### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB14
`/** * Example taken from "A Term Rewriting Approach to the Automated Termination * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf) * and converted to Java. */public class PastaB14 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x == y && x > 0) {            while (y > 0) {                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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
PastaB14.main([Ljava/lang/String;)V: Graph of 167 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 19 rules for P and 5 rules for R.

Combined rules. Obtained 2 rules for P and 0 rules for R.

Filtered ground terms:

901_0_main_LE(x1, x2, x3, x4) → 901_0_main_LE(x2, x3, x4)
Cond_901_0_main_LE1(x1, x2, x3, x4, x5) → Cond_901_0_main_LE1(x1, x3, x4, x5)
Cond_901_0_main_LE(x1, x2, x3, x4, x5) → Cond_901_0_main_LE(x1)

Filtered duplicate args:

901_0_main_LE(x1, x2, x3) → 901_0_main_LE(x1, x3)
Cond_901_0_main_LE1(x1, x2, x3, x4) → Cond_901_0_main_LE1(x1, x2, x4)

Combined rules. Obtained 2 rules for P and 0 rules for R.

Finished conversion. Obtained 2 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(0 > 0, 0, 0)
(1): COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0)
(2): 901_0_MAIN_LE(x0[2], x1[2]) → COND_901_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])
(3): COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 901_0_MAIN_LE(x0[3] + -1, x1[3] + -1)

(0) -> (1), if (0 > 0* TRUE)

(1) -> (0), if true

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

(2) -> (3), if ((x1[2] > 0* TRUE)∧(x0[2]* x0[3])∧(x1[2]* x1[3]))

(3) -> (0), if ((x0[3] + -1* 0)∧(x1[3] + -1* 0))

(3) -> (2), if ((x0[3] + -1* x0[2])∧(x1[3] + -1* x1[2]))

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 901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(>(0, 0), 0, 0) the following chains were created:
• We consider the chain 901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(>(0, 0), 0, 0), COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0) which results in the following constraint:

(1)    (>(0, 0)=TRUE901_0_MAIN_LE(0, 0)≥NonInfC∧901_0_MAIN_LE(0, 0)≥COND_901_0_MAIN_LE(>(0, 0), 0, 0)∧(UIncreasing(COND_901_0_MAIN_LE(>(0, 0), 0, 0)), ≥))

We solved constraint (1) using rules (I), (II), (IDP_CONSTANT_FOLD).

For Pair COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0) the following chains were created:
• We consider the chain COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0), 901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(>(0, 0), 0, 0) which results in the following constraint:

(2)    (COND_901_0_MAIN_LE(TRUE, 0, 0)≥NonInfC∧COND_901_0_MAIN_LE(TRUE, 0, 0)≥901_0_MAIN_LE(0, 0)∧(UIncreasing(901_0_MAIN_LE(0, 0)), ≥))

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

(3)    ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

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

(4)    ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

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

(5)    ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

• We consider the chain COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0), 901_0_MAIN_LE(x0[2], x1[2]) → COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2]) which results in the following constraint:

(6)    (0=x0[2]0=x1[2]COND_901_0_MAIN_LE(TRUE, 0, 0)≥NonInfC∧COND_901_0_MAIN_LE(TRUE, 0, 0)≥901_0_MAIN_LE(0, 0)∧(UIncreasing(901_0_MAIN_LE(0, 0)), ≥))

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

(7)    (COND_901_0_MAIN_LE(TRUE, 0, 0)≥NonInfC∧COND_901_0_MAIN_LE(TRUE, 0, 0)≥901_0_MAIN_LE(0, 0)∧(UIncreasing(901_0_MAIN_LE(0, 0)), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

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

(9)    ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

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

(10)    ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

For Pair 901_0_MAIN_LE(x0, x1) → COND_901_0_MAIN_LE1(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 901_0_MAIN_LE(x0[2], x1[2]) → COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2]), COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1)) which results in the following constraint:

(11)    (>(x1[2], 0)=TRUEx0[2]=x0[3]x1[2]=x1[3]901_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧901_0_MAIN_LE(x0[2], x1[2])≥COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(12)    (>(x1[2], 0)=TRUE901_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧901_0_MAIN_LE(x0[2], x1[2])≥COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(13)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(14)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(15)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (15) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(16)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(17)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair COND_901_0_MAIN_LE1(TRUE, x0, x1) → 901_0_MAIN_LE(+(x0, -1), +(x1, -1)) the following chains were created:
• We consider the chain COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1)) which results in the following constraint:

(18)    (COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3])≥901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))∧(UIncreasing(901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))), ≥))

We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(19)    ((UIncreasing(901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))), ≥)∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(20)    ((UIncreasing(901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))), ≥)∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(21)    ((UIncreasing(901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))), ≥)∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(22)    ((UIncreasing(901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(>(0, 0), 0, 0)

• COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0)
• ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)
• ((UIncreasing(901_0_MAIN_LE(0, 0)), ≥)∧[1 + (-1)bso_10] ≥ 0)

• 901_0_MAIN_LE(x0, x1) → COND_901_0_MAIN_LE1(>(x1, 0), x0, x1)
• (x1[2] ≥ 0 ⇒ (UIncreasing(COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• COND_901_0_MAIN_LE1(TRUE, x0, x1) → 901_0_MAIN_LE(+(x0, -1), +(x1, -1))
• ((UIncreasing(901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 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(901_0_MAIN_LE(x1, x2)) = [1] + x2
POL(0) = 0
POL(COND_901_0_MAIN_LE(x1, x2, x3)) = [2]
POL(>(x1, x2)) = [-1]
POL(COND_901_0_MAIN_LE1(x1, x2, x3)) = [1] + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(>(0, 0), 0, 0)
COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0)
COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 901_0_MAIN_LE(+(x0[3], -1), +(x1[3], -1))

The following pairs are in Pbound:

901_0_MAIN_LE(0, 0) → COND_901_0_MAIN_LE(>(0, 0), 0, 0)
901_0_MAIN_LE(x0[2], x1[2]) → COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])

The following pairs are in P:

901_0_MAIN_LE(x0[2], x1[2]) → COND_901_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])

There are no usable rules.

### (7) 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:
(2): 901_0_MAIN_LE(x0[2], x1[2]) → COND_901_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_901_0_MAIN_LE(TRUE, 0, 0) → 901_0_MAIN_LE(0, 0)
(3): COND_901_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 901_0_MAIN_LE(x0[3] + -1, x1[3] + -1)

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.