### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB12
`/** * 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 PastaB12 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0 || y > 0) {            if (x > 0) {                x--;            } else if (y > 0) {                y--;            } else {                continue;            }        }    }}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:
PastaB12.main([Ljava/lang/String;)V: Graph of 186 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: PastaB12.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:
330_0_main_GT(EOS(STATIC_330), matching1, i49, matching2) → 337_0_main_GT(EOS(STATIC_337), 0, i49, 0) | &&(=(matching1, 0), =(matching2, 0))
330_0_main_GT(EOS(STATIC_330), i56, i49, i56) → 338_0_main_GT(EOS(STATIC_338), i56, i49, i56)
337_0_main_GT(EOS(STATIC_337), matching1, i49, matching2) → 351_0_main_Load(EOS(STATIC_351), 0, i49) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
351_0_main_Load(EOS(STATIC_351), matching1, i49) → 361_0_main_LE(EOS(STATIC_361), 0, i49, i49) | =(matching1, 0)
361_0_main_LE(EOS(STATIC_361), matching1, i60, i60) → 370_0_main_LE(EOS(STATIC_370), 0, i60, i60) | =(matching1, 0)
370_0_main_LE(EOS(STATIC_370), matching1, i60, i60) → 384_0_main_Load(EOS(STATIC_384), 0, i60) | &&(>(i60, 0), =(matching1, 0))
384_0_main_Load(EOS(STATIC_384), matching1, i60) → 397_0_main_LE(EOS(STATIC_397), 0, i60, 0) | =(matching1, 0)
397_0_main_LE(EOS(STATIC_397), matching1, i60, matching2) → 413_0_main_Load(EOS(STATIC_413), 0, i60) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
413_0_main_Load(EOS(STATIC_413), matching1, i60) → 421_0_main_LE(EOS(STATIC_421), 0, i60, i60) | =(matching1, 0)
421_0_main_LE(EOS(STATIC_421), matching1, i60, i60) → 433_0_main_Inc(EOS(STATIC_433), 0, i60) | &&(>(i60, 0), =(matching1, 0))
433_0_main_Inc(EOS(STATIC_433), matching1, i60) → 443_0_main_JMP(EOS(STATIC_443), 0, +(i60, -1)) | &&(>(i60, 0), =(matching1, 0))
443_0_main_JMP(EOS(STATIC_443), matching1, i72) → 453_0_main_Load(EOS(STATIC_453), 0, i72) | =(matching1, 0)
453_0_main_Load(EOS(STATIC_453), matching1, i72) → 322_0_main_Load(EOS(STATIC_322), 0, i72) | =(matching1, 0)
322_0_main_Load(EOS(STATIC_322), i13, i49) → 330_0_main_GT(EOS(STATIC_330), i13, i49, i13)
338_0_main_GT(EOS(STATIC_338), i56, i49, i56) → 354_0_main_Load(EOS(STATIC_354), i56, i49) | >(i56, 0)
354_0_main_Load(EOS(STATIC_354), i56, i49) → 363_0_main_LE(EOS(STATIC_363), i56, i49, i56)
363_0_main_LE(EOS(STATIC_363), i56, i49, i56) → 372_0_main_Inc(EOS(STATIC_372), i56, i49) | >(i56, 0)
372_0_main_Inc(EOS(STATIC_372), i56, i49) → 386_0_main_JMP(EOS(STATIC_386), +(i56, -1), i49) | >(i56, 0)
386_0_main_JMP(EOS(STATIC_386), i63, i49) → 404_0_main_Load(EOS(STATIC_404), i63, i49)
404_0_main_Load(EOS(STATIC_404), i63, i49) → 322_0_main_Load(EOS(STATIC_322), i63, i49)
R rules:

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

P rules:
330_0_main_GT(EOS(STATIC_330), 0, x1, 0) → 330_0_main_GT(EOS(STATIC_330), 0, +(x1, -1), 0) | >(x1, 0)
330_0_main_GT(EOS(STATIC_330), x0, x1, x0) → 330_0_main_GT(EOS(STATIC_330), +(x0, -1), x1, +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

330_0_main_GT(x1, x2, x3, x4) → 330_0_main_GT(x2, x3, x4)
EOS(x1) → EOS
Cond_330_0_main_GT1(x1, x2, x3, x4, x5) → Cond_330_0_main_GT1(x1, x3, x4, x5)
Cond_330_0_main_GT(x1, x2, x3, x4, x5) → Cond_330_0_main_GT(x1, x4)

Filtered duplicate args:

330_0_main_GT(x1, x2, x3) → 330_0_main_GT(x2, x3)
Cond_330_0_main_GT1(x1, x2, x3, x4) → Cond_330_0_main_GT1(x1, x3, x4)

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

P rules:
330_0_main_GT(x1, 0) → 330_0_main_GT(+(x1, -1), 0) | >(x1, 0)
330_0_main_GT(x1, x0) → 330_0_main_GT(x1, +(x0, -1)) | >(x0, 0)
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
330_0_MAIN_GT(x1, 0) → COND_330_0_MAIN_GT(>(x1, 0), x1, 0)
COND_330_0_MAIN_GT(TRUE, x1, 0) → 330_0_MAIN_GT(+(x1, -1), 0)
330_0_MAIN_GT(x1, x0) → COND_330_0_MAIN_GT1(>(x0, 0), x1, x0)
COND_330_0_MAIN_GT1(TRUE, x1, x0) → 330_0_MAIN_GT(x1, +(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(x1[0] > 0, x1[0], 0)
(1): COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(x1[1] + -1, 0)
(2): 330_0_MAIN_GT(x1[2], x0[2]) → COND_330_0_MAIN_GT1(x0[2] > 0, x1[2], x0[2])
(3): COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 330_0_MAIN_GT(x1[3], x0[3] + -1)

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

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

(1) -> (2), if (x1[1] + -1* x1[2]0* x0[2])

(2) -> (3), if (x0[2] > 0x1[2]* x1[3]x0[2]* x0[3])

(3) -> (0), if (x1[3]* x1[0]x0[3] + -1* 0)

(3) -> (2), if (x1[3]* x1[2]x0[3] + -1* x0[2])

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@59a8f100 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 330_0_MAIN_GT(x1, 0) → COND_330_0_MAIN_GT(>(x1, 0), x1, 0) the following chains were created:
• We consider the chain 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0), COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(1)    (>(x1[0], 0)=TRUEx1[0]=x1[1]330_0_MAIN_GT(x1[0], 0)≥NonInfC∧330_0_MAIN_GT(x1[0], 0)≥COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(2)    (>(x1[0], 0)=TRUE330_0_MAIN_GT(x1[0], 0)≥NonInfC∧330_0_MAIN_GT(x1[0], 0)≥COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 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_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_330_0_MAIN_GT(TRUE, x1, 0) → 330_0_MAIN_GT(+(x1, -1), 0) the following chains were created:
• We consider the chain COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(7)    (COND_330_0_MAIN_GT(TRUE, x1[1], 0)≥NonInfC∧COND_330_0_MAIN_GT(TRUE, x1[1], 0)≥330_0_MAIN_GT(+(x1[1], -1), 0)∧(UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥))

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

(8)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_12] = 0∧[(-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_12] = 0∧[(-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_12] = 0∧[(-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_12] = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair 330_0_MAIN_GT(x1, x0) → COND_330_0_MAIN_GT1(>(x0, 0), x1, x0) the following chains were created:
• We consider the chain 330_0_MAIN_GT(x1[2], x0[2]) → COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2]), COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 330_0_MAIN_GT(x1[3], +(x0[3], -1)) which results in the following constraint:

(12)    (>(x0[2], 0)=TRUEx1[2]=x1[3]x0[2]=x0[3]330_0_MAIN_GT(x1[2], x0[2])≥NonInfC∧330_0_MAIN_GT(x1[2], x0[2])≥COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])∧(UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥))

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

(13)    (>(x0[2], 0)=TRUE330_0_MAIN_GT(x1[2], x0[2])≥NonInfC∧330_0_MAIN_GT(x1[2], x0[2])≥COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])∧(UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥))

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

(14)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(15)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(16)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(17)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(18)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_330_0_MAIN_GT1(TRUE, x1, x0) → 330_0_MAIN_GT(x1, +(x0, -1)) the following chains were created:
• We consider the chain COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 330_0_MAIN_GT(x1[3], +(x0[3], -1)) which results in the following constraint:

(19)    (COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3])≥330_0_MAIN_GT(x1[3], +(x0[3], -1))∧(UIncreasing(330_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥))

We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(20)    ((UIncreasing(330_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[bni_16] = 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(21)    ((UIncreasing(330_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[bni_16] = 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(22)    ((UIncreasing(330_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[bni_16] = 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (22) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(23)    ((UIncreasing(330_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 330_0_MAIN_GT(x1, 0) → COND_330_0_MAIN_GT(>(x1, 0), x1, 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_330_0_MAIN_GT(TRUE, x1, 0) → 330_0_MAIN_GT(+(x1, -1), 0)
• ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_12] = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

• 330_0_MAIN_GT(x1, x0) → COND_330_0_MAIN_GT1(>(x0, 0), x1, x0)
• (x0[2] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• COND_330_0_MAIN_GT1(TRUE, x1, x0) → 330_0_MAIN_GT(x1, +(x0, -1))
• ((UIncreasing(330_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 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(330_0_MAIN_GT(x1, x2)) = [-1] + x2
POL(0) = 0
POL(COND_330_0_MAIN_GT(x1, x2, x3)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_330_0_MAIN_GT1(x1, x2, x3)) = [-1] + x3

The following pairs are in P>:

COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 330_0_MAIN_GT(x1[3], +(x0[3], -1))

The following pairs are in Pbound:

330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)
330_0_MAIN_GT(x1[2], x0[2]) → COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])

The following pairs are in P:

330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)
COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0)
330_0_MAIN_GT(x1[2], x0[2]) → COND_330_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])

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): 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(x1[0] > 0, x1[0], 0)
(1): COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(x1[1] + -1, 0)
(2): 330_0_MAIN_GT(x1[2], x0[2]) → COND_330_0_MAIN_GT1(x0[2] > 0, x1[2], x0[2])

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

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

(1) -> (2), if (x1[1] + -1* x1[2]0* x0[2])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (11) 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_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(x1[1] + -1, 0)
(0): 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(x1[0] > 0, x1[0], 0)

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

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

The set Q is empty.

### (12) 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@59a8f100 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 COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0) the following chains were created:
• We consider the chain COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(1)    (COND_330_0_MAIN_GT(TRUE, x1[1], 0)≥NonInfC∧COND_330_0_MAIN_GT(TRUE, x1[1], 0)≥330_0_MAIN_GT(+(x1[1], -1), 0)∧(UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥))

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

(2)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_6] = 0∧[2 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_6] = 0∧[2 + (-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_6] = 0∧[2 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_6] = 0∧0 = 0∧[2 + (-1)bso_7] ≥ 0)

For Pair 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0) the following chains were created:
• We consider the chain 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0), COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(6)    (>(x1[0], 0)=TRUEx1[0]=x1[1]330_0_MAIN_GT(x1[0], 0)≥NonInfC∧330_0_MAIN_GT(x1[0], 0)≥COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(7)    (>(x1[0], 0)=TRUE330_0_MAIN_GT(x1[0], 0)≥NonInfC∧330_0_MAIN_GT(x1[0], 0)≥COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(8)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0)
• ((UIncreasing(330_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[bni_6] = 0∧0 = 0∧[2 + (-1)bso_7] ≥ 0)

• 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 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(COND_330_0_MAIN_GT(x1, x2, x3)) = [-1] + [2]x2
POL(0) = 0
POL(330_0_MAIN_GT(x1, x2)) = [-1] + [2]x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [2]

The following pairs are in P>:

COND_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(+(x1[1], -1), 0)

The following pairs are in Pbound:

330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)

The following pairs are in P:

330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(>(x1[0], 0), x1[0], 0)

There are no usable rules.

### (14) 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): 330_0_MAIN_GT(x1[0], 0) → COND_330_0_MAIN_GT(x1[0] > 0, x1[0], 0)

The set Q is empty.

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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

### (17) 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_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(x1[1] + -1, 0)

The set Q is empty.

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) 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_330_0_MAIN_GT(TRUE, x1[1], 0) → 330_0_MAIN_GT(x1[1] + -1, 0)
(3): COND_330_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 330_0_MAIN_GT(x1[3], x0[3] + -1)

The set Q is empty.

### (21) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.