### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PlusSwap
`public class PlusSwap{  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    int z;    int res = 0;    while (y > 0) {      z = x;      x = y-1;      y = z;      res++;    }    res = res + 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:
PlusSwap.main([Ljava/lang/String;)V: Graph of 169 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 14 rules for P and 6 rules for R.

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

Filtered ground terms:

913_0_main_LE(x1, x2, x3, x4) → 913_0_main_LE(x2, x3, x4)
Cond_913_0_main_LE(x1, x2, x3, x4, x5) → Cond_913_0_main_LE(x1, x3, x4, x5)
930_0_main_Return(x1) → 930_0_main_Return

Filtered duplicate args:

913_0_main_LE(x1, x2, x3) → 913_0_main_LE(x1, x3)
Cond_913_0_main_LE(x1, x2, x3, x4) → Cond_913_0_main_LE(x1, x2, x4)

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

Finished conversion. Obtained 1 rules for P and 1 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:

Integer

The ITRS R consists of the following rules:
913_0_main_LE(x0, 0) → Cond_913_0_main_LE(x0 >= 0, x0, 0)
Cond_913_0_main_LE(TRUE, x0, 0) → 930_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(x1[0] > 0, x0[0], x1[0])
(1): COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(x1[1] - 1, x0[1])

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

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

The set Q consists of the following terms:
913_0_main_LE(x0, 0)
Cond_913_0_main_LE(TRUE, x0, 0)

### (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 913_0_MAIN_LE(x0, x1) → COND_913_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:

(1)    (>(x1[0], 0)=TRUEx0[0]=x0[1]x1[0]=x1[1]913_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧913_0_MAIN_LE(x0[0], x1[0])≥COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))

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

(2)    (>(x1[0], 0)=TRUE913_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧913_0_MAIN_LE(x0[0], x1[0])≥COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])∧(UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥))

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

(3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(6)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_20] = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)

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

(7)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_20] = 0∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair COND_913_0_MAIN_LE(TRUE, x0, x1) → 913_0_MAIN_LE(-(x1, 1), x0) the following chains were created:
• We consider the chain 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]), 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]), 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0]), COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1]) which results in the following constraint:

(8)    (>(x1[0], 0)=TRUEx0[0]=x0[1]x1[0]=x1[1]-(x1[1], 1)=x0[0]1x0[1]=x1[0]1>(x1[0]1, 0)=TRUEx0[0]1=x0[1]1x1[0]1=x1[1]1-(x1[1]1, 1)=x0[0]2x0[1]1=x1[0]2>(x1[0]2, 0)=TRUEx0[0]2=x0[1]2x1[0]2=x1[1]2COND_913_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥NonInfC∧COND_913_0_MAIN_LE(TRUE, x0[1]1, x1[1]1)≥913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)∧(UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))

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

(9)    (>(x1[0], 0)=TRUE>(x1[0]1, 0)=TRUE>(-(x1[0], 1), 0)=TRUECOND_913_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥NonInfC∧COND_913_0_MAIN_LE(TRUE, -(x1[0], 1), x1[0]1)≥913_0_MAIN_LE(-(x1[0]1, 1), -(x1[0], 1))∧(UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥))

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

(10)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

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

(11)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

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

(12)    (x1[0] + [-1] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-2)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

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

(13)    (x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

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

(14)    ([1] + x1[0] ≥ 0∧x1[0]1 + [-1] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

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

(15)    ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 913_0_MAIN_LE(x0, x1) → COND_913_0_MAIN_LE(>(x1, 0), x0, x1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])), ≥)∧[bni_20] = 0∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)

• COND_913_0_MAIN_LE(TRUE, x0, x1) → 913_0_MAIN_LE(-(x1, 1), x0)
• ([1] + x1[0] ≥ 0∧x1[0]1 ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(913_0_MAIN_LE(-(x1[1]1, 1), x0[1]1)), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[0]1 + [bni_22]x1[0] ≥ 0∧[1 + (-1)bso_23] ≥ 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(913_0_main_LE(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(0) = 0
POL(Cond_913_0_main_LE(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(>=(x1, x2)) = [-1]
POL(930_0_main_Return) = [-1]
POL(913_0_MAIN_LE(x1, x2)) = [-1] + x2 + x1
POL(COND_913_0_MAIN_LE(x1, x2, x3)) = [-1] + x3 + x2
POL(>(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]

The following pairs are in P>:

COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in Pbound:

COND_913_0_MAIN_LE(TRUE, x0[1], x1[1]) → 913_0_MAIN_LE(-(x1[1], 1), x0[1])

The following pairs are in P:

913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(>(x1[0], 0), x0[0], x1[0])

There are no usable 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

The ITRS R consists of the following rules:
913_0_main_LE(x0, 0) → Cond_913_0_main_LE(x0 >= 0, x0, 0)
Cond_913_0_main_LE(TRUE, x0, 0) → 930_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 913_0_MAIN_LE(x0[0], x1[0]) → COND_913_0_MAIN_LE(x1[0] > 0, x0[0], x1[0])

The set Q consists of the following terms:
913_0_main_LE(x0, 0)
Cond_913_0_main_LE(TRUE, x0, 0)

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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