### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: AckermannR
`public class AckermannR {    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) {	Random.args = args;	ack(Random.random(),Random.random());    }}public class Random {  static String[] args;  static int index = 0;  public static int random() {      if (index >= args.length)	  return 0;      String string = args[index];      index++;      return string.length();  }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
AckermannR.main([Ljava/lang/String;)V: Graph of 167 nodes with 0 SCCs.

AckermannR.ack(II)I: Graph of 52 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: AckermannR.ack(II)I
SCC calls the following helper methods: AckermannR.ack(II)I
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 20 rules for R.

P rules:
377_0_ack_GT(EOS(STATIC_377), i71, i67, i71) → 380_0_ack_GT(EOS(STATIC_380), i71, i67, i71)
380_0_ack_GT(EOS(STATIC_380), i71, i67, i71) → 386_0_ack_Load(EOS(STATIC_386), i71, i67) | >(i71, 0)
386_0_ack_Load(EOS(STATIC_386), i71, i67) → 392_0_ack_GT(EOS(STATIC_392), i71, i67, i67)
392_0_ack_GT(EOS(STATIC_392), i71, matching1, matching2) → 399_0_ack_GT(EOS(STATIC_399), i71, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
392_0_ack_GT(EOS(STATIC_392), i71, i75, i75) → 400_0_ack_GT(EOS(STATIC_400), i71, i75, i75)
399_0_ack_GT(EOS(STATIC_399), i71, matching1, matching2) → 406_0_ack_Load(EOS(STATIC_406), i71, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
406_0_ack_Load(EOS(STATIC_406), i71, matching1) → 417_0_ack_ConstantStackPush(EOS(STATIC_417), i71, 0, i71) | =(matching1, 0)
417_0_ack_ConstantStackPush(EOS(STATIC_417), i71, matching1, i71) → 437_0_ack_IntArithmetic(EOS(STATIC_437), i71, 0, i71, 1) | =(matching1, 0)
437_0_ack_IntArithmetic(EOS(STATIC_437), i71, matching1, i71, matching2) → 467_0_ack_ConstantStackPush(EOS(STATIC_467), i71, 0, -(i71, 1)) | &&(&&(>(i71, 0), =(matching1, 0)), =(matching2, 1))
467_0_ack_ConstantStackPush(EOS(STATIC_467), i71, matching1, i88) → 480_0_ack_InvokeMethod(EOS(STATIC_480), i71, 0, i88, 1) | =(matching1, 0)
480_0_ack_InvokeMethod(EOS(STATIC_480), i71, matching1, i88, matching2) → 493_1_ack_InvokeMethod(493_0_ack_Load(EOS(STATIC_493), i88, 1), i71, 0, i88, 1) | &&(=(matching1, 0), =(matching2, 1))
372_0_ack_Load(EOS(STATIC_372), i66, i67) → 377_0_ack_GT(EOS(STATIC_377), i66, i67, i66)
400_0_ack_GT(EOS(STATIC_400), i71, i75, i75) → 408_0_ack_Load(EOS(STATIC_408), i71, i75) | >(i75, 0)
408_0_ack_Load(EOS(STATIC_408), i71, i75) → 419_0_ack_ConstantStackPush(EOS(STATIC_419), i71, i75, i71)
419_0_ack_ConstantStackPush(EOS(STATIC_419), i71, i75, i71) → 439_0_ack_IntArithmetic(EOS(STATIC_439), i71, i75, i71, 1)
439_0_ack_IntArithmetic(EOS(STATIC_439), i71, i75, i71, matching1) → 469_0_ack_Load(EOS(STATIC_469), i71, i75, -(i71, 1)) | &&(>(i71, 0), =(matching1, 1))
482_0_ack_Load(EOS(STATIC_482), i75, i89, i71) → 495_0_ack_ConstantStackPush(EOS(STATIC_495), i89, i71, i75)
495_0_ack_ConstantStackPush(EOS(STATIC_495), i89, i71, i75) → 508_0_ack_IntArithmetic(EOS(STATIC_508), i89, i71, i75, 1)
508_0_ack_IntArithmetic(EOS(STATIC_508), i89, i71, i75, matching1) → 512_0_ack_InvokeMethod(EOS(STATIC_512), i89, i71, -(i75, 1)) | &&(>(i75, 0), =(matching1, 1))
512_0_ack_InvokeMethod(EOS(STATIC_512), i89, i71, i93) → 527_1_ack_InvokeMethod(527_0_ack_Load(EOS(STATIC_527), i71, i93), i89, i71, i93)
527_1_ack_InvokeMethod(537_0_ack_Return(EOS(STATIC_537), i108, matching1, i76), i89, i108, matching2) → 566_0_ack_Return(EOS(STATIC_566), i89, i108, 0, i108, 0, i76) | &&(=(matching1, 0), =(matching2, 0))
527_1_ack_InvokeMethod(653_0_ack_Return(EOS(STATIC_653), i172, matching1, i128), i89, i172, matching2) → 700_0_ack_Return(EOS(STATIC_700), i89, i172, 0, i172, 0, i128) | &&(=(matching1, 0), =(matching2, 0))
527_1_ack_InvokeMethod(716_0_ack_Return(EOS(STATIC_716), i76), i89, i207, i208) → 756_0_ack_Return(EOS(STATIC_756), i89, i207, i208, i76)
527_1_ack_InvokeMethod(762_0_ack_Return(EOS(STATIC_762), i76), i89, i249, i250) → 801_0_ack_Return(EOS(STATIC_801), i89, i249, i250, i76)
566_0_ack_Return(EOS(STATIC_566), i89, i108, matching1, i108, matching2, i76) → 574_0_ack_InvokeMethod(EOS(STATIC_574), i89, i76) | &&(=(matching1, 0), =(matching2, 0))
574_0_ack_InvokeMethod(EOS(STATIC_574), i89, i76) → 657_0_ack_InvokeMethod(EOS(STATIC_657), i89, i76)
657_0_ack_InvokeMethod(EOS(STATIC_657), i89, i128) → 663_1_ack_InvokeMethod(663_0_ack_Load(EOS(STATIC_663), i89, i128), i89, i128)
700_0_ack_Return(EOS(STATIC_700), i89, i172, matching1, i172, matching2, i128) → 566_0_ack_Return(EOS(STATIC_566), i89, i172, 0, i172, 0, i128) | &&(=(matching1, 0), =(matching2, 0))
756_0_ack_Return(EOS(STATIC_756), i89, i207, i208, i76) → 642_0_ack_Return(EOS(STATIC_642), i89, i207, i208, i76)
642_0_ack_Return(EOS(STATIC_642), i89, i143, i144, i128) → 657_0_ack_InvokeMethod(EOS(STATIC_657), i89, i128)
801_0_ack_Return(EOS(STATIC_801), i89, i249, i250, i76) → 642_0_ack_Return(EOS(STATIC_642), i89, i249, i250, i76)
R rules:
377_0_ack_GT(EOS(STATIC_377), matching1, i67, matching2) → 379_0_ack_GT(EOS(STATIC_379), 0, i67, 0) | &&(=(matching1, 0), =(matching2, 0))
379_0_ack_GT(EOS(STATIC_379), matching1, i67, matching2) → 384_0_ack_Load(EOS(STATIC_384), 0, i67) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
384_0_ack_Load(EOS(STATIC_384), matching1, i67) → 391_0_ack_ConstantStackPush(EOS(STATIC_391), 0, i67, i67) | =(matching1, 0)
391_0_ack_ConstantStackPush(EOS(STATIC_391), matching1, i67, i67) → 397_0_ack_IntArithmetic(EOS(STATIC_397), 0, i67, i67, 1) | =(matching1, 0)
397_0_ack_IntArithmetic(EOS(STATIC_397), matching1, i67, i67, matching2) → 405_0_ack_Return(EOS(STATIC_405), 0, i67, +(i67, 1)) | &&(&&(>=(i67, 0), =(matching1, 0)), =(matching2, 1))
493_1_ack_InvokeMethod(405_0_ack_Return(EOS(STATIC_405), matching1, matching2, i76), i71, matching3, matching4, matching5) → 531_0_ack_Return(EOS(STATIC_531), i71, 0, 0, 1, 0, 1, i76) | &&(&&(&&(&&(=(matching1, 0), =(matching2, 1)), =(matching3, 0)), =(matching4, 0)), =(matching5, 1))
493_1_ack_InvokeMethod(716_0_ack_Return(EOS(STATIC_716), i76), i71, matching1, i203, matching2) → 753_0_ack_Return(EOS(STATIC_753), i71, 0, i203, 1, i76) | &&(=(matching1, 0), =(matching2, 1))
493_1_ack_InvokeMethod(762_0_ack_Return(EOS(STATIC_762), i76), i71, matching1, i244, matching2) → 799_0_ack_Return(EOS(STATIC_799), i71, 0, i244, 1, i76) | &&(=(matching1, 0), =(matching2, 1))
531_0_ack_Return(EOS(STATIC_531), i71, matching1, matching2, matching3, matching4, matching5, i76) → 537_0_ack_Return(EOS(STATIC_537), i71, 0, i76) | &&(&&(&&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 1)), =(matching4, 0)), =(matching5, 1))
537_0_ack_Return(EOS(STATIC_537), i71, matching1, i76) → 653_0_ack_Return(EOS(STATIC_653), i71, 0, i76) | =(matching1, 0)
641_0_ack_Return(EOS(STATIC_641), i71, matching1, i138, matching2, i128) → 653_0_ack_Return(EOS(STATIC_653), i71, 0, i128) | &&(=(matching1, 0), =(matching2, 1))
663_1_ack_InvokeMethod(405_0_ack_Return(EOS(STATIC_405), matching1, i190, i76), matching2, i190) → 712_0_ack_Return(EOS(STATIC_712), 0, i190, 0, i190, i76) | &&(=(matching1, 0), =(matching2, 0))
663_1_ack_InvokeMethod(716_0_ack_Return(EOS(STATIC_716), i76), i213, i214) → 758_0_ack_Return(EOS(STATIC_758), i213, i214, i76)
663_1_ack_InvokeMethod(762_0_ack_Return(EOS(STATIC_762), i76), i255, i256) → 804_0_ack_Return(EOS(STATIC_804), i255, i256, i76)
712_0_ack_Return(EOS(STATIC_712), matching1, i190, matching2, i190, i76) → 716_0_ack_Return(EOS(STATIC_716), i76) | &&(=(matching1, 0), =(matching2, 0))
716_0_ack_Return(EOS(STATIC_716), i76) → 762_0_ack_Return(EOS(STATIC_762), i76)
753_0_ack_Return(EOS(STATIC_753), i71, matching1, i203, matching2, i76) → 641_0_ack_Return(EOS(STATIC_641), i71, 0, i203, 1, i76) | &&(=(matching1, 0), =(matching2, 1))
758_0_ack_Return(EOS(STATIC_758), i213, i214, i76) → 762_0_ack_Return(EOS(STATIC_762), i76)
799_0_ack_Return(EOS(STATIC_799), i71, matching1, i244, matching2, i76) → 641_0_ack_Return(EOS(STATIC_641), i71, 0, i244, 1, i76) | &&(=(matching1, 0), =(matching2, 1))
804_0_ack_Return(EOS(STATIC_804), i255, i256, i76) → 758_0_ack_Return(EOS(STATIC_758), i255, i256, i76)

Combined rules. Obtained 6 conditional rules for P and 7 conditional rules for R.

P rules:
377_0_ack_GT(EOS(STATIC_377), x0, 0, x0) → 493_1_ack_InvokeMethod(377_0_ack_GT(EOS(STATIC_377), -(x0, 1), 1, -(x0, 1)), x0, 0, -(x0, 1), 1) | >(x0, 0)
377_0_ack_GT(EOS(STATIC_377), x0, x1, x0) → 527_1_ack_InvokeMethod(377_0_ack_GT(EOS(STATIC_377), x0, -(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
527_1_ack_InvokeMethod(537_0_ack_Return(EOS(STATIC_537), x0, 0, x2), x3, x0, 0) → 663_1_ack_InvokeMethod(377_0_ack_GT(EOS(STATIC_377), x3, x2, x3), x3, x2)
527_1_ack_InvokeMethod(653_0_ack_Return(EOS(STATIC_653), x0, 0, x2), x3, x0, 0) → 663_1_ack_InvokeMethod(377_0_ack_GT(EOS(STATIC_377), x3, x2, x3), x3, x2)
527_1_ack_InvokeMethod(716_0_ack_Return(EOS(STATIC_716), x0), x1, x2, x3) → 663_1_ack_InvokeMethod(377_0_ack_GT(EOS(STATIC_377), x1, x0, x1), x1, x0)
527_1_ack_InvokeMethod(762_0_ack_Return(EOS(STATIC_762), x0), x1, x2, x3) → 663_1_ack_InvokeMethod(377_0_ack_GT(EOS(STATIC_377), x1, x0, x1), x1, x0)
R rules:
377_0_ack_GT(EOS(STATIC_377), 0, x1, 0) → 405_0_ack_Return(EOS(STATIC_405), 0, x1, +(x1, 1)) | >(+(x1, 1), 0)
493_1_ack_InvokeMethod(405_0_ack_Return(EOS(STATIC_405), 0, 1, x2), x3, 0, 0, 1) → 653_0_ack_Return(EOS(STATIC_653), x3, 0, x2)
663_1_ack_InvokeMethod(716_0_ack_Return(EOS(STATIC_716), x0), x1, x2) → 762_0_ack_Return(EOS(STATIC_762), x0)
663_1_ack_InvokeMethod(762_0_ack_Return(EOS(STATIC_762), x0), x1, x2) → 762_0_ack_Return(EOS(STATIC_762), x0)
663_1_ack_InvokeMethod(405_0_ack_Return(EOS(STATIC_405), 0, x1, x2), 0, x1) → 762_0_ack_Return(EOS(STATIC_762), x2)
493_1_ack_InvokeMethod(716_0_ack_Return(EOS(STATIC_716), x0), x1, 0, x3, 1) → 653_0_ack_Return(EOS(STATIC_653), x1, 0, x0)
493_1_ack_InvokeMethod(762_0_ack_Return(EOS(STATIC_762), x0), x1, 0, x3, 1) → 653_0_ack_Return(EOS(STATIC_653), x1, 0, x0)

Filtered ground terms:

377_0_ack_GT(x1, x2, x3, x4) → 377_0_ack_GT(x2, x3, x4)
762_0_ack_Return(x1, x2) → 762_0_ack_Return(x2)
716_0_ack_Return(x1, x2) → 716_0_ack_Return(x2)
653_0_ack_Return(x1, x2, x3, x4) → 653_0_ack_Return(x2, x4)
537_0_ack_Return(x1, x2, x3, x4) → 537_0_ack_Return(x2, x4)
Cond_377_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_377_0_ack_GT1(x1, x3, x4, x5)
493_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 493_1_ack_InvokeMethod(x1, x2, x4)
Cond_377_0_ack_GT(x1, x2, x3, x4, x5) → Cond_377_0_ack_GT(x1, x3, x4, x5)
405_0_ack_Return(x1, x2, x3, x4) → 405_0_ack_Return(x3, x4)

Filtered duplicate args:

377_0_ack_GT(x1, x2, x3) → 377_0_ack_GT(x2, x3)
Cond_377_0_ack_GT(x1, x2, x3, x4) → Cond_377_0_ack_GT(x1, x3, x4)
Cond_377_0_ack_GT1(x1, x2, x3, x4) → Cond_377_0_ack_GT1(x1, x3, x4)

Filtered unneeded arguments:

493_1_ack_InvokeMethod(x1, x2, x3) → 493_1_ack_InvokeMethod(x1, x3)
527_1_ack_InvokeMethod(x1, x2, x3, x4) → 527_1_ack_InvokeMethod(x1, x2, x4)
653_0_ack_Return(x1, x2) → 653_0_ack_Return(x2)
537_0_ack_Return(x1, x2) → 537_0_ack_Return(x2)

Combined rules. Obtained 6 conditional rules for P and 7 conditional rules for R.

P rules:
377_0_ack_GT(0, x0) → 493_1_ack_InvokeMethod(377_0_ack_GT(1, -(x0, 1)), -(x0, 1)) | >(x0, 0)
377_0_ack_GT(x1, x0) → 527_1_ack_InvokeMethod(377_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) | &&(>(x1, 0), >(x0, 0))
527_1_ack_InvokeMethod(537_0_ack_Return(x2), x3, 0) → 663_1_ack_InvokeMethod(377_0_ack_GT(x2, x3), x3, x2)
527_1_ack_InvokeMethod(653_0_ack_Return(x2), x3, 0) → 663_1_ack_InvokeMethod(377_0_ack_GT(x2, x3), x3, x2)
527_1_ack_InvokeMethod(716_0_ack_Return(x0), x1, x3) → 663_1_ack_InvokeMethod(377_0_ack_GT(x0, x1), x1, x0)
527_1_ack_InvokeMethod(762_0_ack_Return(x0), x1, x3) → 663_1_ack_InvokeMethod(377_0_ack_GT(x0, x1), x1, x0)
R rules:
377_0_ack_GT(x1, 0) → 405_0_ack_Return(x1, +(x1, 1)) | >(x1, -1)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x2), 0) → 653_0_ack_Return(x2)
663_1_ack_InvokeMethod(716_0_ack_Return(x0), x1, x2) → 762_0_ack_Return(x0)
663_1_ack_InvokeMethod(762_0_ack_Return(x0), x1, x2) → 762_0_ack_Return(x0)
663_1_ack_InvokeMethod(405_0_ack_Return(x1, x2), 0, x1) → 762_0_ack_Return(x2)
493_1_ack_InvokeMethod(716_0_ack_Return(x0), x3) → 653_0_ack_Return(x0)
493_1_ack_InvokeMethod(762_0_ack_Return(x0), x3) → 653_0_ack_Return(x0)

Performed bisimulation on rules. Used the following equivalence classes: {[653_0_ack_Return_1, 716_0_ack_Return_1, 762_0_ack_Return_1, 537_0_ack_Return_1]=653_0_ack_Return_1}

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

P rules:
377_0_ACK_GT(0, x0) → COND_377_0_ACK_GT(>(x0, 0), 0, x0)
COND_377_0_ACK_GT(TRUE, 0, x0) → 377_0_ACK_GT(1, -(x0, 1))
377_0_ACK_GT(x1, x0) → COND_377_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
COND_377_0_ACK_GT1(TRUE, x1, x0) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
COND_377_0_ACK_GT1(TRUE, x1, x0) → 377_0_ACK_GT(-(x1, 1), x0)
527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2), x3, 0) → 377_0_ACK_GT(x2, x3)
527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0), x1, x3) → 377_0_ACK_GT(x0, x1)
R rules:
377_0_ack_GT(x1, 0) → Cond_377_0_ack_GT(>(x1, -1), x1, 0)
Cond_377_0_ack_GT(TRUE, x1, 0) → 405_0_ack_Return(x1, +(x1, 1))
493_1_ack_InvokeMethod(405_0_ack_Return(1, x2), 0) → 653_0_ack_Return(x2)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2) → 653_0_ack_Return(x0)
663_1_ack_InvokeMethod(405_0_ack_Return(x1, x2), 0, x1) → 653_0_ack_Return(x2)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x3) → 653_0_ack_Return(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:
377_0_ack_GT(x1, 0) → Cond_377_0_ack_GT(x1 > -1, x1, 0)
Cond_377_0_ack_GT(TRUE, x1, 0) → 405_0_ack_Return(x1, x1 + 1)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x2), 0) → 653_0_ack_Return(x2)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2) → 653_0_ack_Return(x0)
663_1_ack_InvokeMethod(405_0_ack_Return(x1, x2), 0, x1) → 653_0_ack_Return(x2)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x3) → 653_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 377_0_ACK_GT(0, x0[0]) → COND_377_0_ACK_GT(x0[0] > 0, 0, x0[0])
(1): COND_377_0_ACK_GT(TRUE, 0, x0[1]) → 377_0_ACK_GT(1, x0[1] - 1)
(2): 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_377_0_ACK_GT1(TRUE, x1[3], x0[3]) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(x1[4] - 1, x0[4])
(5): 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0) → 377_0_ACK_GT(x2[5], x3[5])
(6): 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6]) → 377_0_ACK_GT(x0[6], x1[6])

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

(1) -> (0), if (1* 0x0[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 (377_0_ack_GT(x1[3] - 1, x0[3]) →* 653_0_ack_Return(x2[5])∧x0[3] - 1* x3[5]x1[3] - 1* 0)

(3) -> (6), if (377_0_ack_GT(x1[3] - 1, x0[3]) →* 653_0_ack_Return(x0[6])∧x0[3] - 1* x1[6]x1[3] - 1* x3[6])

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

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

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

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

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

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

The set Q consists of the following terms:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (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@1fa53fac 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 377_0_ACK_GT(0, x0) → COND_377_0_ACK_GT(>(x0, 0), 0, x0) the following chains were created:
• We consider the chain 377_0_ACK_GT(0, x0[0]) → COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_377_0_ACK_GT(TRUE, 0, x0[1]) → 377_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]377_0_ACK_GT(0, x0[0])≥NonInfC∧377_0_ACK_GT(0, x0[0])≥COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))

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

(2)    (>(x0[0], 0)=TRUE377_0_ACK_GT(0, x0[0])≥NonInfC∧377_0_ACK_GT(0, x0[0])≥COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_377_0_ACK_GT(TRUE, 0, x0) → 377_0_ACK_GT(1, -(x0, 1)) the following chains were created:
• We consider the chain COND_377_0_ACK_GT(TRUE, 0, x0[1]) → 377_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

(7)    (COND_377_0_ACK_GT(TRUE, 0, x0[1])≥NonInfC∧COND_377_0_ACK_GT(TRUE, 0, x0[1])≥377_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(377_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(8)    ((UIncreasing(377_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧[1 + (-1)bso_26] ≥ 0)

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

(9)    ((UIncreasing(377_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧[1 + (-1)bso_26] ≥ 0)

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

(10)    ((UIncreasing(377_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧[1 + (-1)bso_26] ≥ 0)

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

(11)    ((UIncreasing(377_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

For Pair 377_0_ACK_GT(x1, x0) → COND_377_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_377_0_ACK_GT1(TRUE, x1[3], x0[3]) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:

(12)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]377_0_ACK_GT(x1[2], x0[2])≥NonInfC∧377_0_ACK_GT(x1[2], x0[2])≥COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(13)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE377_0_ACK_GT(x1[2], x0[2])≥NonInfC∧377_0_ACK_GT(x1[2], x0[2])≥COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(14)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(15)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(16)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(17)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(18)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

• We consider the chain 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(19)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]377_0_ACK_GT(x1[2], x0[2])≥NonInfC∧377_0_ACK_GT(x1[2], x0[2])≥COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(20)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE377_0_ACK_GT(x1[2], x0[2])≥NonInfC∧377_0_ACK_GT(x1[2], x0[2])≥COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(21)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(22)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(23)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(24)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(25)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_377_0_ACK_GT1(TRUE, x1, x0) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1)) the following chains were created:
• We consider the chain COND_377_0_ACK_GT1(TRUE, x1[3], x0[3]) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1)) which results in the following constraint:

(26)    (COND_377_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_377_0_ACK_GT1(TRUE, x1[3], x0[3])≥527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))∧(UIncreasing(527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥))

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

(27)    ((UIncreasing(527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧[1 + (-1)bso_30] ≥ 0)

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

(28)    ((UIncreasing(527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧[1 + (-1)bso_30] ≥ 0)

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

(29)    ((UIncreasing(527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧[1 + (-1)bso_30] ≥ 0)

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

(30)    ((UIncreasing(527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_30] ≥ 0)

For Pair COND_377_0_ACK_GT1(TRUE, x1, x0) → 377_0_ACK_GT(-(x1, 1), x0) the following chains were created:
• We consider the chain COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(31)    (COND_377_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_377_0_ACK_GT1(TRUE, x1[4], x0[4])≥377_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(32)    ((UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧[(-1)bso_32] ≥ 0)

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

(33)    ((UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧[(-1)bso_32] ≥ 0)

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

(34)    ((UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧[(-1)bso_32] ≥ 0)

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

(35)    ((UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

For Pair 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2), x3, 0) → 377_0_ACK_GT(x2, x3) the following chains were created:
• We consider the chain 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0) → 377_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:

(36)    (527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0)≥NonInfC∧527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0)≥377_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(377_0_ACK_GT(x2[5], x3[5])), ≥))

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

(37)    ((UIncreasing(377_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)

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

(38)    ((UIncreasing(377_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)

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

(39)    ((UIncreasing(377_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)

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

(40)    ((UIncreasing(377_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

For Pair 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0), x1, x3) → 377_0_ACK_GT(x0, x1) the following chains were created:
• We consider the chain 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6]) → 377_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:

(41)    (527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6])≥NonInfC∧527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6])≥377_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(377_0_ACK_GT(x0[6], x1[6])), ≥))

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

(42)    ((UIncreasing(377_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

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

(43)    ((UIncreasing(377_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

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

(44)    ((UIncreasing(377_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)

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

(45)    ((UIncreasing(377_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 377_0_ACK_GT(0, x0) → COND_377_0_ACK_GT(>(x0, 0), 0, x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)

• COND_377_0_ACK_GT(TRUE, 0, x0) → 377_0_ACK_GT(1, -(x0, 1))
• ((UIncreasing(377_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_25] = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

• 377_0_ACK_GT(x1, x0) → COND_377_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

• COND_377_0_ACK_GT1(TRUE, x1, x0) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1, 1), x0), -(x0, 1), -(x1, 1))
• ((UIncreasing(527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))), ≥)∧[bni_29] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_30] ≥ 0)

• COND_377_0_ACK_GT1(TRUE, x1, x0) → 377_0_ACK_GT(-(x1, 1), x0)
• ((UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_31] = 0∧0 = 0∧0 = 0∧[(-1)bso_32] ≥ 0)

• 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2), x3, 0) → 377_0_ACK_GT(x2, x3)
• ((UIncreasing(377_0_ACK_GT(x2[5], x3[5])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

• 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0), x1, x3) → 377_0_ACK_GT(x0, x1)
• ((UIncreasing(377_0_ACK_GT(x0[6], x1[6])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 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(377_0_ack_GT(x1, x2)) = [-1]x2 + [-1]x1
POL(0) = 0
POL(Cond_377_0_ack_GT(x1, x2, x3)) = [1]
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(405_0_ack_Return(x1, x2)) = x2 + [-1]x1
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(493_1_ack_InvokeMethod(x1, x2)) = [-1]
POL(653_0_ack_Return(x1)) = x1
POL(663_1_ack_InvokeMethod(x1, x2, x3)) = [-1]
POL(377_0_ACK_GT(x1, x2)) = [-1] + x2
POL(COND_377_0_ACK_GT(x1, x2, x3)) = [-1] + x3
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_377_0_ACK_GT1(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(527_1_ACK_INVOKEMETHOD(x1, x2, x3)) = [-1] + x2

The following pairs are in P>:

COND_377_0_ACK_GT(TRUE, 0, x0[1]) → 377_0_ACK_GT(1, -(x0[1], 1))
COND_377_0_ACK_GT1(TRUE, x1[3], x0[3]) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), -(x1[3], 1))

The following pairs are in Pbound:

377_0_ACK_GT(0, x0[0]) → COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])
377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P:

377_0_ACK_GT(0, x0[0]) → COND_377_0_ACK_GT(>(x0[0], 0), 0, x0[0])
377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4])
527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0) → 377_0_ACK_GT(x2[5], x3[5])
527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6]) → 377_0_ACK_GT(x0[6], x1[6])

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

405_0_ack_Return(x1, +(x1, 1))1Cond_377_0_ack_GT(TRUE, x1, 0)1

### (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:
377_0_ack_GT(x1, 0) → Cond_377_0_ack_GT(x1 > -1, x1, 0)
Cond_377_0_ack_GT(TRUE, x1, 0) → 405_0_ack_Return(x1, x1 + 1)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x2), 0) → 653_0_ack_Return(x2)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2) → 653_0_ack_Return(x0)
663_1_ack_InvokeMethod(405_0_ack_Return(x1, x2), 0, x1) → 653_0_ack_Return(x2)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x3) → 653_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 377_0_ACK_GT(0, x0[0]) → COND_377_0_ACK_GT(x0[0] > 0, 0, x0[0])
(2): 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(x1[4] - 1, x0[4])
(5): 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0) → 377_0_ACK_GT(x2[5], x3[5])
(6): 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6]) → 377_0_ACK_GT(x0[6], x1[6])

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

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

(6) -> (0), if (x0[6]* 0x1[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:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (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:
377_0_ack_GT(x1, 0) → Cond_377_0_ack_GT(x1 > -1, x1, 0)
Cond_377_0_ack_GT(TRUE, x1, 0) → 405_0_ack_Return(x1, x1 + 1)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x2), 0) → 653_0_ack_Return(x2)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2) → 653_0_ack_Return(x0)
663_1_ack_InvokeMethod(405_0_ack_Return(x1, x2), 0, x1) → 653_0_ack_Return(x2)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x3) → 653_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(4): COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(x1[4] - 1, x0[4])
(2): 377_0_ACK_GT(x1[2], x0[2]) → COND_377_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:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (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_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(x1[4] - 1, x0[4])
(2): 377_0_ACK_GT(x1[2], x0[2]) → COND_377_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:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (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@1fa53fac 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_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
• We consider the chain COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(1)    (COND_377_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_377_0_ACK_GT1(TRUE, x1[4], x0[4])≥377_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(377_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(377_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(377_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(377_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(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

For Pair 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
• We consider the chain 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_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]377_0_ACK_GT(x1[2], x0[2])≥NonInfC∧377_0_ACK_GT(x1[2], x0[2])≥COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_377_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)=TRUE377_0_ACK_GT(x1[2], x0[2])≥NonInfC∧377_0_ACK_GT(x1[2], x0[2])≥COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_377_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_377_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_377_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_377_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_377_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_377_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_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4])
• ((UIncreasing(377_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[bni_9] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

• 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_377_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_377_0_ACK_GT1(x1, x2, x3)) = [-1] + x3 + x2
POL(377_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_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in Pbound:

377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

The following pairs are in P:

377_0_ACK_GT(x1[2], x0[2]) → COND_377_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): 377_0_ACK_GT(x1[2], x0[2]) → COND_377_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

The set Q consists of the following terms:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (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_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(x1[4] - 1, x0[4])

The set Q consists of the following terms:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (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:
377_0_ack_GT(x1, 0) → Cond_377_0_ack_GT(x1 > -1, x1, 0)
Cond_377_0_ack_GT(TRUE, x1, 0) → 405_0_ack_Return(x1, x1 + 1)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x2), 0) → 653_0_ack_Return(x2)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2) → 653_0_ack_Return(x0)
663_1_ack_InvokeMethod(405_0_ack_Return(x1, x2), 0, x1) → 653_0_ack_Return(x2)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x3) → 653_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(1): COND_377_0_ACK_GT(TRUE, 0, x0[1]) → 377_0_ACK_GT(1, x0[1] - 1)
(3): COND_377_0_ACK_GT1(TRUE, x1[3], x0[3]) → 527_1_ACK_INVOKEMETHOD(377_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x1[3] - 1)
(4): COND_377_0_ACK_GT1(TRUE, x1[4], x0[4]) → 377_0_ACK_GT(x1[4] - 1, x0[4])
(5): 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x2[5]), x3[5], 0) → 377_0_ACK_GT(x2[5], x3[5])
(6): 527_1_ACK_INVOKEMETHOD(653_0_ack_Return(x0[6]), x1[6], x3[6]) → 377_0_ACK_GT(x0[6], x1[6])

(3) -> (5), if (377_0_ack_GT(x1[3] - 1, x0[3]) →* 653_0_ack_Return(x2[5])∧x0[3] - 1* x3[5]x1[3] - 1* 0)

(3) -> (6), if (377_0_ack_GT(x1[3] - 1, x0[3]) →* 653_0_ack_Return(x0[6])∧x0[3] - 1* x1[6]x1[3] - 1* x3[6])

The set Q consists of the following terms:
377_0_ack_GT(x0, 0)
Cond_377_0_ack_GT(TRUE, x0, 0)
493_1_ack_InvokeMethod(405_0_ack_Return(1, x0), 0)
663_1_ack_InvokeMethod(653_0_ack_Return(x0), x1, x2)
663_1_ack_InvokeMethod(405_0_ack_Return(x0, x1), 0, x0)
493_1_ack_InvokeMethod(653_0_ack_Return(x0), x1)

### (23) IDependencyGraphProof (EQUIVALENT transformation)

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