### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB6
`/** * 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 PastaB6 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0 && 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:
Graph of 189 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:
Load548(i36, i45) → Cond_Load548(i45 > 0 && i36 > 0, i36, i45)
Cond_Load548(TRUE, i36, i45) → Load548(i36 + -1, i45 + -1)
The set Q consists of the following terms:

### (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

The ITRS R consists of the following rules:
Load548(i36, i45) → Cond_Load548(i45 > 0 && i36 > 0, i36, i45)
Cond_Load548(TRUE, i36, i45) → Load548(i36 + -1, i45 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD548(i36[0], i45[0]) → COND_LOAD548(i45[0] > 0 && i36[0] > 0, i36[0], i45[0])
(1): COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(i36[1] + -1, i45[1] + -1)

(0) -> (1), if ((i45[0] > 0 && i36[0] > 0* TRUE)∧(i36[0]* i36[1])∧(i45[0]* i45[1]))

(1) -> (0), if ((i45[1] + -1* i45[0])∧(i36[1] + -1* i36[0]))

The set Q consists of the following terms:

### (7) 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.

### (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): LOAD548(i36[0], i45[0]) → COND_LOAD548(i45[0] > 0 && i36[0] > 0, i36[0], i45[0])
(1): COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(i36[1] + -1, i45[1] + -1)

(0) -> (1), if ((i45[0] > 0 && i36[0] > 0* TRUE)∧(i36[0]* i36[1])∧(i45[0]* i45[1]))

(1) -> (0), if ((i45[1] + -1* i45[0])∧(i36[1] + -1* i36[0]))

The set Q consists of the following terms:

### (9) 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 LOAD548(i36, i45) → COND_LOAD548(&&(>(i45, 0), >(i36, 0)), i36, i45) the following chains were created:
• We consider the chain LOAD548(i36[0], i45[0]) → COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0]), COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(+(i36[1], -1), +(i45[1], -1)) which results in the following constraint:

(1)    (&&(>(i45[0], 0), >(i36[0], 0))=TRUEi36[0]=i36[1]i45[0]=i45[1]LOAD548(i36[0], i45[0])≥NonInfC∧LOAD548(i36[0], i45[0])≥COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])∧(UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥))

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

(2)    (>(i45[0], 0)=TRUE>(i36[0], 0)=TRUELOAD548(i36[0], i45[0])≥NonInfC∧LOAD548(i36[0], i45[0])≥COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])∧(UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥))

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

(3)    (i45[0] + [-1] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(4)    (i45[0] + [-1] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(5)    (i45[0] + [-1] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(6)    (i45[0] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(7)    (i45[0] ≥ 0∧i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[(5)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

For Pair COND_LOAD548(TRUE, i36, i45) → LOAD548(+(i36, -1), +(i45, -1)) the following chains were created:
• We consider the chain COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(+(i36[1], -1), +(i45[1], -1)) which results in the following constraint:

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

(9)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧[4 + (-1)bso_12] ≥ 0)

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

(10)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧[4 + (-1)bso_12] ≥ 0)

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

(11)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧[4 + (-1)bso_12] ≥ 0)

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

(12)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD548(i36, i45) → COND_LOAD548(&&(>(i45, 0), >(i36, 0)), i36, i45)
• (i45[0] ≥ 0∧i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[(5)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

• COND_LOAD548(TRUE, i36, i45) → LOAD548(+(i36, -1), +(i45, -1))
• ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_12] ≥ 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(LOAD548(x1, x2)) = [1] + [2]x2 + [2]x1
POL(COND_LOAD548(x1, x2, x3)) = [1] + [2]x3 + [2]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(+(i36[1], -1), +(i45[1], -1))

The following pairs are in Pbound:

LOAD548(i36[0], i45[0]) → COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])

The following pairs are in P:

LOAD548(i36[0], i45[0]) → COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])

There are no usable rules.

### (11) 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): LOAD548(i36[0], i45[0]) → COND_LOAD548(i45[0] > 0 && i36[0] > 0, i36[0], i45[0])

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) 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_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(i36[1] + -1, i45[1] + -1)

The set Q consists of the following terms: