### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Duplicate
`public class Duplicate{  public static int round (int x) {    if (x % 2 == 0) return x;    else return x+1;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    while ((x > y) && (y > 2)) {      x++;      y = 2*y;    }  }}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:
Duplicate.main([Ljava/lang/String;)V: Graph of 163 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 15 rules for P and 5 rules for R.

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

Filtered ground terms:

Filtered duplicate args:

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

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

### (4) 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): 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(x1[0] > 2 && x1[0] < x0[0] && x0[0] >= 0, x1[0], x0[0])

(0) -> (1), if ((x1[0] > 2 && x1[0] < x0[0] && x0[0] >= 0* TRUE)∧(x1[0]* x1[1])∧(x0[0]* x0[1]))

(1) -> (0), if ((2 * x1[1]* x1[0])∧(x0[1] + 1* x0[0]))

The set Q is empty.

### (5) 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 723_0_MAIN_LOAD(x1, x0) → COND_723_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >=(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0]), COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1)) which results in the following constraint:

(1)    (&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]723_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧723_0_MAIN_LOAD(x1[0], x0[0])≥COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥))

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)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] + [(-1)bni_14]x1[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(4)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] + [(-1)bni_14]x1[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(5)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] + [(-1)bni_14]x1[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(6)    ([1] + x1[0] + x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(7)    ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_723_0_MAIN_LOAD(TRUE, x1, x0) → 723_0_MAIN_LOAD(*(2, x1), +(x0, 1)) the following chains were created:
• We consider the chain 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0]), COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1)), 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 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)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

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

(11)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

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

(12)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

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

(13)    ([1] + x1[0] + x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

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

(14)    ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 723_0_MAIN_LOAD(x1, x0) → COND_723_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >=(x0, 0)), x1, x0)
• ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

• ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[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) = [1]
POL(FALSE) = 0
POL(723_0_MAIN_LOAD(x1, x2)) = [2] + x2 + [-1]x1
POL(COND_723_0_MAIN_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2 + x1
POL(&&(x1, x2)) = [2]
POL(>(x1, x2)) = [-1]
POL(2) = [2]
POL(<(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(*(x1, x2)) = x1·x2
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:

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

### (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: