### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB5
`/** * 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 PastaB5 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        while (x > 0 && (x % 2) == 0) {            x--;        }    }}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:
PastaB5.main([Ljava/lang/String;)V: Graph of 113 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: PastaB5.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 11 rules for P and 0 rules for R.

P rules:
146_0_main_LE(EOS(STATIC_146), i22, i22) → 149_0_main_LE(EOS(STATIC_149), i22, i22)
149_0_main_LE(EOS(STATIC_149), i22, i22) → 158_0_main_Load(EOS(STATIC_158), i22) | >(i22, 0)
158_0_main_Load(EOS(STATIC_158), i22) → 166_0_main_ConstantStackPush(EOS(STATIC_166), i22, i22)
166_0_main_ConstantStackPush(EOS(STATIC_166), i22, i22) → 170_0_main_IntArithmetic(EOS(STATIC_170), i22, i22, 2)
170_0_main_IntArithmetic(EOS(STATIC_170), i22, i22, matching1) → 174_0_main_NE(EOS(STATIC_174), i22, %(i22, 2)) | =(matching1, 2)
174_0_main_NE(EOS(STATIC_174), i22, matching1) → 183_0_main_NE(EOS(STATIC_183), i22, 0) | =(matching1, 0)
183_0_main_NE(EOS(STATIC_183), i22, matching1) → 195_0_main_Inc(EOS(STATIC_195), i22) | =(matching1, 0)
195_0_main_Inc(EOS(STATIC_195), i22) → 204_0_main_JMP(EOS(STATIC_204), +(i22, -1)) | >(i22, 0)
143_0_main_Load(EOS(STATIC_143), i18) → 146_0_main_LE(EOS(STATIC_146), i18, i18)
R rules:

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

P rules:
146_0_main_LE(EOS(STATIC_146), x0, x0) → 146_0_main_LE(EOS(STATIC_146), +(x0, -1), +(x0, -1)) | &&(>(x0, 0), =(0, %(x0, 2)))
R rules:

Filtered ground terms:

146_0_main_LE(x1, x2, x3) → 146_0_main_LE(x2, x3)
EOS(x1) → EOS
Cond_146_0_main_LE(x1, x2, x3, x4) → Cond_146_0_main_LE(x1, x3, x4)

Filtered duplicate args:

146_0_main_LE(x1, x2) → 146_0_main_LE(x2)
Cond_146_0_main_LE(x1, x2, x3) → Cond_146_0_main_LE(x1, x3)

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

P rules:
146_0_main_LE(x0) → 146_0_main_LE(+(x0, -1)) | &&(>(x0, 0), =(0, %(x0, 2)))
R rules:

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

P rules:
146_0_MAIN_LE(x0) → COND_146_0_MAIN_LE(&&(>(x0, 0), =(0, %(x0, 2))), x0)
COND_146_0_MAIN_LE(TRUE, x0) → 146_0_MAIN_LE(+(x0, -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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 146_0_MAIN_LE(x0[0]) → COND_146_0_MAIN_LE(x0[0] > 0 && 0 = x0[0] % 2, x0[0])
(1): COND_146_0_MAIN_LE(TRUE, x0[1]) → 146_0_MAIN_LE(x0[1] + -1)

(0) -> (1), if (x0[0] > 0 && 0 = x0[0] % 2x0[0]* x0[1])

(1) -> (0), if (x0[1] + -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.IdpCand1ShapeHeuristic@3ff23f8b 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 146_0_MAIN_LE(x0) → COND_146_0_MAIN_LE(&&(>(x0, 0), =(0, %(x0, 2))), x0) the following chains were created:
• We consider the chain 146_0_MAIN_LE(x0[0]) → COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0]), COND_146_0_MAIN_LE(TRUE, x0[1]) → 146_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:

(1)    (&&(>(x0[0], 0), =(0, %(x0[0], 2)))=TRUEx0[0]=x0[1]146_0_MAIN_LE(x0[0])≥NonInfC∧146_0_MAIN_LE(x0[0])≥COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])∧(UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE>=(0, %(x0[0], 2))=TRUE<=(0, %(x0[0], 2))=TRUE146_0_MAIN_LE(x0[0])≥NonInfC∧146_0_MAIN_LE(x0[0])≥COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])∧(UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x0[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(7)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_146_0_MAIN_LE(TRUE, x0) → 146_0_MAIN_LE(+(x0, -1)) the following chains were created:
• We consider the chain COND_146_0_MAIN_LE(TRUE, x0[1]) → 146_0_MAIN_LE(+(x0[1], -1)) which results in the following constraint:

(8)    (COND_146_0_MAIN_LE(TRUE, x0[1])≥NonInfC∧COND_146_0_MAIN_LE(TRUE, x0[1])≥146_0_MAIN_LE(+(x0[1], -1))∧(UIncreasing(146_0_MAIN_LE(+(x0[1], -1))), ≥))

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

(9)    ((UIncreasing(146_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(146_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(146_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_12] = 0∧[2 + (-1)bso_13] ≥ 0)

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

(12)    ((UIncreasing(146_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 146_0_MAIN_LE(x0) → COND_146_0_MAIN_LE(&&(>(x0, 0), =(0, %(x0, 2))), x0)
• (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_146_0_MAIN_LE(TRUE, x0) → 146_0_MAIN_LE(+(x0, -1))
• ((UIncreasing(146_0_MAIN_LE(+(x0[1], -1))), ≥)∧[bni_12] = 0∧0 = 0∧[2 + (-1)bso_13] ≥ 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(146_0_MAIN_LE(x1)) = [2]x1
POL(COND_146_0_MAIN_LE(x1, x2)) = [2]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(=(x1, x2)) = [-1]
POL(2) = [2]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}

The following pairs are in P>:

COND_146_0_MAIN_LE(TRUE, x0[1]) → 146_0_MAIN_LE(+(x0[1], -1))

The following pairs are in Pbound:

146_0_MAIN_LE(x0[0]) → COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[0])

The following pairs are in P:

146_0_MAIN_LE(x0[0]) → COND_146_0_MAIN_LE(&&(>(x0[0], 0), =(0, %(x0[0], 2))), x0[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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 146_0_MAIN_LE(x0[0]) → COND_146_0_MAIN_LE(x0[0] > 0 && 0 = x0[0] % 2, x0[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_146_0_MAIN_LE(TRUE, x0[1]) → 146_0_MAIN_LE(x0[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.