i42[0] ≥ 0∧i38[0] ≥ 0∧[2] + i42[0] ≥ 0 ⇒ (UIncreasing(LOAD653(/(i42[1], 2), +(i38[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i42[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i42[0] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD653(&&(>(i42[0], 1), >(+(i38[0], 1), 0)), i42[0], i38[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]i42[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• (i42[0] ≥ 0∧i38[0] ≥ 0∧[2] + i42[0] ≥ 0 ⇒ (UIncreasing(LOAD653(/(i42[1], 2), +(i38[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i42[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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) = [3]
POL(LOAD653(x1, x2)) = [-1] + x1
POL(COND_LOAD653(x1, x2, x3)) = [1] + x2 + [-1]x1
POL(&&(x1, x2)) = [2]
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(+(x1, x2)) = x1 + x2
POL(0) = 0
POL(2) = [2]

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

POL(/(x1, 2)1 @ {LOAD653_2/0}) = max{x1, [-1]x1} + [-1]

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

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD653(i42[0], i38[0]) → COND_LOAD653(i42[0] > 1 && i38[0] + 1 > 0, i42[0], i38[0])

The set Q consists of the following terms:

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) 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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: