### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA9
/**
* 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 PastaA9 {
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();
int z = Random.random();

if (y > 0) {
while (x >= z) {
z += 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:
PastaA9.main([Ljava/lang/String;)V: Graph of 228 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 10 rules for P and 3 rules for R.

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

Filtered ground terms:

1029_0_main_LT(x1, x2, x3, x4, x5, x6) → 1029_0_main_LT(x2, x3, x4, x5, x6)
Cond_1029_0_main_LT(x1, x2, x3, x4, x5, x6, x7) → Cond_1029_0_main_LT(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

1029_0_main_LT(x1, x2, x3, x4, x5) → 1029_0_main_LT(x2, x4, x5)
Cond_1029_0_main_LT(x1, x2, x3, x4, x5, x6) → Cond_1029_0_main_LT(x1, x3, x5, x6)

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): 1029_0_MAIN_LT(x1[0], x0[0], x2[0]) → COND_1029_0_MAIN_LT(x2[0] >= 0 && x2[0] <= x0[0] && x1[0] > 0, x1[0], x0[0], x2[0])
(1): COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1]) → 1029_0_MAIN_LT(x1[1], x0[1], x2[1] + x1[1])

(0) -> (1), if ((x2[0] >= 0 && x2[0] <= x0[0] && x1[0] > 0* TRUE)∧(x1[0]* x1[1])∧(x0[0]* x0[1])∧(x2[0]* x2[1]))

(1) -> (0), if ((x1[1]* x1[0])∧(x0[1]* x0[0])∧(x2[1] + x1[1]* x2[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 1029_0_MAIN_LT(x1, x0, x2) → COND_1029_0_MAIN_LT(&&(&&(>=(x2, 0), <=(x2, x0)), >(x1, 0)), x1, x0, x2) the following chains were created:
• We consider the chain 1029_0_MAIN_LT(x1[0], x0[0], x2[0]) → COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0]), COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1]) → 1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1])) which results in the following constraint:

(1)    (&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]x2[0]=x2[1]1029_0_MAIN_LT(x1[0], x0[0], x2[0])≥NonInfC∧1029_0_MAIN_LT(x1[0], x0[0], x2[0])≥COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])∧(UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥))

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

(2)    (>(x1[0], 0)=TRUE>=(x2[0], 0)=TRUE<=(x2[0], x0[0])=TRUE1029_0_MAIN_LT(x1[0], x0[0], x2[0])≥NonInfC∧1029_0_MAIN_LT(x1[0], x0[0], x2[0])≥COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])∧(UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥))

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

(3)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[0] + [bni_16]x0[0] ≥ 0∧[1 + (-1)bso_17] + x1[0] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[1 + (-1)bso_17] + x1[0] ≥ 0)

For Pair COND_1029_0_MAIN_LT(TRUE, x1, x0, x2) → 1029_0_MAIN_LT(x1, x0, +(x2, x1)) the following chains were created:
• We consider the chain 1029_0_MAIN_LT(x1[0], x0[0], x2[0]) → COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0]), COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1]) → 1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1])), 1029_0_MAIN_LT(x1[0], x0[0], x2[0]) → COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0]) which results in the following constraint:

(8)    (&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]x2[0]=x2[1]x1[1]=x1[0]1x0[1]=x0[0]1+(x2[1], x1[1])=x2[0]1COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1])≥NonInfC∧COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1])≥1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))∧(UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥))

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

(9)    (>(x1[0], 0)=TRUE>=(x2[0], 0)=TRUE<=(x2[0], x0[0])=TRUECOND_1029_0_MAIN_LT(TRUE, x1[0], x0[0], x2[0])≥NonInfC∧COND_1029_0_MAIN_LT(TRUE, x1[0], x0[0], x2[0])≥1029_0_MAIN_LT(x1[0], x0[0], +(x2[0], x1[0]))∧(UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥))

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

(10)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x2[0] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(11)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x2[0] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(12)    (x1[0] + [-1] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x2[0] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(13)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x2[0] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(14)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1029_0_MAIN_LT(x1, x0, x2) → COND_1029_0_MAIN_LT(&&(&&(>=(x2, 0), <=(x2, x0)), >(x1, 0)), x1, x0, x2)
• (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[1 + (-1)bso_17] + x1[0] ≥ 0)

• COND_1029_0_MAIN_LT(TRUE, x1, x0, x2) → 1029_0_MAIN_LT(x1, x0, +(x2, x1))
• (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))), ≥)∧[(-2)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [(-1)bni_18]x1[0] ≥ 0∧[(-1)bso_19] ≥ 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) = [1]
POL(1029_0_MAIN_LT(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(COND_1029_0_MAIN_LT(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

1029_0_MAIN_LT(x1[0], x0[0], x2[0]) → COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])

The following pairs are in Pbound:

1029_0_MAIN_LT(x1[0], x0[0], x2[0]) → COND_1029_0_MAIN_LT(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], 0)), x1[0], x0[0], x2[0])

The following pairs are in P:

COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1]) → 1029_0_MAIN_LT(x1[1], x0[1], +(x2[1], x1[1]))

At least the following rules have been oriented under context sensitive arithmetic replacement:

FALSE1&&(FALSE, FALSE)1

### (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:
(1): COND_1029_0_MAIN_LT(TRUE, x1[1], x0[1], x2[1]) → 1029_0_MAIN_LT(x1[1], x0[1], x2[1] + x1[1])

The set Q is empty.

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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