### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB16
`/** * 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 PastaB16 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0) {            while (y > 0) {                y--;            }            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:
PastaB16.main([Ljava/lang/String;)V: Graph of 180 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: PastaB16.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 14 rules for P and 0 rules for R.

P rules:
313_0_main_LE(EOS(STATIC_313), i55, i49, i55) → 320_0_main_LE(EOS(STATIC_320), i55, i49, i55)
320_0_main_LE(EOS(STATIC_320), i55, i49, i55) → 336_0_main_Load(EOS(STATIC_336), i55, i49) | >(i55, 0)
336_0_main_Load(EOS(STATIC_336), i55, i49) → 346_0_main_LE(EOS(STATIC_346), i55, i49, i49)
346_0_main_LE(EOS(STATIC_346), i55, matching1, matching2) → 353_0_main_LE(EOS(STATIC_353), i55, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
346_0_main_LE(EOS(STATIC_346), i55, i62, i62) → 354_0_main_LE(EOS(STATIC_354), i55, i62, i62)
353_0_main_LE(EOS(STATIC_353), i55, matching1, matching2) → 363_0_main_Inc(EOS(STATIC_363), i55, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
363_0_main_Inc(EOS(STATIC_363), i55, matching1) → 375_0_main_JMP(EOS(STATIC_375), +(i55, -1), 0) | &&(>(i55, 0), =(matching1, 0))
375_0_main_JMP(EOS(STATIC_375), i66, matching1) → 391_0_main_Load(EOS(STATIC_391), i66, 0) | =(matching1, 0)
305_0_main_Load(EOS(STATIC_305), i13, i49) → 313_0_main_LE(EOS(STATIC_313), i13, i49, i13)
354_0_main_LE(EOS(STATIC_354), i55, i62, i62) → 366_0_main_Inc(EOS(STATIC_366), i55, i62) | >(i62, 0)
366_0_main_Inc(EOS(STATIC_366), i55, i62) → 377_0_main_JMP(EOS(STATIC_377), i55, +(i62, -1)) | >(i62, 0)
377_0_main_JMP(EOS(STATIC_377), i55, i67) → 396_0_main_Load(EOS(STATIC_396), i55, i67)
R rules:

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

P rules:
346_0_main_LE(EOS(STATIC_346), x0, 0, 0) → 346_0_main_LE(EOS(STATIC_346), +(x0, -1), 0, 0) | >(x0, 1)
346_0_main_LE(EOS(STATIC_346), x0, x1, x1) → 346_0_main_LE(EOS(STATIC_346), x0, +(x1, -1), +(x1, -1)) | >(x1, 0)
R rules:

Filtered ground terms:

346_0_main_LE(x1, x2, x3, x4) → 346_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_346_0_main_LE1(x1, x2, x3, x4, x5) → Cond_346_0_main_LE1(x1, x3, x4, x5)
Cond_346_0_main_LE(x1, x2, x3, x4, x5) → Cond_346_0_main_LE(x1, x3)

Filtered duplicate args:

346_0_main_LE(x1, x2, x3) → 346_0_main_LE(x1, x3)
Cond_346_0_main_LE1(x1, x2, x3, x4) → Cond_346_0_main_LE1(x1, x2, x4)

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

P rules:
346_0_main_LE(x0, 0) → 346_0_main_LE(+(x0, -1), 0) | >(x0, 1)
346_0_main_LE(x0, x1) → 346_0_main_LE(x0, +(x1, -1)) | >(x1, 0)
R rules:

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

P rules:
346_0_MAIN_LE(x0, 0) → COND_346_0_MAIN_LE(>(x0, 1), x0, 0)
COND_346_0_MAIN_LE(TRUE, x0, 0) → 346_0_MAIN_LE(+(x0, -1), 0)
346_0_MAIN_LE(x0, x1) → COND_346_0_MAIN_LE1(>(x1, 0), x0, x1)
COND_346_0_MAIN_LE1(TRUE, x0, x1) → 346_0_MAIN_LE(x0, +(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): 346_0_MAIN_LE(x0[0], 0) → COND_346_0_MAIN_LE(x0[0] > 1, x0[0], 0)
(1): COND_346_0_MAIN_LE(TRUE, x0[1], 0) → 346_0_MAIN_LE(x0[1] + -1, 0)
(2): 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])
(3): COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], x1[3] + -1)

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

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

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

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

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

(3) -> (2), if (x0[3]* x0[2]x1[3] + -1* x1[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@72f1cbd7 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 346_0_MAIN_LE(x0, 0) → COND_346_0_MAIN_LE(>(x0, 1), x0, 0) the following chains were created:
• We consider the chain 346_0_MAIN_LE(x0[0], 0) → COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0), COND_346_0_MAIN_LE(TRUE, x0[1], 0) → 346_0_MAIN_LE(+(x0[1], -1), 0) which results in the following constraint:

(1)    (>(x0[0], 1)=TRUEx0[0]=x0[1]346_0_MAIN_LE(x0[0], 0)≥NonInfC∧346_0_MAIN_LE(x0[0], 0)≥COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)∧(UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥))

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

(2)    (>(x0[0], 1)=TRUE346_0_MAIN_LE(x0[0], 0)≥NonInfC∧346_0_MAIN_LE(x0[0], 0)≥COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)∧(UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥))

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

(3)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] + [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] + [-2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] + [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] + [-2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] + [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 ⇒ (UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥)∧[(3)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_346_0_MAIN_LE(TRUE, x0, 0) → 346_0_MAIN_LE(+(x0, -1), 0) the following chains were created:
• We consider the chain COND_346_0_MAIN_LE(TRUE, x0[1], 0) → 346_0_MAIN_LE(+(x0[1], -1), 0) which results in the following constraint:

(7)    (COND_346_0_MAIN_LE(TRUE, x0[1], 0)≥NonInfC∧COND_346_0_MAIN_LE(TRUE, x0[1], 0)≥346_0_MAIN_LE(+(x0[1], -1), 0)∧(UIncreasing(346_0_MAIN_LE(+(x0[1], -1), 0)), ≥))

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

(8)    ((UIncreasing(346_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(346_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(346_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(346_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[bni_12] = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

For Pair 346_0_MAIN_LE(x0, x1) → COND_346_0_MAIN_LE1(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2]), COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1)) which results in the following constraint:

(12)    (>(x1[2], 0)=TRUEx0[2]=x0[3]x1[2]=x1[3]346_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧346_0_MAIN_LE(x0[2], x1[2])≥COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(13)    (>(x1[2], 0)=TRUE346_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧346_0_MAIN_LE(x0[2], x1[2])≥COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(14)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] + [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)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] + [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)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] + [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)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_14] = 0∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x1[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)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_14] = 0∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_346_0_MAIN_LE1(TRUE, x0, x1) → 346_0_MAIN_LE(x0, +(x1, -1)) the following chains were created:
• We consider the chain COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1)) which results in the following constraint:

(19)    (COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3])≥346_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥))

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

(20)    ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[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(346_0_MAIN_LE(x0[3], +(x1[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(346_0_MAIN_LE(x0[3], +(x1[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(346_0_MAIN_LE(x0[3], +(x1[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.
• 346_0_MAIN_LE(x0, 0) → COND_346_0_MAIN_LE(>(x0, 1), x0, 0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)), ≥)∧[(3)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_346_0_MAIN_LE(TRUE, x0, 0) → 346_0_MAIN_LE(+(x0, -1), 0)
• ((UIncreasing(346_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[bni_12] = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

• 346_0_MAIN_LE(x0, x1) → COND_346_0_MAIN_LE1(>(x1, 0), x0, x1)
• (x1[2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_14] = 0∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• COND_346_0_MAIN_LE1(TRUE, x0, x1) → 346_0_MAIN_LE(x0, +(x1, -1))
• ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[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(346_0_MAIN_LE(x1, x2)) = [1] + x2 + x1
POL(0) = 0
POL(COND_346_0_MAIN_LE(x1, x2, x3)) = [1] + x2
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_346_0_MAIN_LE1(x1, x2, x3)) = [1] + x3 + x2

The following pairs are in P>:

COND_346_0_MAIN_LE(TRUE, x0[1], 0) → 346_0_MAIN_LE(+(x0[1], -1), 0)
COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1))

The following pairs are in Pbound:

346_0_MAIN_LE(x0[0], 0) → COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)

The following pairs are in P:

346_0_MAIN_LE(x0[0], 0) → COND_346_0_MAIN_LE(>(x0[0], 1), x0[0], 0)
346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[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): 346_0_MAIN_LE(x0[0], 0) → COND_346_0_MAIN_LE(x0[0] > 1, x0[0], 0)
(2): 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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_346_0_MAIN_LE(TRUE, x0[1], 0) → 346_0_MAIN_LE(x0[1] + -1, 0)
(2): 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])
(3): COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], x1[3] + -1)

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

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

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

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(3): COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], x1[3] + -1)
(2): 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])

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

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

The set Q is empty.

### (15) 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@72f1cbd7 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_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1)) the following chains were created:
• We consider the chain COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1)) which results in the following constraint:

(1)    (COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3])≥346_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥))

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

(2)    ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[bni_7] = 0∧[(-1)bso_8] ≥ 0)

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

(3)    ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[bni_7] = 0∧[(-1)bso_8] ≥ 0)

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

(4)    ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[bni_7] = 0∧[(-1)bso_8] ≥ 0)

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

(5)    ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[bni_7] = 0∧0 = 0∧[(-1)bso_8] ≥ 0)

For Pair 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2]) the following chains were created:
• We consider the chain 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2]), COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1)) which results in the following constraint:

(6)    (>(x1[2], 0)=TRUEx0[2]=x0[3]x1[2]=x1[3]346_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧346_0_MAIN_LE(x0[2], x1[2])≥COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(7)    (>(x1[2], 0)=TRUE346_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧346_0_MAIN_LE(x0[2], x1[2])≥COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

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

(8)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x1[2] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(9)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x1[2] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(10)    (x1[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x1[2] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(11)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x1[2] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1))
• ((UIncreasing(346_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[bni_7] = 0∧0 = 0∧[(-1)bso_8] ≥ 0)

• 346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])
• (x1[2] ≥ 0 ⇒ (UIncreasing(COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]x1[2] ≥ 0∧[2 + (-1)bso_10] ≥ 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_346_0_MAIN_LE1(x1, x2, x3)) = [-1] + [2]x3
POL(346_0_MAIN_LE(x1, x2)) = [1] + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])

The following pairs are in Pbound:

346_0_MAIN_LE(x0[2], x1[2]) → COND_346_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])

The following pairs are in P:

COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], +(x1[3], -1))

There are no usable rules.

### (16) 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:
(3): COND_346_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 346_0_MAIN_LE(x0[3], x1[3] + -1)

The set Q is empty.

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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