### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Ackermann
`/** * The classical Ackermann function. * * All calls terminate. * * Julia + BinTerm prove that all calls terminate * * Note that we have to express the basic cases as m <= 0 and n <= 0 * in order to prove termination. * * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A> */public class Ackermann {    public static int ack(int m, int n) {	if (m <= 0) return n + 1;	else if (n <= 0) return ack(m - 1,1);	else return ack(m - 1,ack(m,n - 1));    }    public static void main(String[] args) {	ack(10,12);    }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
Ackermann.main([Ljava/lang/String;)V: Graph of 34 nodes with 0 SCCs.

Ackermann.ack(II)I: Graph of 55 nodes with 0 SCCs.

### (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: Ackermann.ack(II)I
SCC calls the following helper methods: Ackermann.ack(II)I
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 25 rules for R.

P rules:
375_0_ack_GT(EOS(STATIC_375), i40, i38, i40) → 379_0_ack_GT(EOS(STATIC_379), i40, i38, i40)
379_0_ack_GT(EOS(STATIC_379), i40, i38, i40) → 382_0_ack_Load(EOS(STATIC_382), i40, i38) | >(i40, 0)
382_0_ack_Load(EOS(STATIC_382), i40, i38) → 386_0_ack_GT(EOS(STATIC_386), i40, i38, i38)
386_0_ack_GT(EOS(STATIC_386), i40, i43, i43) → 390_0_ack_GT(EOS(STATIC_390), i40, i43, i43)
386_0_ack_GT(EOS(STATIC_386), i40, i44, i44) → 391_0_ack_GT(EOS(STATIC_391), i40, i44, i44)
390_0_ack_GT(EOS(STATIC_390), i40, i43, i43) → 394_0_ack_Load(EOS(STATIC_394), i40, i43) | <=(i43, 0)
394_0_ack_Load(EOS(STATIC_394), i40, i43) → 401_0_ack_ConstantStackPush(EOS(STATIC_401), i40, i43, i40)
401_0_ack_ConstantStackPush(EOS(STATIC_401), i40, i43, i40) → 408_0_ack_IntArithmetic(EOS(STATIC_408), i40, i43, i40, 1)
408_0_ack_IntArithmetic(EOS(STATIC_408), i40, i43, i40, matching1) → 413_0_ack_ConstantStackPush(EOS(STATIC_413), i40, i43, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
413_0_ack_ConstantStackPush(EOS(STATIC_413), i40, i43, i46) → 416_0_ack_InvokeMethod(EOS(STATIC_416), i40, i43, i46, 1)
416_0_ack_InvokeMethod(EOS(STATIC_416), i40, i43, i46, matching1) → 419_1_ack_InvokeMethod(419_0_ack_Load(EOS(STATIC_419), i46, 1), i40, i43, i46, 1) | =(matching1, 1)
373_0_ack_Load(EOS(STATIC_373), i37, i38) → 375_0_ack_GT(EOS(STATIC_375), i37, i38, i37)
391_0_ack_GT(EOS(STATIC_391), i40, i44, i44) → 396_0_ack_Load(EOS(STATIC_396), i40, i44) | >(i44, 0)
396_0_ack_Load(EOS(STATIC_396), i40, i44) → 402_0_ack_ConstantStackPush(EOS(STATIC_402), i40, i44, i40)
402_0_ack_ConstantStackPush(EOS(STATIC_402), i40, i44, i40) → 410_0_ack_IntArithmetic(EOS(STATIC_410), i40, i44, i40, 1)
410_0_ack_IntArithmetic(EOS(STATIC_410), i40, i44, i40, matching1) → 414_0_ack_Load(EOS(STATIC_414), i40, i44, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
418_0_ack_Load(EOS(STATIC_418), i44, i47, i40) → 421_0_ack_ConstantStackPush(EOS(STATIC_421), i47, i40, i44)
421_0_ack_ConstantStackPush(EOS(STATIC_421), i47, i40, i44) → 424_0_ack_IntArithmetic(EOS(STATIC_424), i47, i40, i44, 1)
424_0_ack_IntArithmetic(EOS(STATIC_424), i47, i40, i44, matching1) → 426_0_ack_InvokeMethod(EOS(STATIC_426), i47, i40, -(i44, 1)) | &&(>(i44, 0), =(matching1, 1))
426_0_ack_InvokeMethod(EOS(STATIC_426), i47, i40, i48) → 433_1_ack_InvokeMethod(433_0_ack_Load(EOS(STATIC_433), i40, i48), i47, i40, i48)
433_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), i54, matching1, i45), i47, i54, matching2) → 450_0_ack_Return(EOS(STATIC_450), i47, i54, 0, i54, 0, i45) | &&(=(matching1, 0), =(matching2, 0))
433_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), i81, matching1, i62), i47, i81, matching2) → 513_0_ack_Return(EOS(STATIC_513), i47, i81, 0, i81, 0, i62) | &&(=(matching1, 0), =(matching2, 0))
433_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), i45), i47, i97, i98) → 550_0_ack_Return(EOS(STATIC_550), i47, i97, i98, i45)
433_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), i45), i47, i112, i113) → 572_0_ack_Return(EOS(STATIC_572), i47, i112, i113, i45)
450_0_ack_Return(EOS(STATIC_450), i47, i54, matching1, i54, matching2, i45) → 454_0_ack_InvokeMethod(EOS(STATIC_454), i47, i45) | &&(=(matching1, 0), =(matching2, 0))
454_0_ack_InvokeMethod(EOS(STATIC_454), i47, i45) → 496_0_ack_InvokeMethod(EOS(STATIC_496), i47, i45)
496_0_ack_InvokeMethod(EOS(STATIC_496), i47, i62) → 502_1_ack_InvokeMethod(502_0_ack_Load(EOS(STATIC_502), i47, i62), i47, i62)
513_0_ack_Return(EOS(STATIC_513), i47, i81, matching1, i81, matching2, i62) → 450_0_ack_Return(EOS(STATIC_450), i47, i81, 0, i81, 0, i62) | &&(=(matching1, 0), =(matching2, 0))
550_0_ack_Return(EOS(STATIC_550), i47, i97, i98, i45) → 488_0_ack_Return(EOS(STATIC_488), i47, i97, i98, i45)
488_0_ack_Return(EOS(STATIC_488), i47, i68, i69, i62) → 496_0_ack_InvokeMethod(EOS(STATIC_496), i47, i62)
572_0_ack_Return(EOS(STATIC_572), i47, i112, i113, i45) → 488_0_ack_Return(EOS(STATIC_488), i47, i112, i113, i45)
R rules:
375_0_ack_GT(EOS(STATIC_375), matching1, i38, matching2) → 378_0_ack_GT(EOS(STATIC_378), 0, i38, 0) | &&(=(matching1, 0), =(matching2, 0))
378_0_ack_GT(EOS(STATIC_378), matching1, i38, matching2) → 380_0_ack_Load(EOS(STATIC_380), 0, i38) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
380_0_ack_Load(EOS(STATIC_380), matching1, i38) → 384_0_ack_ConstantStackPush(EOS(STATIC_384), 0, i38, i38) | =(matching1, 0)
384_0_ack_ConstantStackPush(EOS(STATIC_384), matching1, i38, i38) → 388_0_ack_IntArithmetic(EOS(STATIC_388), 0, i38, i38, 1) | =(matching1, 0)
388_0_ack_IntArithmetic(EOS(STATIC_388), matching1, i38, i38, matching2) → 393_0_ack_Return(EOS(STATIC_393), 0, i38, +(i38, 1)) | &&(=(matching1, 0), =(matching2, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), matching1, matching2, i45), i40, i43, matching3, matching4) → 434_0_ack_Return(EOS(STATIC_434), i40, i43, 0, 1, 0, 1, i45) | &&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 1))
419_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), i45), i40, i43, i95, matching1) → 548_0_ack_Return(EOS(STATIC_548), i40, i43, i95, 1, i45) | =(matching1, 1)
419_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), i45), i40, i43, i110, matching1) → 570_0_ack_Return(EOS(STATIC_570), i40, i43, i110, 1, i45) | =(matching1, 1)
434_0_ack_Return(EOS(STATIC_434), i40, i43, matching1, matching2, matching3, matching4, i45) → 438_0_ack_Return(EOS(STATIC_438), i40, i43, i45) | &&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 1))
438_0_ack_Return(EOS(STATIC_438), i40, i43, i45) → 494_0_ack_Return(EOS(STATIC_494), i40, i43, i45)
487_0_ack_Return(EOS(STATIC_487), i40, i43, i66, matching1, i62) → 494_0_ack_Return(EOS(STATIC_494), i40, i43, i62) | =(matching1, 1)
502_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), matching1, i86, i45), matching2, i86) → 523_0_ack_Return(EOS(STATIC_523), 0, i86, 0, i86, i45) | &&(=(matching1, 0), =(matching2, 0))
502_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), i87, i88, i45), i87, i88) → 524_0_ack_Return(EOS(STATIC_524), i87, i88, i87, i88, i45)
502_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), i90, i91, i89), i90, i91) → 527_0_ack_Return(EOS(STATIC_527), i90, i91, i90, i91, i89)
502_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), i45), i101, i102) → 552_0_ack_Return(EOS(STATIC_552), i101, i102, i45)
502_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), i45), i116, i117) → 574_0_ack_Return(EOS(STATIC_574), i116, i117, i45)
523_0_ack_Return(EOS(STATIC_523), matching1, i86, matching2, i86, i45) → 528_0_ack_Return(EOS(STATIC_528), i45) | &&(=(matching1, 0), =(matching2, 0))
524_0_ack_Return(EOS(STATIC_524), i87, i88, i87, i88, i45) → 531_0_ack_Return(EOS(STATIC_531), i45)
527_0_ack_Return(EOS(STATIC_527), i90, i91, i90, i91, i89) → 524_0_ack_Return(EOS(STATIC_524), i90, i91, i90, i91, i89)
528_0_ack_Return(EOS(STATIC_528), i45) → 531_0_ack_Return(EOS(STATIC_531), i45)
531_0_ack_Return(EOS(STATIC_531), i45) → 555_0_ack_Return(EOS(STATIC_555), i45)
548_0_ack_Return(EOS(STATIC_548), i40, i43, i95, matching1, i45) → 487_0_ack_Return(EOS(STATIC_487), i40, i43, i95, 1, i45) | =(matching1, 1)
552_0_ack_Return(EOS(STATIC_552), i101, i102, i45) → 555_0_ack_Return(EOS(STATIC_555), i45)
570_0_ack_Return(EOS(STATIC_570), i40, i43, i110, matching1, i45) → 487_0_ack_Return(EOS(STATIC_487), i40, i43, i110, 1, i45) | =(matching1, 1)
574_0_ack_Return(EOS(STATIC_574), i116, i117, i45) → 552_0_ack_Return(EOS(STATIC_552), i116, i117, i45)

Combined rules. Obtained 6 conditional rules for P and 9 conditional rules for R.

P rules:
375_0_ack_GT(EOS(STATIC_375), x0, x1, x0) → 419_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), -(x0, 1), 1, -(x0, 1)), x0, x1, -(x0, 1), 1) | &&(<=(x1, 0), >(x0, 0))
375_0_ack_GT(EOS(STATIC_375), x0, x1, x0) → 433_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x0, -(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
433_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), x0, 0, x2), x3, x0, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x3, x2, x3), x3, x2)
433_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), x0, 0, x2), x3, x0, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x3, x2, x3), x3, x2)
433_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), x0), x1, x2, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x1, x0, x1), x1, x0)
433_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), x0), x1, x2, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(EOS(STATIC_375), x1, x0, x1), x1, x0)
R rules:
375_0_ack_GT(EOS(STATIC_375), 0, x1, 0) → 393_0_ack_Return(EOS(STATIC_393), 0, x1, +(x1, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), 0, 1, x2), x3, x4, 0, 1) → 494_0_ack_Return(EOS(STATIC_494), x3, x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), x0), x1, x2) → 555_0_ack_Return(EOS(STATIC_555), x0)
502_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), x0), x1, x2) → 555_0_ack_Return(EOS(STATIC_555), x0)
502_1_ack_InvokeMethod(438_0_ack_Return(EOS(STATIC_438), x0, x1, x2), x0, x1) → 555_0_ack_Return(EOS(STATIC_555), x2)
502_1_ack_InvokeMethod(494_0_ack_Return(EOS(STATIC_494), x0, x1, x2), x0, x1) → 555_0_ack_Return(EOS(STATIC_555), x2)
502_1_ack_InvokeMethod(393_0_ack_Return(EOS(STATIC_393), 0, x1, x2), 0, x1) → 555_0_ack_Return(EOS(STATIC_555), x2)
419_1_ack_InvokeMethod(531_0_ack_Return(EOS(STATIC_531), x0), x1, x2, x3, 1) → 494_0_ack_Return(EOS(STATIC_494), x1, x2, x0)
419_1_ack_InvokeMethod(555_0_ack_Return(EOS(STATIC_555), x0), x1, x2, x3, 1) → 494_0_ack_Return(EOS(STATIC_494), x1, x2, x0)

Filtered ground terms:

375_0_ack_GT(x1, x2, x3, x4) → 375_0_ack_GT(x2, x3, x4)
555_0_ack_Return(x1, x2) → 555_0_ack_Return(x2)
531_0_ack_Return(x1, x2) → 531_0_ack_Return(x2)
494_0_ack_Return(x1, x2, x3, x4) → 494_0_ack_Return(x2, x3, x4)
438_0_ack_Return(x1, x2, x3, x4) → 438_0_ack_Return(x2, x3, x4)
Cond_375_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_375_0_ack_GT1(x1, x3, x4, x5)
419_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 419_1_ack_InvokeMethod(x1, x2, x3, x4)
Cond_375_0_ack_GT(x1, x2, x3, x4, x5) → Cond_375_0_ack_GT(x1, x3, x4, x5)
393_0_ack_Return(x1, x2, x3, x4) → 393_0_ack_Return(x3, x4)

Filtered duplicate args:

375_0_ack_GT(x1, x2, x3) → 375_0_ack_GT(x2, x3)
Cond_375_0_ack_GT(x1, x2, x3, x4) → Cond_375_0_ack_GT(x1, x3, x4)
Cond_375_0_ack_GT1(x1, x2, x3, x4) → Cond_375_0_ack_GT1(x1, x3, x4)

Filtered unneeded arguments:

419_1_ack_InvokeMethod(x1, x2, x3, x4) → 419_1_ack_InvokeMethod(x1, x3, x4)
433_1_ack_InvokeMethod(x1, x2, x3, x4) → 433_1_ack_InvokeMethod(x1, x2, x4)
438_0_ack_Return(x1, x2, x3) → 438_0_ack_Return(x2, x3)
494_0_ack_Return(x1, x2, x3) → 494_0_ack_Return(x2, x3)

Combined rules. Obtained 6 conditional rules for P and 9 conditional rules for R.

P rules:
375_0_ack_GT(x1, x0) → 419_1_ack_InvokeMethod(375_0_ack_GT(1, -(x0, 1)), x1, -(x0, 1)) | &&(<=(x1, 0), >(x0, 0))
375_0_ack_GT(x1, x0) → 433_1_ack_InvokeMethod(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
433_1_ack_InvokeMethod(438_0_ack_Return(0, x2), x3, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(x2, x3), x3, x2)
433_1_ack_InvokeMethod(494_0_ack_Return(0, x2), x3, 0) → 502_1_ack_InvokeMethod(375_0_ack_GT(x2, x3), x3, x2)
433_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(x0, x1), x1, x0)
433_1_ack_InvokeMethod(555_0_ack_Return(x0), x1, x3) → 502_1_ack_InvokeMethod(375_0_ack_GT(x0, x1), x1, x0)
R rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, +(x1, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 555_0_ack_Return(x0)
502_1_ack_InvokeMethod(555_0_ack_Return(x0), x1, x2) → 555_0_ack_Return(x0)
502_1_ack_InvokeMethod(438_0_ack_Return(x1, x2), x0, x1) → 555_0_ack_Return(x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 555_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 555_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)
419_1_ack_InvokeMethod(555_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

Performed bisimulation on rules. Used the following equivalence classes: {[494_0_ack_Return_2, 438_0_ack_Return_2]=494_0_ack_Return_2, [531_0_ack_Return_1, 555_0_ack_Return_1]=531_0_ack_Return_1}

Finished conversion. Obtained 7 rules for P and 6 rules for R. System has predefined symbols.

P rules:
375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
COND_375_0_ACK_GT(TRUE, x1, x0) → 375_0_ACK_GT(1, -(x0, 1))
375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_375_0_ACK_GT1(TRUE, x1, x0) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
COND_375_0_ACK_GT1(TRUE, x1, x0) → 375_0_ACK_GT(-(x1, 1), x0)
433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2), x3, 0) → 375_0_ACK_GT(x2, x3)
433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0), x1, x3) → 375_0_ACK_GT(x0, x1)
R rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, +(x1, 1))
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

### (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, Boolean

The ITRS R consists of the following rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(0): 375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, x0[1] - 1)
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(5): 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
(6): 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6])

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

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

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

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

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

(3) -> (5), if (375_0_ack_GT(x1[3] - 1, x0[3]) →* 494_0_ack_Return(0, x2[5])∧x0[3] - 1* x3[5]x1[3] - 1* 0)

(3) -> (6), if (375_0_ack_GT(x1[3] - 1, x0[3]) →* 531_0_ack_Return(x0[6])∧x0[3] - 1* x1[6]x1[3] - 1* x3[6])

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

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

(5) -> (0), if (x2[5]* x1[0]x3[5]* x0[0])

(5) -> (2), if (x2[5]* x1[2]x3[5]* x0[2])

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

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

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (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@1f65b124 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 375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

(1)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]375_0_ACK_GT(x1[0], x0[0])≥NonInfC∧375_0_ACK_GT(x1[0], x0[0])≥COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_375_0_ACK_GT(&&(<=(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)    (<=(x1[0], 0)=TRUE>(x0[0], 0)=TRUE375_0_ACK_GT(x1[0], x0[0])≥NonInfC∧375_0_ACK_GT(x1[0], x0[0])≥COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_375_0_ACK_GT(&&(<=(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)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(4)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(5)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

For Pair COND_375_0_ACK_GT(TRUE, x1, x0) → 375_0_ACK_GT(1, -(x0, 1)) the following chains were created:
• We consider the chain COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

(8)    (COND_375_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_375_0_ACK_GT(TRUE, x1[1], x0[1])≥375_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(9)    ((UIncreasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

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

(10)    ((UIncreasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

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

(11)    ((UIncreasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

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

(12)    ((UIncreasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

For Pair 375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:

(13)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(14)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(15)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(16)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(17)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(18)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(19)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

• We consider the chain 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(20)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(21)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(22)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(23)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(25)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(26)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

For Pair COND_375_0_ACK_GT1(TRUE, x1, x0) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) the following chains were created:
• We consider the chain COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:

(27)    (COND_375_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_375_0_ACK_GT1(TRUE, x1[3], x0[3])≥433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))∧(UIncreasing(433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥))

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

(28)    ((UIncreasing(433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)

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

(29)    ((UIncreasing(433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)

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

(30)    ((UIncreasing(433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧[1 + (-1)bso_29] ≥ 0)

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

(31)    ((UIncreasing(433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_29] ≥ 0)

For Pair COND_375_0_ACK_GT1(TRUE, x1, x0) → 375_0_ACK_GT(-(x1, 1), x0) the following chains were created:
• We consider the chain COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(32)    (COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥375_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(33)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧[(-1)bso_31] ≥ 0)

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

(34)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧[(-1)bso_31] ≥ 0)

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

(35)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧[(-1)bso_31] ≥ 0)

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

(36)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2), x3, 0) → 375_0_ACK_GT(x2, x3) the following chains were created:
• We consider the chain 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:

(37)    (433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0)≥NonInfC∧433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0)≥375_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(375_0_ACK_GT(x2[5], x3[5])), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(38)    ((UIncreasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧[(-1)bso_33] ≥ 0)

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

(39)    ((UIncreasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧[(-1)bso_33] ≥ 0)

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

(40)    ((UIncreasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧[(-1)bso_33] ≥ 0)

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

(41)    ((UIncreasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bso_33] ≥ 0)

For Pair 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0), x1, x3) → 375_0_ACK_GT(x0, x1) the following chains were created:
• We consider the chain 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:

(42)    (433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6])≥NonInfC∧433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6])≥375_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(375_0_ACK_GT(x0[6], x1[6])), ≥))

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

(43)    ((UIncreasing(375_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧[(-1)bso_35] ≥ 0)

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

(44)    ((UIncreasing(375_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧[(-1)bso_35] ≥ 0)

We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(45)    ((UIncreasing(375_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧[(-1)bso_35] ≥ 0)

We simplified constraint (45) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(46)    ((UIncreasing(375_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

• COND_375_0_ACK_GT(TRUE, x1, x0) → 375_0_ACK_GT(1, -(x0, 1))
• ((UIncreasing(375_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

• 375_0_ACK_GT(x1, x0) → COND_375_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] ≥ 0)

• COND_375_0_ACK_GT1(TRUE, x1, x0) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
• ((UIncreasing(433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_28] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_29] ≥ 0)

• COND_375_0_ACK_GT1(TRUE, x1, x0) → 375_0_ACK_GT(-(x1, 1), x0)
• ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

• 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2), x3, 0) → 375_0_ACK_GT(x2, x3)
• ((UIncreasing(375_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bso_33] ≥ 0)

• 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0), x1, x3) → 375_0_ACK_GT(x0, x1)
• ((UIncreasing(375_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_34] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 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(375_0_ack_GT(x1, x2)) = [2] + [2]x2 + x1
POL(0) = 0
POL(393_0_ack_Return(x1, x2)) = [-1] + [-1]x2 + [2]x1
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(419_1_ack_InvokeMethod(x1, x2, x3)) = [-1]
POL(494_0_ack_Return(x1, x2)) = x2
POL(502_1_ack_InvokeMethod(x1, x2, x3)) = [-1]
POL(531_0_ack_Return(x1)) = x1
POL(375_0_ACK_GT(x1, x2)) = [-1] + x2
POL(COND_375_0_ACK_GT(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_375_0_ACK_GT1(x1, x2, x3)) = [-1] + x3
POL(433_1_ACK_INVOKEMETHOD(x1, x2, x3)) = [-1] + x2

The following pairs are in P>:

COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, -(x0[1], 1))
COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))

The following pairs are in Pbound:

375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P:

375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4])
433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6])

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, Boolean

The ITRS R consists of the following rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(0): 375_0_ACK_GT(x1[0], x0[0]) → COND_375_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(5): 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
(6): 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6])

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

(5) -> (0), if (x2[5]* x1[0]x3[5]* x0[0])

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

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

(5) -> (2), if (x2[5]* x1[2]x3[5]* x0[2])

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

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

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

### (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, Boolean

The ITRS R consists of the following rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

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

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

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (12) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (13) 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

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

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

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (14) 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@1f65b124 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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
• We consider the chain COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(1)    (COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_375_0_ACK_GT1(TRUE, x1[4], x0[4])≥375_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(2)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)

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

(3)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)

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

(4)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧[1 + (-1)bso_10] ≥ 0)

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

(5)    ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

For Pair 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
• We consider the chain 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(6)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE375_0_ACK_GT(x1[2], x0[2])≥NonInfC∧375_0_ACK_GT(x1[2], x0[2])≥COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(8)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(9)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(11)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(12)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_11 + bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4])
• ((UIncreasing(375_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

• 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_11 + bni_11] + [bni_11]x1[2] + [bni_11]x0[2] ≥ 0∧[(-1)bso_12] ≥ 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_375_0_ACK_GT1(x1, x2, x3)) = [-1] + x3 + x2
POL(375_0_ACK_GT(x1, x2)) = [-1] + x1 + x2
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in Pbound:

375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P:

375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 375_0_ACK_GT(x1[2], x0[2]) → COND_375_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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

### (19) 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_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (22) 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

The ITRS R consists of the following rules:
375_0_ack_GT(x1, 0) → 393_0_ack_Return(x1, x1 + 1)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x2), x4, 0) → 494_0_ack_Return(x4, x2)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2) → 531_0_ack_Return(x0)
502_1_ack_InvokeMethod(494_0_ack_Return(x1, x2), x0, x1) → 531_0_ack_Return(x2)
502_1_ack_InvokeMethod(393_0_ack_Return(x1, x2), 0, x1) → 531_0_ack_Return(x2)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x2, x3) → 494_0_ack_Return(x2, x0)

The integer pair graph contains the following rules and edges:
(1): COND_375_0_ACK_GT(TRUE, x1[1], x0[1]) → 375_0_ACK_GT(1, x0[1] - 1)
(3): COND_375_0_ACK_GT1(TRUE, x1[3], x0[3]) → 433_1_ACK_INVOKEMETHOD(375_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_375_0_ACK_GT1(TRUE, x1[4], x0[4]) → 375_0_ACK_GT(x1[4] - 1, x0[4])
(5): 433_1_ACK_INVOKEMETHOD(494_0_ack_Return(0, x2[5]), x3[5], 0) → 375_0_ACK_GT(x2[5], x3[5])
(6): 433_1_ACK_INVOKEMETHOD(531_0_ack_Return(x0[6]), x1[6], x3[6]) → 375_0_ACK_GT(x0[6], x1[6])

(3) -> (5), if (375_0_ack_GT(x1[3] - 1, x0[3]) →* 494_0_ack_Return(0, x2[5])∧x0[3] - 1* x3[5]x1[3] - 1* 0)

(3) -> (6), if (375_0_ack_GT(x1[3] - 1, x0[3]) →* 531_0_ack_Return(x0[6])∧x0[3] - 1* x1[6]x1[3] - 1* x3[6])

The set Q consists of the following terms:
375_0_ack_GT(x0, 0)
419_1_ack_InvokeMethod(393_0_ack_Return(1, x0), x1, 0)
502_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)
502_1_ack_InvokeMethod(494_0_ack_Return(x0, x1), x2, x0)
502_1_ack_InvokeMethod(393_0_ack_Return(x0, x1), 0, x0)
419_1_ack_InvokeMethod(531_0_ack_Return(x0), x1, x2)

### (23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 5 less nodes.