### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB18
`/** * 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 PastaB18 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0 && y > 0) {            if (x > y) {                while (x > 0) {                    x--;                }            } else {                while (y > 0) {                    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:
PastaB18.main([Ljava/lang/String;)V: Graph of 198 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: PastaB18.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 32 rules for P and 0 rules for R.

P rules:
303_0_main_LE(EOS(STATIC_303), i51, i47, i51) → 308_0_main_LE(EOS(STATIC_308), i51, i47, i51)
308_0_main_LE(EOS(STATIC_308), i51, i47, i51) → 320_0_main_Load(EOS(STATIC_320), i51, i47) | >(i51, 0)
320_0_main_Load(EOS(STATIC_320), i51, i47) → 330_0_main_LE(EOS(STATIC_330), i51, i47, i47)
330_0_main_LE(EOS(STATIC_330), i51, i54, i54) → 333_0_main_LE(EOS(STATIC_333), i51, i54, i54)
333_0_main_LE(EOS(STATIC_333), i51, i54, i54) → 347_0_main_Load(EOS(STATIC_347), i51, i54) | >(i54, 0)
358_0_main_Load(EOS(STATIC_358), i51, i54, i51) → 367_0_main_LE(EOS(STATIC_367), i51, i54, i51, i54)
367_0_main_LE(EOS(STATIC_367), i51, i54, i51, i54) → 375_0_main_LE(EOS(STATIC_375), i51, i54, i51, i54)
367_0_main_LE(EOS(STATIC_367), i51, i54, i51, i54) → 376_0_main_LE(EOS(STATIC_376), i51, i54, i51, i54)
375_0_main_LE(EOS(STATIC_375), i51, i54, i51, i54) → 389_0_main_Load(EOS(STATIC_389), i51, i54) | <=(i51, i54)
439_0_main_Load(EOS(STATIC_439), i51, i68) → 456_0_main_LE(EOS(STATIC_456), i51, i68, i68)
456_0_main_LE(EOS(STATIC_456), i51, matching1, matching2) → 464_0_main_LE(EOS(STATIC_464), i51, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
456_0_main_LE(EOS(STATIC_456), i51, i79, i79) → 465_0_main_LE(EOS(STATIC_465), i51, i79, i79)
464_0_main_LE(EOS(STATIC_464), i51, matching1, matching2) → 477_0_main_Load(EOS(STATIC_477), i51, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
297_0_main_Load(EOS(STATIC_297), i18, i47) → 303_0_main_LE(EOS(STATIC_303), i18, i47, i18)
465_0_main_LE(EOS(STATIC_465), i51, i79, i79) → 479_0_main_Inc(EOS(STATIC_479), i51, i79) | >(i79, 0)
479_0_main_Inc(EOS(STATIC_479), i51, i79) → 491_0_main_JMP(EOS(STATIC_491), i51, +(i79, -1)) | >(i79, 0)
491_0_main_JMP(EOS(STATIC_491), i51, i88) → 499_0_main_Load(EOS(STATIC_499), i51, i88)
376_0_main_LE(EOS(STATIC_376), i51, i54, i51, i54) → 391_0_main_Load(EOS(STATIC_391), i51, i54) | >(i51, i54)
450_0_main_Load(EOS(STATIC_450), i72, i54) → 459_0_main_LE(EOS(STATIC_459), i72, i54, i72)
459_0_main_LE(EOS(STATIC_459), matching1, i54, matching2) → 467_0_main_LE(EOS(STATIC_467), 0, i54, 0) | &&(=(matching1, 0), =(matching2, 0))
459_0_main_LE(EOS(STATIC_459), i81, i54, i81) → 468_0_main_LE(EOS(STATIC_468), i81, i54, i81)
467_0_main_LE(EOS(STATIC_467), matching1, i54, matching2) → 484_0_main_Load(EOS(STATIC_484), 0, i54) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
468_0_main_LE(EOS(STATIC_468), i81, i54, i81) → 486_0_main_Inc(EOS(STATIC_486), i81, i54) | >(i81, 0)
486_0_main_Inc(EOS(STATIC_486), i81, i54) → 494_0_main_JMP(EOS(STATIC_494), +(i81, -1), i54) | >(i81, 0)
494_0_main_JMP(EOS(STATIC_494), i89, i54) → 503_0_main_Load(EOS(STATIC_503), i89, i54)
R rules:

Combined rules. Obtained 6 conditional rules for P and 0 conditional rules for R.

P rules:
303_0_main_LE(EOS(STATIC_303), x0, x1, x0) → 456_0_main_LE(EOS(STATIC_456), x0, x1, x1) | &&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0))
456_0_main_LE(EOS(STATIC_456), x0, 0, 0) → 303_0_main_LE(EOS(STATIC_303), x0, 0, x0)
456_0_main_LE(EOS(STATIC_456), x0, x1, x1) → 456_0_main_LE(EOS(STATIC_456), x0, +(x1, -1), +(x1, -1)) | >(x1, 0)
303_0_main_LE(EOS(STATIC_303), x0, x1, x0) → 459_0_main_LE(EOS(STATIC_459), x0, x1, x0) | &&(&&(>(x1, 0), <(x1, x0)), >(x0, 0))
459_0_main_LE(EOS(STATIC_459), 0, x1, 0) → 303_0_main_LE(EOS(STATIC_303), 0, x1, 0)
459_0_main_LE(EOS(STATIC_459), x0, x1, x0) → 459_0_main_LE(EOS(STATIC_459), +(x0, -1), x1, +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

459_0_main_LE(x1, x2, x3, x4) → 459_0_main_LE(x2, x3, x4)
Cond_459_0_main_LE(x1, x2, x3, x4, x5) → Cond_459_0_main_LE(x1, x3, x4, x5)
303_0_main_LE(x1, x2, x3, x4) → 303_0_main_LE(x2, x3, x4)
Cond_303_0_main_LE1(x1, x2, x3, x4, x5) → Cond_303_0_main_LE1(x1, x3, x4, x5)
456_0_main_LE(x1, x2, x3, x4) → 456_0_main_LE(x2, x3, x4)
Cond_456_0_main_LE(x1, x2, x3, x4, x5) → Cond_456_0_main_LE(x1, x3, x4, x5)
Cond_303_0_main_LE(x1, x2, x3, x4, x5) → Cond_303_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

303_0_main_LE(x1, x2, x3) → 303_0_main_LE(x2, x3)
Cond_303_0_main_LE(x1, x2, x3, x4) → Cond_303_0_main_LE(x1, x3, x4)
456_0_main_LE(x1, x2, x3) → 456_0_main_LE(x1, x3)
Cond_456_0_main_LE(x1, x2, x3, x4) → Cond_456_0_main_LE(x1, x2, x4)
Cond_303_0_main_LE1(x1, x2, x3, x4) → Cond_303_0_main_LE1(x1, x3, x4)
459_0_main_LE(x1, x2, x3) → 459_0_main_LE(x2, x3)
Cond_459_0_main_LE(x1, x2, x3, x4) → Cond_459_0_main_LE(x1, x3, x4)

Combined rules. Obtained 6 conditional rules for P and 0 conditional rules for R.

P rules:
303_0_main_LE(x1, x0) → 456_0_main_LE(x0, x1) | &&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0))
456_0_main_LE(x0, 0) → 303_0_main_LE(0, x0)
456_0_main_LE(x0, x1) → 456_0_main_LE(x0, +(x1, -1)) | >(x1, 0)
303_0_main_LE(x1, x0) → 459_0_main_LE(x1, x0) | &&(&&(>(x1, 0), <(x1, x0)), >(x0, 0))
459_0_main_LE(x1, 0) → 303_0_main_LE(x1, 0)
459_0_main_LE(x1, x0) → 459_0_main_LE(x1, +(x0, -1)) | >(x0, 0)
R rules:

Performed bisimulation on rules. Used the following equivalence classes: {[456_0_main_LE_2, 459_0_main_LE_2]=456_0_main_LE_2, [Cond_456_0_main_LE_3, Cond_459_0_main_LE_3]=Cond_456_0_main_LE_3}

Finished conversion. Obtained 8 rules for P and 0 rules for R. System has predefined symbols.

P rules:
303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0)
COND_303_0_MAIN_LE(TRUE, x1, x0) → 456_0_MAIN_LE(x0, x1)
456_0_MAIN_LE(x0, 0) → 303_0_MAIN_LE(0, x0)
456_0_MAIN_LE(x0, x1) → COND_456_0_MAIN_LE(>(x1, 0), x0, x1)
COND_456_0_MAIN_LE(TRUE, x0, x1) → 456_0_MAIN_LE(x0, +(x1, -1))
303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0)
COND_303_0_MAIN_LE1(TRUE, x1, x0) → 456_0_MAIN_LE(x1, x0)
456_0_MAIN_LE(x1, 0) → 303_0_MAIN_LE(x1, 0)
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): 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(x1[0] >= x0[0] && x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1])
(2): 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2])
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], x1[4] + -1)
(5): 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0, x1[5], x0[5])
(6): COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6])
(7): 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0)

(0) -> (1), if (x1[0] >= x0[0] && x1[0] > 0 && x0[0] > 0x1[0]* x1[1]x0[0]* x0[1])

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

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

(1) -> (7), if (x0[1]* x1[7]x1[1]* 0)

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

(2) -> (5), if (0* x1[5]x0[2]* x0[5])

(3) -> (4), if (x1[3] > 0x0[3]* x0[4]x1[3]* x1[4])

(4) -> (2), if (x0[4]* x0[2]x1[4] + -1* 0)

(4) -> (3), if (x0[4]* x0[3]x1[4] + -1* x1[3])

(4) -> (7), if (x0[4]* x1[7]x1[4] + -1* 0)

(5) -> (6), if (x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0x1[5]* x1[6]x0[5]* x0[6])

(6) -> (2), if (x1[6]* x0[2]x0[6]* 0)

(6) -> (3), if (x1[6]* x0[3]x0[6]* x1[3])

(6) -> (7), if (x1[6]* x1[7]x0[6]* 0)

(7) -> (0), if (x1[7]* x1[0]0* x0[0])

(7) -> (5), if (x1[7]* x1[5]0* x0[5])

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.IdpDefaultShapeHeuristic@31068b41 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]), COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]) which results in the following constraint:

(1)    (&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]303_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧303_0_MAIN_LE(x1[0], x0[0])≥COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE>=(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUE303_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧303_0_MAIN_LE(x1[0], x0[0])≥COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

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

(6)    (x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_25] + [bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[(-1)bso_26] + x1[0] ≥ 0)

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

(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_25 + bni_25] + [(2)bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[1 + (-1)bso_26] + x0[0] + x1[0] ≥ 0)

For Pair COND_303_0_MAIN_LE(TRUE, x1, x0) → 456_0_MAIN_LE(x0, x1) the following chains were created:
• We consider the chain COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]), 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]) which results in the following constraint:

(8)    (x0[1]=x0[2]x1[1]=0COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:

(9)    (COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥456_0_MAIN_LE(x0[1], 0)∧(UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(10)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

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

(11)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

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

(12)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

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

(13)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• We consider the chain COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(14)    (x0[1]=x0[3]x1[1]=x1[3]COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(15)    (COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(16)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

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

(17)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

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

(18)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(19)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• We consider the chain COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1]), 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0) which results in the following constraint:

(20)    (x0[1]=x1[7]x1[1]=0COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, x1[1], x0[1])≥456_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

We simplified constraint (20) using rules (III), (IV) which results in the following new constraint:

(21)    (COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥NonInfC∧COND_303_0_MAIN_LE(TRUE, 0, x0[1])≥456_0_MAIN_LE(x0[1], 0)∧(UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(22)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

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

(23)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(24)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(25)    ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

For Pair 456_0_MAIN_LE(x0, 0) → 303_0_MAIN_LE(0, x0) the following chains were created:
• We consider the chain 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]), 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(26)    (0=x1[0]x0[2]=x0[0]456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (26) using rule (IV) which results in the following new constraint:

(27)    (456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(28)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(29)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(30)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (30) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(31)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

• We consider the chain 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]), 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:

(32)    (0=x1[5]x0[2]=x0[5]456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (32) using rule (IV) which results in the following new constraint:

(33)    (456_0_MAIN_LE(x0[2], 0)≥NonInfC∧456_0_MAIN_LE(x0[2], 0)≥303_0_MAIN_LE(0, x0[2])∧(UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥))

We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(34)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (34) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(35)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(36)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(37)    ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

For Pair 456_0_MAIN_LE(x0, x1) → COND_456_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:

(38)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

We simplified constraint (38) using rule (IV) which results in the following new constraint:

(39)    (>(x1[3], 0)=TRUE456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(40)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[3] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(41)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[3] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(42)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[3] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(43)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_31] = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(44)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_31] = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧[(-1)bso_32] ≥ 0)

For Pair COND_456_0_MAIN_LE(TRUE, x0, x1) → 456_0_MAIN_LE(x0, +(x1, -1)) the following chains were created:
• We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]) which results in the following constraint:

(45)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x0[2]+(x1[4], -1)=0COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (45) using rules (III), (IV) which results in the following new constraint:

(46)    (>(x1[3], 0)=TRUE+(x1[3], -1)=0COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (46) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(47)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (47) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(48)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (48) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(49)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (49) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(50)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(51)    (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

• We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(52)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x0[3]1+(x1[4], -1)=x1[3]1COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (52) using rules (III), (IV) which results in the following new constraint:

(53)    (>(x1[3], 0)=TRUECOND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(54)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(55)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(56)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (56) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(57)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(58)    (x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

• We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0) which results in the following constraint:

(59)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x1[7]+(x1[4], -1)=0COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (59) using rules (III), (IV) which results in the following new constraint:

(60)    (>(x1[3], 0)=TRUE+(x1[3], -1)=0COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (60) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(61)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (61) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(62)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (62) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(63)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x0[3] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (63) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(64)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(65)    (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]), COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]) which results in the following constraint:

(66)    (&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0))=TRUEx1[5]=x1[6]x0[5]=x0[6]303_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧303_0_MAIN_LE(x1[5], x0[5])≥COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))

We simplified constraint (66) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(67)    (>(x0[5], 0)=TRUE>(x1[5], 0)=TRUE<(x1[5], x0[5])=TRUE303_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧303_0_MAIN_LE(x1[5], x0[5])≥COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))

We simplified constraint (67) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(68)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[(-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (68) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(69)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[(-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (69) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(70)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[(-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(71)    (x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_35] + [bni_35]x0[5] + [bni_35]x1[5] ≥ 0∧[1 + (-1)bso_36] + x0[5] ≥ 0)

We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(72)    (x1[5] + x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_35] + [(2)bni_35]x1[5] + [bni_35]x0[5] ≥ 0∧[1 + (-1)bso_36] + x1[5] + x0[5] ≥ 0)

We simplified constraint (72) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(73)    ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(2)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x1[5] + [bni_35]x0[5] ≥ 0∧[2 + (-1)bso_36] + x1[5] + x0[5] ≥ 0)

For Pair COND_303_0_MAIN_LE1(TRUE, x1, x0) → 456_0_MAIN_LE(x1, x0) the following chains were created:
• We consider the chain COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]), 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2]) which results in the following constraint:

(74)    (x1[6]=x0[2]x0[6]=0COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (74) using rules (III), (IV) which results in the following new constraint:

(75)    (COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥456_0_MAIN_LE(x1[6], 0)∧(UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (75) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(76)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (76) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(77)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (77) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(78)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (78) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(79)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• We consider the chain COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(80)    (x1[6]=x0[3]x0[6]=x1[3]COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (80) using rule (IV) which results in the following new constraint:

(81)    (COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (81) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(82)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (82) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(83)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (83) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(84)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (84) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(85)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• We consider the chain COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6]), 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0) which results in the following constraint:

(86)    (x1[6]=x1[7]x0[6]=0COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6])≥456_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (86) using rules (III), (IV) which results in the following new constraint:

(87)    (COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥NonInfC∧COND_303_0_MAIN_LE1(TRUE, x1[6], 0)≥456_0_MAIN_LE(x1[6], 0)∧(UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (87) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(88)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (88) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(89)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (89) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(90)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)

We simplified constraint (90) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(91)    ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

For Pair 456_0_MAIN_LE(x1, 0) → 303_0_MAIN_LE(x1, 0) the following chains were created:
• We consider the chain 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0), 303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(92)    (x1[7]=x1[0]0=x0[0]456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (92) using rule (IV) which results in the following new constraint:

(93)    (456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (93) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(94)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (94) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(95)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (95) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(96)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (96) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(97)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

• We consider the chain 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0), 303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:

(98)    (x1[7]=x1[5]0=x0[5]456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (98) using rule (IV) which results in the following new constraint:

(99)    (456_0_MAIN_LE(x1[7], 0)≥NonInfC∧456_0_MAIN_LE(x1[7], 0)≥303_0_MAIN_LE(x1[7], 0)∧(UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (99) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(100)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (100) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(101)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (101) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(102)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)

We simplified constraint (102) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(103)    ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_25 + bni_25] + [(2)bni_25]x0[0] + [bni_25]x1[0] ≥ 0∧[1 + (-1)bso_26] + x0[0] + x1[0] ≥ 0)

• COND_303_0_MAIN_LE(TRUE, x1, x0) → 456_0_MAIN_LE(x0, x1)
• ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
• ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)
• ((UIncreasing(456_0_MAIN_LE(x0[1], x1[1])), ≥)∧[bni_27] = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• 456_0_MAIN_LE(x0, 0) → 303_0_MAIN_LE(0, x0)
• ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)
• ((UIncreasing(303_0_MAIN_LE(0, x0[2])), ≥)∧[bni_29] = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

• 456_0_MAIN_LE(x0, x1) → COND_456_0_MAIN_LE(>(x1, 0), x0, x1)
• (x1[3] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_31] = 0∧[(-1)bni_31 + (-1)Bound*bni_31] ≥ 0∧0 = 0∧[(-1)bso_32] ≥ 0)

• COND_456_0_MAIN_LE(TRUE, x0, x1) → 456_0_MAIN_LE(x0, +(x1, -1))
• (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
• (x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)
• (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_33] = 0∧[(-1)bni_33 + (-1)Bound*bni_33] ≥ 0∧0 = 0∧[(-1)bso_34] ≥ 0)

• 303_0_MAIN_LE(x1, x0) → COND_303_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0)
• ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(2)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x1[5] + [bni_35]x0[5] ≥ 0∧[2 + (-1)bso_36] + x1[5] + x0[5] ≥ 0)

• COND_303_0_MAIN_LE1(TRUE, x1, x0) → 456_0_MAIN_LE(x1, x0)
• ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
• ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
• ((UIncreasing(456_0_MAIN_LE(x1[6], x0[6])), ≥)∧[bni_37] = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

• 456_0_MAIN_LE(x1, 0) → 303_0_MAIN_LE(x1, 0)
• ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
• ((UIncreasing(303_0_MAIN_LE(x1[7], 0)), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 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) = [1]
POL(303_0_MAIN_LE(x1, x2)) = [-1] + x2 + x1
POL(COND_303_0_MAIN_LE(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = 0
POL(>=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(456_0_MAIN_LE(x1, x2)) = [-1] + x1
POL(COND_456_0_MAIN_LE(x1, x2, x3)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_303_0_MAIN_LE1(x1, x2, x3)) = [-1] + x2 + [-1]x1
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])

The following pairs are in Pbound:

303_0_MAIN_LE(x1[0], x0[0]) → COND_303_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
303_0_MAIN_LE(x1[5], x0[5]) → COND_303_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])

The following pairs are in P:

COND_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1])
456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2])
456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))
COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6])
456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0)

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1

### (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_303_0_MAIN_LE(TRUE, x1[1], x0[1]) → 456_0_MAIN_LE(x0[1], x1[1])
(2): 456_0_MAIN_LE(x0[2], 0) → 303_0_MAIN_LE(0, x0[2])
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], x1[4] + -1)
(6): COND_303_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 456_0_MAIN_LE(x1[6], x0[6])
(7): 456_0_MAIN_LE(x1[7], 0) → 303_0_MAIN_LE(x1[7], 0)

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

(4) -> (2), if (x0[4]* x0[2]x1[4] + -1* 0)

(6) -> (2), if (x1[6]* x0[2]x0[6]* 0)

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

(4) -> (3), if (x0[4]* x0[3]x1[4] + -1* x1[3])

(6) -> (3), if (x1[6]* x0[3]x0[6]* x1[3])

(3) -> (4), if (x1[3] > 0x0[3]* x0[4]x1[3]* x1[4])

(1) -> (7), if (x0[1]* x1[7]x1[1]* 0)

(4) -> (7), if (x0[4]* x1[7]x1[4] + -1* 0)

(6) -> (7), if (x1[6]* x1[7]x0[6]* 0)

The set Q is empty.

### (9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

### (10) 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:
(4): COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], x1[4] + -1)
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

(4) -> (3), if (x0[4]* x0[3]x1[4] + -1* x1[3])

(3) -> (4), if (x1[3] > 0x0[3]* x0[4]x1[3]* x1[4])

The set Q is empty.

### (11) 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.IdpDefaultShapeHeuristic@31068b41 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)) the following chains were created:
• We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)), 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(1)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x0[3]1+(x1[4], -1)=x1[3]1COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[4], x1[4])≥456_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

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

(2)    (>(x1[3], 0)=TRUECOND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_456_0_MAIN_LE(TRUE, x0[3], x1[3])≥456_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

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

(3)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(4)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(5)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(6)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

(7)    (x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) the following chains were created:
• We consider the chain 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:

(8)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

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

(9)    (>(x1[3], 0)=TRUE456_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧456_0_MAIN_LE(x0[3], x1[3])≥COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

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

(10)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(13)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

(14)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))
• (x1[3] ≥ 0 ⇒ (UIncreasing(456_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

• 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
• (x1[3] ≥ 0 ⇒ (UIncreasing(COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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_456_0_MAIN_LE(x1, x2, x3)) = [-1] + x3
POL(456_0_MAIN_LE(x1, x2)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))

The following pairs are in Pbound:

COND_456_0_MAIN_LE(TRUE, x0[4], x1[4]) → 456_0_MAIN_LE(x0[4], +(x1[4], -1))
456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

The following pairs are in P:

456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

There are no usable rules.

### (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:
(3): 456_0_MAIN_LE(x0[3], x1[3]) → COND_456_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

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.