### (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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
PastaB3.main([Ljava/lang/String;)V: Graph of 182 nodes with 1 SCC.

### (3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

### (4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: PastaB3.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 10 rules for P and 0 rules for R.

P rules:
328_0_main_Load(EOS(STATIC_328), i51, i47, i51) → 333_0_main_LE(EOS(STATIC_333), i51, i47, i51, i47)
333_0_main_LE(EOS(STATIC_333), i51, i47, i51, i47) → 343_0_main_LE(EOS(STATIC_343), i51, i47, i51, i47)
343_0_main_LE(EOS(STATIC_343), i51, i47, i51, i47) → 362_0_main_Load(EOS(STATIC_362), i51, i47) | >(i51, i47)
362_0_main_Load(EOS(STATIC_362), i51, i47) → 372_0_main_Load(EOS(STATIC_372), i51, i47, i51)
372_0_main_Load(EOS(STATIC_372), i51, i47, i51) → 382_0_main_IntArithmetic(EOS(STATIC_382), i51, i51, i47)
382_0_main_IntArithmetic(EOS(STATIC_382), i51, i51, i47) → 395_0_main_Store(EOS(STATIC_395), i51, +(i51, i47)) | &&(>(i51, 0), >=(i47, 0))
395_0_main_Store(EOS(STATIC_395), i51, i55) → 404_0_main_JMP(EOS(STATIC_404), i51, i55)
404_0_main_JMP(EOS(STATIC_404), i51, i55) → 416_0_main_Load(EOS(STATIC_416), i51, i55)
416_0_main_Load(EOS(STATIC_416), i51, i55) → 319_0_main_Load(EOS(STATIC_319), i51, i55)
319_0_main_Load(EOS(STATIC_319), i51, i47) → 328_0_main_Load(EOS(STATIC_328), i51, i47, i51)
R rules:

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

P rules:
328_0_main_Load(EOS(STATIC_328), x0, x1, x0) → 328_0_main_Load(EOS(STATIC_328), x0, +(x0, x1), x0) | &&(&&(>(+(x1, 1), 0), <(x1, x0)), >(x0, 0))
R rules:

Filtered ground terms:

328_0_main_Load(x1, x2, x3, x4) → 328_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_328_0_main_Load(x1, x2, x3, x4, x5) → Cond_328_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

328_0_main_Load(x1, x2, x3) → 328_0_main_Load(x2, x3)
Cond_328_0_main_Load(x1, x2, x3, x4) → Cond_328_0_main_Load(x1, x3, x4)

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

P rules:
328_0_main_Load(x1, x0) → 328_0_main_Load(+(x0, x1), x0) | &&(&&(>(x1, -1), <(x1, x0)), >(x0, 0))
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
328_0_MAIN_LOAD(x1, x0) → COND_328_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, 0)), x1, x0)
COND_328_0_MAIN_LOAD(TRUE, x1, x0) → 328_0_MAIN_LOAD(+(x0, x1), x0)
R 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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 328_0_MAIN_LOAD(x1[0], x0[0]) → COND_328_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0] && x0[0] > 0, x1[0], x0[0])
(1): COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 328_0_MAIN_LOAD(x0[1] + x1[1], x0[1])

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

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

The set Q is empty.

### (7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5cf6930 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 328_0_MAIN_LOAD(x1, x0) → COND_328_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 328_0_MAIN_LOAD(x1[0], x0[0]) → COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0]), COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1]) which results in the following constraint:

(1)    (&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]328_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧328_0_MAIN_LOAD(x1[0], x0[0])≥COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE>(x1[0], -1)=TRUE<(x1[0], x0[0])=TRUE328_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧328_0_MAIN_LOAD(x1[0], x0[0])≥COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[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_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] + x0[0] ≥ 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_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] + x0[0] ≥ 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_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[(-1)bso_14] + x0[0] ≥ 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_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 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_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] + x1[0] + x0[0] ≥ 0)

For Pair COND_328_0_MAIN_LOAD(TRUE, x1, x0) → 328_0_MAIN_LOAD(+(x0, x1), x0) the following chains were created:
• We consider the chain 328_0_MAIN_LOAD(x1[0], x0[0]) → COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0]), COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1]), 328_0_MAIN_LOAD(x1[0], x0[0]) → COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(8)    (&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]+(x0[1], x1[1])=x1[0]1x0[1]=x0[0]1COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])∧(UIncreasing(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[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], -1)=TRUE<(x1[0], x0[0])=TRUECOND_328_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_328_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥328_0_MAIN_LOAD(+(x0[0], x1[0]), x0[0])∧(UIncreasing(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[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(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 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(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 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(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 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(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 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(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 328_0_MAIN_LOAD(x1, x0) → COND_328_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, 0)), x1, x0)
• (x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] + x1[0] + x0[0] ≥ 0)

• COND_328_0_MAIN_LOAD(TRUE, x1, x0) → 328_0_MAIN_LOAD(+(x0, x1), x0)
• (x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 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(328_0_MAIN_LOAD(x1, x2)) = [-1] + x2 + [-1]x1
POL(COND_328_0_MAIN_LOAD(x1, x2, x3)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(<(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

328_0_MAIN_LOAD(x1[0], x0[0]) → COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in Pbound:

328_0_MAIN_LOAD(x1[0], x0[0]) → COND_328_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P:

COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 328_0_MAIN_LOAD(+(x0[1], x1[1]), x0[1])

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

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_328_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 328_0_MAIN_LOAD(x0[1] + x1[1], x0[1])

The set Q is empty.

### (9) IDependencyGraphProof (EQUIVALENT transformation)

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