(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB3
`/** * 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 PastaB3 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        if (x > 0) {            while (x > y) {                y = 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:
PastaB3.main([Ljava/lang/String;)V: Graph of 161 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:

689_0_main_LE(x1, x2, x3, x4, x5) → 689_0_main_LE(x2, x3, x4, x5)
Cond_689_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_689_0_main_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:

689_0_main_LE(x1, x2, x3, x4) → 689_0_main_LE(x3, x4)
Cond_689_0_main_LE(x1, x2, x3, x4, x5) → Cond_689_0_main_LE(x1, x4, x5)

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): 689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(x1[0] >= 0 && x1[0] < x0[0] && x0[0] > 0, x0[0], x1[0])
(1): COND_689_0_MAIN_LE(TRUE, x0[1], x1[1]) → 689_0_MAIN_LE(x0[1], x0[1] + x1[1])

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

(1) -> (0), if ((x0[1]* x0[0])∧(x0[1] + x1[1]* x1[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 689_0_MAIN_LE(x0, x1) → COND_689_0_MAIN_LE(&&(&&(>=(x1, 0), <(x1, x0)), >(x0, 0)), x0, x1) the following chains were created:
• We consider the chain 689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0]), COND_689_0_MAIN_LE(TRUE, x0[1], x1[1]) → 689_0_MAIN_LE(x0[1], +(x0[1], x1[1])) which results in the following constraint:

(1)    (&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0))=TRUEx0[0]=x0[1]x1[0]=x1[1]689_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧689_0_MAIN_LE(x0[0], x1[0])≥COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE>=(x1[0], 0)=TRUE<(x1[0], x0[0])=TRUE689_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧689_0_MAIN_LE(x0[0], x1[0])≥COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [(-1)bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(7)    (x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_689_0_MAIN_LE(TRUE, x0, x1) → 689_0_MAIN_LE(x0, +(x0, x1)) the following chains were created:
• We consider the chain 689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0]), COND_689_0_MAIN_LE(TRUE, x0[1], x1[1]) → 689_0_MAIN_LE(x0[1], +(x0[1], x1[1])), 689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0]) which results in the following constraint:

(8)    (&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0))=TRUEx0[0]=x0[1]x1[0]=x1[1]x0[1]=x0[0]1+(x0[1], x1[1])=x1[0]1COND_689_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_689_0_MAIN_LE(TRUE, x0[1], x1[1])≥689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))∧(UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥))

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

(9)    (>(x0[0], 0)=TRUE>=(x1[0], 0)=TRUE<(x1[0], x0[0])=TRUECOND_689_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_689_0_MAIN_LE(TRUE, x0[0], x1[0])≥689_0_MAIN_LE(x0[0], +(x0[0], x1[0]))∧(UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥))

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

(10)    (x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x0[0] ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x0[0] ≥ 0)

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

(12)    (x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x0[0] ≥ 0)

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

(13)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [(-1)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[1 + (-1)bso_17] + x0[0] ≥ 0)

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

(14)    (x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[1 + (-1)bso_17] + x1[0] + x0[0] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 689_0_MAIN_LE(x0, x1) → COND_689_0_MAIN_LE(&&(&&(>=(x1, 0), <(x1, x0)), >(x0, 0)), x0, x1)
• (x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• COND_689_0_MAIN_LE(TRUE, x0, x1) → 689_0_MAIN_LE(x0, +(x0, x1))
• (x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[1 + (-1)bso_17] + x1[0] + x0[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) = [3]
POL(FALSE) = [2]
POL(689_0_MAIN_LE(x1, x2)) = [-1]x2 + x1
POL(COND_689_0_MAIN_LE(x1, x2, x3)) = [-1]x3 + 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>:

COND_689_0_MAIN_LE(TRUE, x0[1], x1[1]) → 689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))

The following pairs are in Pbound:

689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])
COND_689_0_MAIN_LE(TRUE, x0[1], x1[1]) → 689_0_MAIN_LE(x0[1], +(x0[1], x1[1]))

The following pairs are in P:

689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), >(x0[0], 0)), x0[0], x1[0])

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

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 689_0_MAIN_LE(x0[0], x1[0]) → COND_689_0_MAIN_LE(x1[0] >= 0 && x1[0] < x0[0] && x0[0] > 0, x0[0], x1[0])

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.