### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB15
`/** * 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 PastaB15 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        int z = Random.random();        while (x == y && x > z) {            while (y > z) {                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:
PastaB15.main([Ljava/lang/String;)V: Graph of 257 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: PastaB15.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 20 rules for P and 0 rules for R.

P rules:
742_0_main_Load(EOS(STATIC_742), i125, i126, i87, i125) → 744_0_main_NE(EOS(STATIC_744), i125, i126, i87, i125, i126)
744_0_main_NE(EOS(STATIC_744), i126, i126, i87, i126, i126) → 748_0_main_NE(EOS(STATIC_748), i126, i126, i87, i126, i126)
748_0_main_NE(EOS(STATIC_748), i126, i126, i87, i126, i126) → 752_0_main_Load(EOS(STATIC_752), i126, i126, i87)
755_0_main_Load(EOS(STATIC_755), i126, i126, i87, i126) → 757_0_main_LE(EOS(STATIC_757), i126, i126, i87, i126, i87)
757_0_main_LE(EOS(STATIC_757), i126, i126, i87, i126, i87) → 760_0_main_LE(EOS(STATIC_760), i126, i126, i87, i126, i87)
760_0_main_LE(EOS(STATIC_760), i126, i126, i87, i126, i87) → 766_0_main_Load(EOS(STATIC_766), i126, i126, i87) | >(i126, i87)
795_0_main_Load(EOS(STATIC_795), i134, i135, i87, i135) → 798_0_main_LE(EOS(STATIC_798), i134, i135, i87, i135, i87)
798_0_main_LE(EOS(STATIC_798), i134, i135, i87, i135, i87) → 800_0_main_LE(EOS(STATIC_800), i134, i135, i87, i135, i87)
798_0_main_LE(EOS(STATIC_798), i134, i135, i87, i135, i87) → 801_0_main_LE(EOS(STATIC_801), i134, i135, i87, i135, i87)
800_0_main_LE(EOS(STATIC_800), i134, i135, i87, i135, i87) → 804_0_main_Load(EOS(STATIC_804), i134, i135, i87) | <=(i135, i87)
801_0_main_LE(EOS(STATIC_801), i134, i135, i87, i135, i87) → 806_0_main_Inc(EOS(STATIC_806), i134, i135, i87) | >(i135, i87)
806_0_main_Inc(EOS(STATIC_806), i134, i135, i87) → 808_0_main_Inc(EOS(STATIC_808), +(i134, -1), i135, i87)
808_0_main_Inc(EOS(STATIC_808), i144, i135, i87) → 810_0_main_JMP(EOS(STATIC_810), i144, +(i135, -1), i87)
810_0_main_JMP(EOS(STATIC_810), i144, i145, i87) → 813_0_main_Load(EOS(STATIC_813), i144, i145, i87)
R rules:

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

P rules:
798_0_main_LE(EOS(STATIC_798), x0, x0, x1, x0, x1) → 798_0_main_LE(EOS(STATIC_798), x0, x0, x1, x0, x1) | FALSE
798_0_main_LE(EOS(STATIC_798), x0, x1, x2, x1, x2) → 798_0_main_LE(EOS(STATIC_798), +(x0, -1), +(x1, -1), x2, +(x1, -1), x2) | <(x2, x1)
R rules:

Filtered ground terms:

798_0_main_LE(x1, x2, x3, x4, x5, x6) → 798_0_main_LE(x2, x3, x4, x5, x6)
EOS(x1) → EOS
Cond_798_0_main_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_798_0_main_LE(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

798_0_main_LE(x1, x2, x3, x4, x5) → 798_0_main_LE(x1, x4, x5)
Cond_798_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_798_0_main_LE(x1, x2, x5, x6)

Filtered unneeded arguments:

Cond_798_0_main_LE(x1, x2, x3, x4) → Cond_798_0_main_LE(x1, x3, x4)
798_0_main_LE(x1, x2, x3) → 798_0_main_LE(x2, x3)

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

P rules:
798_0_main_LE(x1, x2) → 798_0_main_LE(+(x1, -1), x2) | <(x2, x1)
R rules:

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

P rules:
798_0_MAIN_LE(x1, x2) → COND_798_0_MAIN_LE(<(x2, x1), x1, x2)
COND_798_0_MAIN_LE(TRUE, x1, x2) → 798_0_MAIN_LE(+(x1, -1), x2)
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): 798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(x2[0] < x1[0], x1[0], x2[0])
(1): COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(x1[1] + -1, x2[1])

(0) -> (1), if (x2[0] < x1[0]x1[0]* x1[1]x2[0]* x2[1])

(1) -> (0), if (x1[1] + -1* x1[0]x2[1]* x2[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@bb51061 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 798_0_MAIN_LE(x1, x2) → COND_798_0_MAIN_LE(<(x2, x1), x1, x2) the following chains were created:
• We consider the chain 798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0]), COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(+(x1[1], -1), x2[1]) which results in the following constraint:

(1)    (<(x2[0], x1[0])=TRUEx1[0]=x1[1]x2[0]=x2[1]798_0_MAIN_LE(x1[0], x2[0])≥NonInfC∧798_0_MAIN_LE(x1[0], x2[0])≥COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])∧(UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥))

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

(2)    (<(x2[0], x1[0])=TRUE798_0_MAIN_LE(x1[0], x2[0])≥NonInfC∧798_0_MAIN_LE(x1[0], x2[0])≥COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])∧(UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥))

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

(3)    (x1[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]x2[0] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

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

(4)    (x1[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]x2[0] + [bni_8]x1[0] ≥ 0∧[1 + (-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] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]x2[0] + [bni_8]x1[0] ≥ 0∧[1 + (-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_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

(8)    (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

For Pair COND_798_0_MAIN_LE(TRUE, x1, x2) → 798_0_MAIN_LE(+(x1, -1), x2) the following chains were created:
• We consider the chain COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(+(x1[1], -1), x2[1]) which results in the following constraint:

(9)    (COND_798_0_MAIN_LE(TRUE, x1[1], x2[1])≥NonInfC∧COND_798_0_MAIN_LE(TRUE, x1[1], x2[1])≥798_0_MAIN_LE(+(x1[1], -1), x2[1])∧(UIncreasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥))

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

(10)    ((UIncreasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

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

(12)    ((UIncreasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

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

(13)    ((UIncreasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 798_0_MAIN_LE(x1, x2) → COND_798_0_MAIN_LE(<(x2, x1), x1, x2)
• (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
• (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

• COND_798_0_MAIN_LE(TRUE, x1, x2) → 798_0_MAIN_LE(+(x1, -1), x2)
• ((UIncreasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-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(798_0_MAIN_LE(x1, x2)) = [1] + [-1]x2 + x1
POL(COND_798_0_MAIN_LE(x1, x2, x3)) = [-1]x3 + x2
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])

The following pairs are in Pbound:

798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])

The following pairs are in P:

COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(+(x1[1], -1), x2[1])

There are no usable rules.

### (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_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(x1[1] + -1, x2[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.