### (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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
PastaB14.main([Ljava/lang/String;)V: Graph of 189 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: PastaB14.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 19 rules for P and 0 rules for R.

P rules:
1160_0_main_Load(EOS(STATIC_1160), i441, i442, i441) → 1162_0_main_NE(EOS(STATIC_1162), i441, i442, i441, i442)
1162_0_main_NE(EOS(STATIC_1162), i442, i442, i442, i442) → 1165_0_main_NE(EOS(STATIC_1165), i442, i442, i442, i442)
1165_0_main_NE(EOS(STATIC_1165), i442, i442, i442, i442) → 1168_0_main_Load(EOS(STATIC_1168), i442, i442)
1168_0_main_Load(EOS(STATIC_1168), i442, i442) → 1170_0_main_LE(EOS(STATIC_1170), i442, i442, i442)
1170_0_main_LE(EOS(STATIC_1170), i447, i447, i447) → 1173_0_main_LE(EOS(STATIC_1173), i447, i447, i447)
1173_0_main_LE(EOS(STATIC_1173), i447, i447, i447) → 1177_0_main_Load(EOS(STATIC_1177), i447, i447) | >(i447, 0)
1296_0_main_Load(EOS(STATIC_1296), i566, i567) → 1298_0_main_LE(EOS(STATIC_1298), i566, i567, i567)
1298_0_main_LE(EOS(STATIC_1298), i566, matching1, matching2) → 1300_0_main_LE(EOS(STATIC_1300), i566, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
1298_0_main_LE(EOS(STATIC_1298), i566, i572, i572) → 1301_0_main_LE(EOS(STATIC_1301), i566, i572, i572)
1300_0_main_LE(EOS(STATIC_1300), i566, matching1, matching2) → 1303_0_main_Load(EOS(STATIC_1303), i566, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
1301_0_main_LE(EOS(STATIC_1301), i566, i572, i572) → 1305_0_main_Inc(EOS(STATIC_1305), i566, i572) | >(i572, 0)
1305_0_main_Inc(EOS(STATIC_1305), i566, i572) → 1453_0_main_Inc(EOS(STATIC_1453), +(i566, -1), i572)
1453_0_main_Inc(EOS(STATIC_1453), i690, i572) → 1454_0_main_JMP(EOS(STATIC_1454), i690, +(i572, -1)) | >(i572, 0)
1454_0_main_JMP(EOS(STATIC_1454), i690, i691) → 1457_0_main_Load(EOS(STATIC_1457), i690, i691)
R rules:

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

P rules:
1298_0_main_LE(EOS(STATIC_1298), 0, x0, x1) → 1298_0_main_LE(EOS(STATIC_1298), 0, 0, 0) | FALSE
1298_0_main_LE(EOS(STATIC_1298), x0, x1, x1) → 1298_0_main_LE(EOS(STATIC_1298), +(x0, -1), +(x1, -1), +(x1, -1)) | >(x1, 0)
R rules:

Filtered ground terms:

1298_0_main_LE(x1, x2, x3, x4) → 1298_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_1298_0_main_LE(x1, x2, x3, x4, x5) → Cond_1298_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

1298_0_main_LE(x1, x2, x3) → 1298_0_main_LE(x1, x3)
Cond_1298_0_main_LE(x1, x2, x3, x4) → Cond_1298_0_main_LE(x1, x2, x4)

Filtered unneeded arguments:

Cond_1298_0_main_LE(x1, x2, x3) → Cond_1298_0_main_LE(x1, x3)
1298_0_main_LE(x1, x2) → 1298_0_main_LE(x2)

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

P rules:
1298_0_main_LE(x1) → 1298_0_main_LE(+(x1, -1)) | >(x1, 0)
R rules:

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

P rules:
1298_0_MAIN_LE(x1) → COND_1298_0_MAIN_LE(>(x1, 0), x1)
COND_1298_0_MAIN_LE(TRUE, x1) → 1298_0_MAIN_LE(+(x1, -1))
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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(x1[0] > 0, x1[0])
(1): COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(x1[1] + -1)

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

(1) -> (0), if (x1[1] + -1* x1[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.IdpCand1ShapeHeuristic@3661eeb Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 1298_0_MAIN_LE(x1) → COND_1298_0_MAIN_LE(>(x1, 0), x1) the following chains were created:
• We consider the chain 1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0]), COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(+(x1[1], -1)) which results in the following constraint:

(1)    (>(x1[0], 0)=TRUEx1[0]=x1[1]1298_0_MAIN_LE(x1[0])≥NonInfC∧1298_0_MAIN_LE(x1[0])≥COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])∧(UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:

(2)    (>(x1[0], 0)=TRUE1298_0_MAIN_LE(x1[0])≥NonInfC∧1298_0_MAIN_LE(x1[0])≥COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])∧(UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥))

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

(3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_1298_0_MAIN_LE(TRUE, x1) → 1298_0_MAIN_LE(+(x1, -1)) the following chains were created:
• We consider the chain COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(+(x1[1], -1)) which results in the following constraint:

(7)    (COND_1298_0_MAIN_LE(TRUE, x1[1])≥NonInfC∧COND_1298_0_MAIN_LE(TRUE, x1[1])≥1298_0_MAIN_LE(+(x1[1], -1))∧(UIncreasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥))

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

(8)    ((UIncreasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(9)    ((UIncreasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(10)    ((UIncreasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1298_0_MAIN_LE(x1) → COND_1298_0_MAIN_LE(>(x1, 0), x1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• COND_1298_0_MAIN_LE(TRUE, x1) → 1298_0_MAIN_LE(+(x1, -1))
• ((UIncreasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 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(1298_0_MAIN_LE(x1)) = [2]x1
POL(COND_1298_0_MAIN_LE(x1, x2)) = [2]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(+(x1[1], -1))

The following pairs are in Pbound:

1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])

The following pairs are in P:

1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])

There are no usable rules.

### (9) 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): 1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(x1[0] > 0, x1[0])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (12) 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_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(x1[1] + -1)

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

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