### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_14 (Sun Microsystems Inc.) Main-Class: AProVEMathRecursive
`public class AProVEMathRecursive {  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    power(x, y);  }  public static int power(int base, int exponent) {               if (exponent <= 0) {      return 1;    } else if (exponent == 1) {      return base;    } else if (base == 2) {      return base << (exponent-1);    } else if (exponent % 2 == 1) {      return base * power(base, exponent-1);    } else {      int halfPower = power(base, exponent/2);      return halfPower * halfPower;    }  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    if (args.length <= index) {      return 0;    }    String string = args[index];    index++;    if (string == null) {      return 0;    }    return string.length();  }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
AProVEMathRecursive.main([Ljava/lang/String;)V: Graph of 149 nodes with 0 SCCs.

AProVEMathRecursive.power(II)I: Graph of 125 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: AProVEMathRecursive.power(II)I
SCC calls the following helper methods: AProVEMathRecursive.power(II)I
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 59 rules for P and 70 rules for R.

P rules:
342_0_power_GT(EOS(STATIC_342), i56, i62, i62) → 345_0_power_GT(EOS(STATIC_345), i56, i62, i62)
345_0_power_GT(EOS(STATIC_345), i56, i62, i62) → 349_0_power_Load(EOS(STATIC_349), i56, i62) | >(i62, 0)
349_0_power_Load(EOS(STATIC_349), i56, i62) → 353_0_power_ConstantStackPush(EOS(STATIC_353), i56, i62, i62)
353_0_power_ConstantStackPush(EOS(STATIC_353), i56, i62, i62) → 358_0_power_NE(EOS(STATIC_358), i56, i62, i62, 1)
358_0_power_NE(EOS(STATIC_358), i56, i69, i69, matching1) → 377_0_power_NE(EOS(STATIC_377), i56, i69, i69, 1) | =(matching1, 1)
377_0_power_NE(EOS(STATIC_377), i56, i69, i69, matching1) → 390_0_power_Load(EOS(STATIC_390), i56, i69) | &&(!(=(i69, 1)), =(matching1, 1))
390_0_power_Load(EOS(STATIC_390), i56, i69) → 397_0_power_ConstantStackPush(EOS(STATIC_397), i56, i69, i56)
397_0_power_ConstantStackPush(EOS(STATIC_397), i56, i69, i56) → 404_0_power_NE(EOS(STATIC_404), i56, i69, i56, 2)
404_0_power_NE(EOS(STATIC_404), i78, i69, i78, matching1) → 412_0_power_NE(EOS(STATIC_412), i78, i69, i78, 2) | =(matching1, 2)
404_0_power_NE(EOS(STATIC_404), i79, i69, i79, matching1) → 414_0_power_NE(EOS(STATIC_414), i79, i69, i79, 2) | =(matching1, 2)
412_0_power_NE(EOS(STATIC_412), i78, i69, i78, matching1) → 420_0_power_Load(EOS(STATIC_420), i78, i69) | &&(!(=(i78, 2)), =(matching1, 2))
420_0_power_Load(EOS(STATIC_420), i78, i69) → 432_0_power_ConstantStackPush(EOS(STATIC_432), i78, i69, i69)
432_0_power_ConstantStackPush(EOS(STATIC_432), i78, i69, i69) → 441_0_power_IntArithmetic(EOS(STATIC_441), i78, i69, i69, 2)
441_0_power_IntArithmetic(EOS(STATIC_441), i78, i69, i69, matching1) → 449_0_power_ConstantStackPush(EOS(STATIC_449), i78, i69, %(i69, 2)) | =(matching1, 2)
449_0_power_ConstantStackPush(EOS(STATIC_449), i78, i69, i84) → 456_0_power_NE(EOS(STATIC_456), i78, i69, i84, 1)
456_0_power_NE(EOS(STATIC_456), i78, i69, matching1, matching2) → 463_0_power_NE(EOS(STATIC_463), i78, i69, 0, 1) | &&(=(matching1, 0), =(matching2, 1))
456_0_power_NE(EOS(STATIC_456), i78, i69, matching1, matching2) → 464_0_power_NE(EOS(STATIC_464), i78, i69, 1, 1) | &&(=(matching1, 1), =(matching2, 1))
463_0_power_NE(EOS(STATIC_463), i78, i69, matching1, matching2) → 472_0_power_Load(EOS(STATIC_472), i78, i69) | &&(=(matching1, 0), =(matching2, 1))
482_0_power_Load(EOS(STATIC_482), i69, i78) → 500_0_power_ConstantStackPush(EOS(STATIC_500), i78, i69)
500_0_power_ConstantStackPush(EOS(STATIC_500), i78, i69) → 513_0_power_IntArithmetic(EOS(STATIC_513), i78, i69, 2)
513_0_power_IntArithmetic(EOS(STATIC_513), i78, i69, matching1) → 526_0_power_InvokeMethod(EOS(STATIC_526), i78, /(i69, 2)) | &&(>(i69, 1), =(matching1, 2))
526_0_power_InvokeMethod(EOS(STATIC_526), i78, i98) → 537_1_power_InvokeMethod(537_0_power_Load(EOS(STATIC_537), i78, i98), i78, i98)
339_0_power_Load(EOS(STATIC_339), i56, i57) → 342_0_power_GT(EOS(STATIC_342), i56, i57, i57)
464_0_power_NE(EOS(STATIC_464), i78, i69, matching1, matching2) → 474_0_power_Load(EOS(STATIC_474), i78, i69) | &&(=(matching1, 1), =(matching2, 1))
502_0_power_Load(EOS(STATIC_502), i78, i69, i78, i78) → 515_0_power_ConstantStackPush(EOS(STATIC_515), i78, i69, i78, i78, i69)
515_0_power_ConstantStackPush(EOS(STATIC_515), i78, i69, i78, i78, i69) → 528_0_power_IntArithmetic(EOS(STATIC_528), i78, i69, i78, i78, i69, 1)
528_0_power_IntArithmetic(EOS(STATIC_528), i78, i69, i78, i78, i69, matching1) → 540_0_power_InvokeMethod(EOS(STATIC_540), i78, i69, i78, i78, -(i69, 1)) | &&(>(i69, 0), =(matching1, 1))
540_0_power_InvokeMethod(EOS(STATIC_540), i78, i69, i78, i78, i103) → 550_1_power_InvokeMethod(550_0_power_Load(EOS(STATIC_550), i78, i103), i78, i69, i78, i78, i103)
414_0_power_NE(EOS(STATIC_414), i79, i69, i79, matching1) → 423_0_power_Load(EOS(STATIC_423), i79, i69) | &&(!(=(i79, 2)), =(matching1, 2))
423_0_power_Load(EOS(STATIC_423), i79, i69) → 436_0_power_ConstantStackPush(EOS(STATIC_436), i79, i69, i69)
436_0_power_ConstantStackPush(EOS(STATIC_436), i79, i69, i69) → 445_0_power_IntArithmetic(EOS(STATIC_445), i79, i69, i69, 2)
445_0_power_IntArithmetic(EOS(STATIC_445), i79, i69, i69, matching1) → 453_0_power_ConstantStackPush(EOS(STATIC_453), i79, i69, %(i69, 2)) | =(matching1, 2)
453_0_power_ConstantStackPush(EOS(STATIC_453), i79, i69, i85) → 461_0_power_NE(EOS(STATIC_461), i79, i69, i85, 1)
461_0_power_NE(EOS(STATIC_461), i79, i69, matching1, matching2) → 469_0_power_NE(EOS(STATIC_469), i79, i69, 0, 1) | &&(=(matching1, 0), =(matching2, 1))
461_0_power_NE(EOS(STATIC_461), i79, i69, matching1, matching2) → 470_0_power_NE(EOS(STATIC_470), i79, i69, 1, 1) | &&(=(matching1, 1), =(matching2, 1))
469_0_power_NE(EOS(STATIC_469), i79, i69, matching1, matching2) → 478_0_power_Load(EOS(STATIC_478), i79, i69) | &&(=(matching1, 0), =(matching2, 1))
496_0_power_Load(EOS(STATIC_496), i69, i79) → 508_0_power_ConstantStackPush(EOS(STATIC_508), i79, i69)
508_0_power_ConstantStackPush(EOS(STATIC_508), i79, i69) → 522_0_power_IntArithmetic(EOS(STATIC_522), i79, i69, 2)
522_0_power_IntArithmetic(EOS(STATIC_522), i79, i69, matching1) → 533_0_power_InvokeMethod(EOS(STATIC_533), i79, /(i69, 2)) | &&(>(i69, 1), =(matching1, 2))
533_0_power_InvokeMethod(EOS(STATIC_533), i79, i100) → 543_1_power_InvokeMethod(543_0_power_Load(EOS(STATIC_543), i79, i100), i79, i100)
470_0_power_NE(EOS(STATIC_470), i79, i69, matching1, matching2) → 479_0_power_Load(EOS(STATIC_479), i79, i69) | &&(=(matching1, 1), =(matching2, 1))
510_0_power_Load(EOS(STATIC_510), i79, i69, i79, i79) → 524_0_power_ConstantStackPush(EOS(STATIC_524), i79, i69, i79, i79, i69)
524_0_power_ConstantStackPush(EOS(STATIC_524), i79, i69, i79, i79, i69) → 536_0_power_IntArithmetic(EOS(STATIC_536), i79, i69, i79, i79, i69, 1)
536_0_power_IntArithmetic(EOS(STATIC_536), i79, i69, i79, i79, i69, matching1) → 546_0_power_InvokeMethod(EOS(STATIC_546), i79, i69, i79, i79, -(i69, 1)) | &&(>(i69, 0), =(matching1, 1))
546_0_power_InvokeMethod(EOS(STATIC_546), i79, i69, i79, i79, i104) → 554_1_power_InvokeMethod(554_0_power_Load(EOS(STATIC_554), i79, i104), i79, i69, i79, i79, i104)
R rules:
342_0_power_GT(EOS(STATIC_342), i56, matching1, matching2) → 344_0_power_GT(EOS(STATIC_344), i56, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
344_0_power_GT(EOS(STATIC_344), i56, matching1, matching2) → 347_0_power_ConstantStackPush(EOS(STATIC_347), i56, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
347_0_power_ConstantStackPush(EOS(STATIC_347), i56, matching1) → 351_0_power_Return(EOS(STATIC_351), i56, 0) | =(matching1, 0)
358_0_power_NE(EOS(STATIC_358), i56, matching1, matching2, matching3) → 376_0_power_NE(EOS(STATIC_376), i56, 1, 1, 1) | &&(&&(=(matching1, 1), =(matching2, 1)), =(matching3, 1))
376_0_power_NE(EOS(STATIC_376), i56, matching1, matching2, matching3) → 388_0_power_Load(EOS(STATIC_388), i56, 1) | &&(&&(=(matching1, 1), =(matching2, 1)), =(matching3, 1))
388_0_power_Load(EOS(STATIC_388), i56, matching1) → 394_0_power_Return(EOS(STATIC_394), i56, 1, i56) | =(matching1, 1)
404_0_power_NE(EOS(STATIC_404), matching1, i69, matching2, matching3) → 413_0_power_NE(EOS(STATIC_413), 2, i69, 2, 2) | &&(&&(=(matching1, 2), =(matching2, 2)), =(matching3, 2))
413_0_power_NE(EOS(STATIC_413), matching1, i69, matching2, matching3) → 422_0_power_Load(EOS(STATIC_422), 2, i69) | &&(&&(=(matching1, 2), =(matching2, 2)), =(matching3, 2))
434_0_power_Load(EOS(STATIC_434), matching1, i69, matching2) → 443_0_power_ConstantStackPush(EOS(STATIC_443), 2, i69, 2, i69) | &&(=(matching1, 2), =(matching2, 2))
443_0_power_ConstantStackPush(EOS(STATIC_443), matching1, i69, matching2, i69) → 451_0_power_IntArithmetic(EOS(STATIC_451), 2, i69, 2, i69) | &&(=(matching1, 2), =(matching2, 2))
451_0_power_IntArithmetic(EOS(STATIC_451), matching1, i69, matching2, i69) → 458_0_power_IntArithmetic(EOS(STATIC_458), 2, i69, 2) | &&(&&(>(i69, 0), =(matching1, 2)), =(matching2, 2))
458_0_power_IntArithmetic(EOS(STATIC_458), matching1, i69, matching2) → 466_0_power_Return(EOS(STATIC_466), 2, i69) | &&(=(matching1, 2), =(matching2, 2))
537_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), i115, matching1, i115), i115, matching2) → 583_0_power_Return(EOS(STATIC_583), i115, 1, i115, 1, i115) | &&(=(matching1, 1), =(matching2, 1))
537_1_power_InvokeMethod(862_0_power_Return(EOS(STATIC_862), i337, i338, i316), i337, i338) → 934_0_power_Return(EOS(STATIC_934), i337, i338, i337, i338, i316)
537_1_power_InvokeMethod(924_0_power_Return(EOS(STATIC_924), i367), i406, i407) → 1008_0_power_Return(EOS(STATIC_1008), i406, i407, i367)
543_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), i128, matching1, i128), i128, matching2) → 597_0_power_Return(EOS(STATIC_597), i128, 1, i128, 1, i128) | &&(=(matching1, 1), =(matching2, 1))
543_1_power_InvokeMethod(637_0_power_Return(EOS(STATIC_637), i192, i193), i192, i193) → 721_0_power_Return(EOS(STATIC_721), i192, i193, i192, i193)
543_1_power_InvokeMethod(671_0_power_Return(EOS(STATIC_671)), i250, i251) → 814_0_power_Return(EOS(STATIC_814), i250, i251)
543_1_power_InvokeMethod(872_0_power_Return(EOS(STATIC_872), i357, i358), i357, i358) → 954_0_power_Return(EOS(STATIC_954), i357, i358, i357, i358)
543_1_power_InvokeMethod(943_0_power_Return(EOS(STATIC_943)), i427, i428) → 1024_0_power_Return(EOS(STATIC_1024), i427, i428)
550_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), i141, matching1, i141), i141, i69, i141, i141, matching2) → 614_0_power_Return(EOS(STATIC_614), i141, i69, i141, i141, 1, i141, 1, i141) | &&(=(matching1, 1), =(matching2, 1))
550_1_power_InvokeMethod(862_0_power_Return(EOS(STATIC_862), i343, i344, i316), i343, i69, i343, i343, i344) → 938_0_power_Return(EOS(STATIC_938), i343, i69, i343, i343, i344, i343, i344, i316)
550_1_power_InvokeMethod(924_0_power_Return(EOS(STATIC_924), i367), i413, i69, i413, i413, i414) → 1012_0_power_Return(EOS(STATIC_1012), i413, i69, i413, i413, i414, i367)
554_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), i156, matching1, i156), i156, i69, i156, i156, matching2) → 619_0_power_Return(EOS(STATIC_619), i156, i69, i156, i156, 1, i156, 1, i156) | &&(=(matching1, 1), =(matching2, 1))
554_1_power_InvokeMethod(637_0_power_Return(EOS(STATIC_637), i200, i201), i200, i69, i200, i200, i201) → 724_0_power_Return(EOS(STATIC_724), i200, i69, i200, i200, i201, i200, i201)
554_1_power_InvokeMethod(671_0_power_Return(EOS(STATIC_671)), i256, i69, i256, i256, i257) → 817_0_power_Return(EOS(STATIC_817), i256, i69, i256, i256, i257)
554_1_power_InvokeMethod(872_0_power_Return(EOS(STATIC_872), i363, i364), i363, i69, i363, i363, i364) → 958_0_power_Return(EOS(STATIC_958), i363, i69, i363, i363, i364, i363, i364)
554_1_power_InvokeMethod(943_0_power_Return(EOS(STATIC_943)), i435, i69, i435, i435, i436) → 1029_0_power_Return(EOS(STATIC_1029), i435, i69, i435, i435, i436)
583_0_power_Return(EOS(STATIC_583), i115, matching1, i115, matching2, i115) → 611_0_power_Store(EOS(STATIC_611), i115) | &&(=(matching1, 1), =(matching2, 1))
597_0_power_Return(EOS(STATIC_597), i128, matching1, i128, matching2, i128) → 617_0_power_Store(EOS(STATIC_617), i128) | &&(=(matching1, 1), =(matching2, 1))
611_0_power_Store(EOS(STATIC_611), i115) → 752_0_power_Store(EOS(STATIC_752), i115)
614_0_power_Return(EOS(STATIC_614), i141, i69, i141, i141, matching1, i141, matching2, i141) → 625_0_power_IntArithmetic(EOS(STATIC_625), i141, i69, i141, i141) | &&(=(matching1, 1), =(matching2, 1))
619_0_power_Return(EOS(STATIC_619), i156, i69, i156, i156, matching1, i156, matching2, i156) → 628_0_power_IntArithmetic(EOS(STATIC_628), i156, i69, i156, i156) | &&(=(matching1, 1), =(matching2, 1))
625_0_power_IntArithmetic(EOS(STATIC_625), i141, i69, i141, i141) → 756_0_power_IntArithmetic(EOS(STATIC_756), i141, i69, i141, i141)
628_0_power_IntArithmetic(EOS(STATIC_628), i156, i69, i156, i156) → 637_0_power_Return(EOS(STATIC_637), i156, i69) | &&(>(i156, 1), >(i156, 1))
635_0_power_Load(EOS(STATIC_635), i128, i128) → 643_0_power_IntArithmetic(EOS(STATIC_643), i128, i128)
643_0_power_IntArithmetic(EOS(STATIC_643), i128, i128) → 671_0_power_Return(EOS(STATIC_671)) | &&(>(i128, 1), >(i128, 1))
701_0_power_Return(EOS(STATIC_701), i174, i175, i174, i175, i165) → 752_0_power_Store(EOS(STATIC_752), i165)
704_0_power_Return(EOS(STATIC_704), i180, i69, i180, i180, i181, i180, i181, i165) → 756_0_power_IntArithmetic(EOS(STATIC_756), i180, i69, i180, i165)
721_0_power_Return(EOS(STATIC_721), i192, i193, i192, i193) → 785_0_power_Store(EOS(STATIC_785))
724_0_power_Return(EOS(STATIC_724), i200, i69, i200, i200, i201, i200, i201) → 788_0_power_IntArithmetic(EOS(STATIC_788), i200, i69, i200)
752_0_power_Store(EOS(STATIC_752), i165) → 835_0_power_Store(EOS(STATIC_835), i165)
756_0_power_IntArithmetic(EOS(STATIC_756), i180, i69, i180, i165) → 839_0_power_IntArithmetic(EOS(STATIC_839), i180, i69, i180, i165)
785_0_power_Store(EOS(STATIC_785)) → 851_0_power_Store(EOS(STATIC_851))
788_0_power_IntArithmetic(EOS(STATIC_788), i200, i69, i200) → 856_0_power_IntArithmetic(EOS(STATIC_856), i200, i69, i200)
793_0_power_Return(EOS(STATIC_793), i219, i220, i168) → 835_0_power_Store(EOS(STATIC_835), i168)
796_0_power_Return(EOS(STATIC_796), i226, i69, i226, i226, i227, i168) → 839_0_power_IntArithmetic(EOS(STATIC_839), i226, i69, i226, i168)
814_0_power_Return(EOS(STATIC_814), i250, i251) → 851_0_power_Store(EOS(STATIC_851))
817_0_power_Return(EOS(STATIC_817), i256, i69, i256, i256, i257) → 856_0_power_IntArithmetic(EOS(STATIC_856), i256, i69, i256)
839_0_power_IntArithmetic(EOS(STATIC_839), i226, i69, i226, i168) → 862_0_power_Return(EOS(STATIC_862), i226, i69, *(i226, i168)) | &&(<=(i226, 1), <=(i168, 1))
856_0_power_IntArithmetic(EOS(STATIC_856), i256, i69, i256) → 872_0_power_Return(EOS(STATIC_872), i256, i69) | >(i256, 1)
874_0_power_Load(EOS(STATIC_874), i168, i168) → 884_0_power_IntArithmetic(EOS(STATIC_884), i168, i168)
884_0_power_IntArithmetic(EOS(STATIC_884), i168, i168) → 924_0_power_Return(EOS(STATIC_924), *(i168, i168)) | &&(<=(i168, 1), <=(i168, 1))
904_0_power_IntArithmetic(EOS(STATIC_904)) → 943_0_power_Return(EOS(STATIC_943))
934_0_power_Return(EOS(STATIC_934), i337, i338, i337, i338, i316) → 701_0_power_Return(EOS(STATIC_701), i337, i338, i337, i338, i316)
938_0_power_Return(EOS(STATIC_938), i343, i69, i343, i343, i344, i343, i344, i316) → 704_0_power_Return(EOS(STATIC_704), i343, i69, i343, i343, i344, i343, i344, i316)
954_0_power_Return(EOS(STATIC_954), i357, i358, i357, i358) → 721_0_power_Return(EOS(STATIC_721), i357, i358, i357, i358)
958_0_power_Return(EOS(STATIC_958), i363, i69, i363, i363, i364, i363, i364) → 724_0_power_Return(EOS(STATIC_724), i363, i69, i363, i363, i364, i363, i364)
1008_0_power_Return(EOS(STATIC_1008), i406, i407, i367) → 793_0_power_Return(EOS(STATIC_793), i406, i407, i367)
1012_0_power_Return(EOS(STATIC_1012), i413, i69, i413, i413, i414, i367) → 796_0_power_Return(EOS(STATIC_796), i413, i69, i413, i413, i414, i367)
1024_0_power_Return(EOS(STATIC_1024), i427, i428) → 814_0_power_Return(EOS(STATIC_814), i427, i428)
1029_0_power_Return(EOS(STATIC_1029), i435, i69, i435, i435, i436) → 817_0_power_Return(EOS(STATIC_817), i435, i69, i435, i435, i436)

Combined rules. Obtained 4 conditional rules for P and 17 conditional rules for R.

P rules:
342_0_power_GT(EOS(STATIC_342), x0, x1, x1) → 537_1_power_InvokeMethod(342_0_power_GT(EOS(STATIC_342), x0, /(x1, 2), /(x1, 2)), x0, /(x1, 2)) | &&(&&(>(x1, 1), !(=(x0, 2))), =(0, %(x1, 2)))
342_0_power_GT(EOS(STATIC_342), x0, x1, x1) → 550_1_power_InvokeMethod(342_0_power_GT(EOS(STATIC_342), x0, -(x1, 1), -(x1, 1)), x0, x1, x0, x0, -(x1, 1)) | &&(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), =(1, %(x1, 2)))
342_0_power_GT(EOS(STATIC_342), x0, x1, x1) → 543_1_power_InvokeMethod(342_0_power_GT(EOS(STATIC_342), x0, /(x1, 2), /(x1, 2)), x0, /(x1, 2)) | &&(&&(>(x1, 1), !(=(x0, 2))), =(0, %(x1, 2)))
342_0_power_GT(EOS(STATIC_342), x0, x1, x1) → 554_1_power_InvokeMethod(342_0_power_GT(EOS(STATIC_342), x0, -(x1, 1), -(x1, 1)), x0, x1, x0, x0, -(x1, 1)) | &&(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), =(1, %(x1, 2)))
R rules:
342_0_power_GT(EOS(STATIC_342), x0, 0, 0) → 351_0_power_Return(EOS(STATIC_351), x0, 0)
554_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), x0, 1, x0), x0, x2, x0, x0, 1) → 637_0_power_Return(EOS(STATIC_637), x0, x2) | >(x0, 1)
543_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), x0, 1, x0), x0, 1) → 671_0_power_Return(EOS(STATIC_671)) | >(x0, 1)
550_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), x0, 1, x0), x0, x2, x0, x0, 1) → 862_0_power_Return(EOS(STATIC_862), x0, x2, *(x0, x0)) | <=(x0, 1)
554_1_power_InvokeMethod(671_0_power_Return(EOS(STATIC_671)), x0, x1, x0, x0, x2) → 872_0_power_Return(EOS(STATIC_872), x0, x1) | >(x0, 1)
554_1_power_InvokeMethod(943_0_power_Return(EOS(STATIC_943)), x0, x1, x0, x0, x2) → 872_0_power_Return(EOS(STATIC_872), x0, x1) | >(x0, 1)
554_1_power_InvokeMethod(637_0_power_Return(EOS(STATIC_637), x0, x1), x0, x2, x0, x0, x1) → 872_0_power_Return(EOS(STATIC_872), x0, x2) | >(x0, 1)
554_1_power_InvokeMethod(872_0_power_Return(EOS(STATIC_872), x0, x1), x0, x2, x0, x0, x1) → 872_0_power_Return(EOS(STATIC_872), x0, x2) | >(x0, 1)
537_1_power_InvokeMethod(394_0_power_Return(EOS(STATIC_394), x0, 1, x0), x0, 1) → 924_0_power_Return(EOS(STATIC_924), *(x0, x0)) | <=(x0, 1)
543_1_power_InvokeMethod(671_0_power_Return(EOS(STATIC_671)), x0, x1) → 943_0_power_Return(EOS(STATIC_943))
543_1_power_InvokeMethod(943_0_power_Return(EOS(STATIC_943)), x0, x1) → 943_0_power_Return(EOS(STATIC_943))
543_1_power_InvokeMethod(637_0_power_Return(EOS(STATIC_637), x0, x1), x0, x1) → 943_0_power_Return(EOS(STATIC_943))
543_1_power_InvokeMethod(872_0_power_Return(EOS(STATIC_872), x0, x1), x0, x1) → 943_0_power_Return(EOS(STATIC_943))
537_1_power_InvokeMethod(862_0_power_Return(EOS(STATIC_862), x0, x1, x2), x0, x1) → 924_0_power_Return(EOS(STATIC_924), *(x2, x2)) | <=(x2, 1)
550_1_power_InvokeMethod(862_0_power_Return(EOS(STATIC_862), x0, x1, x2), x0, x3, x0, x0, x1) → 862_0_power_Return(EOS(STATIC_862), x0, x3, *(x0, x2)) | &&(<=(x2, 1), <=(x0, 1))
537_1_power_InvokeMethod(924_0_power_Return(EOS(STATIC_924), x0), x1, x2) → 924_0_power_Return(EOS(STATIC_924), *(x0, x0)) | <=(x0, 1)
550_1_power_InvokeMethod(924_0_power_Return(EOS(STATIC_924), x0), x1, x2, x1, x1, x3) → 862_0_power_Return(EOS(STATIC_862), x1, x2, *(x1, x0)) | &&(<=(x1, 1), <=(x0, 1))

Filtered ground terms:

342_0_power_GT(x1, x2, x3, x4) → 342_0_power_GT(x2, x3, x4)
Cond_342_0_power_GT3(x1, x2, x3, x4, x5) → Cond_342_0_power_GT3(x1, x3, x4, x5)
Cond_342_0_power_GT2(x1, x2, x3, x4, x5) → Cond_342_0_power_GT2(x1, x3, x4, x5)
Cond_342_0_power_GT1(x1, x2, x3, x4, x5) → Cond_342_0_power_GT1(x1, x3, x4, x5)
Cond_342_0_power_GT(x1, x2, x3, x4, x5) → Cond_342_0_power_GT(x1, x3, x4, x5)
862_0_power_Return(x1, x2, x3, x4) → 862_0_power_Return(x2, x3, x4)
924_0_power_Return(x1, x2) → 924_0_power_Return(x2)
943_0_power_Return(x1) → 943_0_power_Return
872_0_power_Return(x1, x2, x3) → 872_0_power_Return(x2, x3)
637_0_power_Return(x1, x2, x3) → 637_0_power_Return(x2, x3)
671_0_power_Return(x1) → 671_0_power_Return
Cond_537_1_power_InvokeMethod(x1, x2, x3, x4) → Cond_537_1_power_InvokeMethod(x1, x2, x3)
394_0_power_Return(x1, x2, x3, x4) → 394_0_power_Return(x2, x4)
Cond_554_1_power_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_554_1_power_InvokeMethod2(x1, x3, x4, x5, x6, x7)
Cond_554_1_power_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_554_1_power_InvokeMethod1(x1, x3, x4, x5, x6, x7)
Cond_550_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_550_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6)
Cond_543_1_power_InvokeMethod(x1, x2, x3, x4) → Cond_543_1_power_InvokeMethod(x1, x2, x3)
Cond_554_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_554_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6)
351_0_power_Return(x1, x2, x3) → 351_0_power_Return(x2)

Filtered duplicate args:

342_0_power_GT(x1, x2, x3) → 342_0_power_GT(x1, x3)
Cond_342_0_power_GT(x1, x2, x3, x4) → Cond_342_0_power_GT(x1, x2, x4)
Cond_342_0_power_GT1(x1, x2, x3, x4) → Cond_342_0_power_GT1(x1, x2, x4)
550_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6) → 550_1_power_InvokeMethod(x1, x3, x5, x6)
Cond_342_0_power_GT2(x1, x2, x3, x4) → Cond_342_0_power_GT2(x1, x2, x4)
Cond_342_0_power_GT3(x1, x2, x3, x4) → Cond_342_0_power_GT3(x1, x2, x4)
554_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6) → 554_1_power_InvokeMethod(x1, x3, x5, x6)
394_0_power_Return(x1, x2) → 394_0_power_Return(x2)
Cond_554_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_554_1_power_InvokeMethod(x1, x2, x4)
Cond_543_1_power_InvokeMethod(x1, x2, x3) → Cond_543_1_power_InvokeMethod(x1, x2)
Cond_550_1_power_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_550_1_power_InvokeMethod(x1, x2, x4)
Cond_554_1_power_InvokeMethod1(x1, x2, x3, x4, x5, x6) → Cond_554_1_power_InvokeMethod1(x1, x3, x5, x6)
Cond_554_1_power_InvokeMethod2(x1, x2, x3, x4, x5, x6) → Cond_554_1_power_InvokeMethod2(x1, x3, x5, x6)
Cond_554_1_power_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_554_1_power_InvokeMethod3(x1, x2, x4)
Cond_554_1_power_InvokeMethod4(x1, x2, x3, x4, x5, x6, x7) → Cond_554_1_power_InvokeMethod4(x1, x2, x4)
Cond_537_1_power_InvokeMethod(x1, x2, x3) → Cond_537_1_power_InvokeMethod(x1, x2)
Cond_537_1_power_InvokeMethod1(x1, x2, x3, x4) → Cond_537_1_power_InvokeMethod1(x1, x2)
Cond_550_1_power_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_550_1_power_InvokeMethod1(x1, x2, x4)
Cond_550_1_power_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_550_1_power_InvokeMethod2(x1, x2, x4, x6, x7)

Filtered unneeded arguments:

550_1_power_InvokeMethod(x1, x2, x3, x4) → 550_1_power_InvokeMethod(x1, x3, x4)
554_1_power_InvokeMethod(x1, x2, x3, x4) → 554_1_power_InvokeMethod(x1, x3, x4)
Cond_554_1_power_InvokeMethod(x1, x2, x3) → Cond_554_1_power_InvokeMethod(x1, x2)
Cond_543_1_power_InvokeMethod(x1, x2) → Cond_543_1_power_InvokeMethod(x1)
Cond_550_1_power_InvokeMethod(x1, x2, x3) → Cond_550_1_power_InvokeMethod(x1, x2)
Cond_554_1_power_InvokeMethod1(x1, x2, x3, x4) → Cond_554_1_power_InvokeMethod1(x1, x3)
Cond_554_1_power_InvokeMethod2(x1, x2, x3, x4) → Cond_554_1_power_InvokeMethod2(x1, x3)
Cond_554_1_power_InvokeMethod3(x1, x2, x3) → Cond_554_1_power_InvokeMethod3(x1, x2)
Cond_554_1_power_InvokeMethod4(x1, x2, x3) → Cond_554_1_power_InvokeMethod4(x1, x2)
Cond_550_1_power_InvokeMethod1(x1, x2, x3) → Cond_550_1_power_InvokeMethod1(x1, x2)
Cond_537_1_power_InvokeMethod2(x1, x2, x3, x4) → Cond_537_1_power_InvokeMethod2(x1, x2)
Cond_550_1_power_InvokeMethod2(x1, x2, x3, x4, x5) → Cond_550_1_power_InvokeMethod2(x1, x2, x4)
637_0_power_Return(x1, x2) → 637_0_power_Return(x1)
872_0_power_Return(x1, x2) → 872_0_power_Return(x1)
862_0_power_Return(x1, x2, x3) → 862_0_power_Return(x1, x3)

Filtered modulo operations.

Combined rules. Obtained 4 conditional rules for P and 17 conditional rules for R.

P rules:
342_0_power_GT(x0, x1) → 537_1_power_InvokeMethod(342_0_power_GT(x0, /(x1, 2)), x0, /(x1, 2)) | &&(>(x1, 1), !(=(x0, 2)))
342_0_power_GT(x0, x1) → 550_1_power_InvokeMethod(342_0_power_GT(x0, -(x1, 1)), x0, -(x1, 1)) | &&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2)))
342_0_power_GT(x0, x1) → 543_1_power_InvokeMethod(342_0_power_GT(x0, /(x1, 2)), x0, /(x1, 2)) | &&(>(x1, 1), !(=(x0, 2)))
342_0_power_GT(x0, x1) → 554_1_power_InvokeMethod(342_0_power_GT(x0, -(x1, 1)), x0, -(x1, 1)) | &&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2)))
R rules:
342_0_power_GT(x0, 0) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(394_0_power_Return(x0), x0, 1) → 637_0_power_Return(x0) | >(x0, 1)
543_1_power_InvokeMethod(394_0_power_Return(x0), x0, 1) → 671_0_power_Return | >(x0, 1)
550_1_power_InvokeMethod(394_0_power_Return(x0), x0, 1) → 862_0_power_Return(x0, *(x0, x0)) | <=(x0, 1)
554_1_power_InvokeMethod(671_0_power_Return, x0, x2) → 872_0_power_Return(x0) | >(x0, 1)
554_1_power_InvokeMethod(943_0_power_Return, x0, x2) → 872_0_power_Return(x0) | >(x0, 1)
554_1_power_InvokeMethod(637_0_power_Return(x0), x0, x1) → 872_0_power_Return(x0) | >(x0, 1)
554_1_power_InvokeMethod(872_0_power_Return(x0), x0, x1) → 872_0_power_Return(x0) | >(x0, 1)
537_1_power_InvokeMethod(394_0_power_Return(x0), x0, 1) → 924_0_power_Return(*(x0, x0)) | <=(x0, 1)
543_1_power_InvokeMethod(671_0_power_Return, x0, x1) → 943_0_power_Return
543_1_power_InvokeMethod(943_0_power_Return, x0, x1) → 943_0_power_Return
543_1_power_InvokeMethod(637_0_power_Return(x0), x0, x1) → 943_0_power_Return
543_1_power_InvokeMethod(872_0_power_Return(x0), x0, x1) → 943_0_power_Return
537_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → 924_0_power_Return(*(x2, x2)) | <=(x2, 1)
550_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → 862_0_power_Return(x0, *(x0, x2)) | &&(<=(x2, 1), <=(x0, 1))
537_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2) → 924_0_power_Return(*(x0, x0)) | <=(x0, 1)
550_1_power_InvokeMethod(924_0_power_Return(x0), x1, x3) → 862_0_power_Return(x1, *(x1, x0)) | &&(<=(x1, 1), <=(x0, 1))

Performed bisimulation on rules. Used the following equivalence classes: {[351_0_power_Return_1, 394_0_power_Return_1, 637_0_power_Return_1, 872_0_power_Return_1]=351_0_power_Return_1, [Cond_554_1_power_InvokeMethod3_4, Cond_554_1_power_InvokeMethod4_4]=Cond_554_1_power_InvokeMethod3_4, [Cond_554_1_power_InvokeMethod1_4, Cond_554_1_power_InvokeMethod2_4]=Cond_554_1_power_InvokeMethod1_4, [671_0_power_Return, 943_0_power_Return]=671_0_power_Return}

Finished conversion. Obtained 8 rules for P and 23 rules for R. System has predefined symbols.

P rules:
342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT(&&(>(x1, 1), !(=(x0, 2))), x0, x1)
COND_342_0_POWER_GT(TRUE, x0, x1) → 342_0_POWER_GT(x0, /(x1, 2))
342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT1(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), x0, x1)
COND_342_0_POWER_GT1(TRUE, x0, x1) → 342_0_POWER_GT(x0, -(x1, 1))
342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT2(&&(>(x1, 1), !(=(x0, 2))), x0, x1)
COND_342_0_POWER_GT2(TRUE, x0, x1) → 342_0_POWER_GT(x0, /(x1, 2))
342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT3(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), x0, x1)
COND_342_0_POWER_GT3(TRUE, x0, x1) → 342_0_POWER_GT(x0, -(x1, 1))
R rules:
342_0_power_GT(x0, 0) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_554_1_power_InvokeMethod(>(x0, 1), 351_0_power_Return(x0), x0, 1)
Cond_554_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 351_0_power_Return(x0)
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_543_1_power_InvokeMethod(>(x0, 1), 351_0_power_Return(x0), x0, 1)
Cond_543_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 671_0_power_Return
550_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_550_1_power_InvokeMethod(<=(x0, 1), 351_0_power_Return(x0), x0, 1)
Cond_550_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 862_0_power_Return(x0, *(x0, x0))
554_1_power_InvokeMethod(671_0_power_Return, x0, x2) → Cond_554_1_power_InvokeMethod1(>(x0, 1), 671_0_power_Return, x0, x2)
Cond_554_1_power_InvokeMethod1(TRUE, 671_0_power_Return, x0, x2) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1) → Cond_554_1_power_InvokeMethod3(>(x0, 1), 351_0_power_Return(x0), x0, x1)
Cond_554_1_power_InvokeMethod3(TRUE, 351_0_power_Return(x0), x0, x1) → 351_0_power_Return(x0)
537_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_537_1_power_InvokeMethod(<=(x0, 1), 351_0_power_Return(x0), x0, 1)
Cond_537_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 924_0_power_Return(*(x0, x0))
543_1_power_InvokeMethod(671_0_power_Return, x0, x1) → 671_0_power_Return
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1) → 671_0_power_Return
537_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → Cond_537_1_power_InvokeMethod1(<=(x2, 1), 862_0_power_Return(x0, x2), x0, x1)
Cond_537_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x2), x0, x1) → 924_0_power_Return(*(x2, x2))
550_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → Cond_550_1_power_InvokeMethod1(&&(<=(x2, 1), <=(x0, 1)), 862_0_power_Return(x0, x2), x0, x1)
Cond_550_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x2), x0, x1) → 862_0_power_Return(x0, *(x0, x2))
537_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2) → Cond_537_1_power_InvokeMethod2(<=(x0, 1), 924_0_power_Return(x0), x1, x2)
Cond_537_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x2) → 924_0_power_Return(*(x0, x0))
550_1_power_InvokeMethod(924_0_power_Return(x0), x1, x3) → Cond_550_1_power_InvokeMethod2(&&(<=(x1, 1), <=(x0, 1)), 924_0_power_Return(x0), x1, x3)
Cond_550_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x3) → 862_0_power_Return(x1, *(x1, 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:
342_0_power_GT(x0, 0) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_554_1_power_InvokeMethod(x0 > 1, 351_0_power_Return(x0), x0, 1)
Cond_554_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 351_0_power_Return(x0)
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_543_1_power_InvokeMethod(x0 > 1, 351_0_power_Return(x0), x0, 1)
Cond_543_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 671_0_power_Return
550_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_550_1_power_InvokeMethod(x0 <= 1, 351_0_power_Return(x0), x0, 1)
Cond_550_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 862_0_power_Return(x0, x0 * x0)
554_1_power_InvokeMethod(671_0_power_Return, x0, x2) → Cond_554_1_power_InvokeMethod1(x0 > 1, 671_0_power_Return, x0, x2)
Cond_554_1_power_InvokeMethod1(TRUE, 671_0_power_Return, x0, x2) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1) → Cond_554_1_power_InvokeMethod3(x0 > 1, 351_0_power_Return(x0), x0, x1)
Cond_554_1_power_InvokeMethod3(TRUE, 351_0_power_Return(x0), x0, x1) → 351_0_power_Return(x0)
537_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_537_1_power_InvokeMethod(x0 <= 1, 351_0_power_Return(x0), x0, 1)
Cond_537_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 924_0_power_Return(x0 * x0)
543_1_power_InvokeMethod(671_0_power_Return, x0, x1) → 671_0_power_Return
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1) → 671_0_power_Return
537_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → Cond_537_1_power_InvokeMethod1(x2 <= 1, 862_0_power_Return(x0, x2), x0, x1)
Cond_537_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x2), x0, x1) → 924_0_power_Return(x2 * x2)
550_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → Cond_550_1_power_InvokeMethod1(x2 <= 1 && x0 <= 1, 862_0_power_Return(x0, x2), x0, x1)
Cond_550_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x2), x0, x1) → 862_0_power_Return(x0, x0 * x2)
537_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2) → Cond_537_1_power_InvokeMethod2(x0 <= 1, 924_0_power_Return(x0), x1, x2)
Cond_537_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x2) → 924_0_power_Return(x0 * x0)
550_1_power_InvokeMethod(924_0_power_Return(x0), x1, x3) → Cond_550_1_power_InvokeMethod2(x1 <= 1 && x0 <= 1, 924_0_power_Return(x0), x1, x3)
Cond_550_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x3) → 862_0_power_Return(x1, x1 * x0)

The integer pair graph contains the following rules and edges:
(0): 342_0_POWER_GT(x0[0], x1[0]) → COND_342_0_POWER_GT(x1[0] > 1 && !(x0[0] = 2), x0[0], x1[0])
(1): COND_342_0_POWER_GT(TRUE, x0[1], x1[1]) → 342_0_POWER_GT(x0[1], x1[1] / 2)
(2): 342_0_POWER_GT(x0[2], x1[2]) → COND_342_0_POWER_GT1(x1[2] > 0 && !(x1[2] = 1) && !(x0[2] = 2), x0[2], x1[2])
(3): COND_342_0_POWER_GT1(TRUE, x0[3], x1[3]) → 342_0_POWER_GT(x0[3], x1[3] - 1)
(4): 342_0_POWER_GT(x0[4], x1[4]) → COND_342_0_POWER_GT2(x1[4] > 1 && !(x0[4] = 2), x0[4], x1[4])
(5): COND_342_0_POWER_GT2(TRUE, x0[5], x1[5]) → 342_0_POWER_GT(x0[5], x1[5] / 2)
(6): 342_0_POWER_GT(x0[6], x1[6]) → COND_342_0_POWER_GT3(x1[6] > 0 && !(x1[6] = 1) && !(x0[6] = 2), x0[6], x1[6])
(7): COND_342_0_POWER_GT3(TRUE, x0[7], x1[7]) → 342_0_POWER_GT(x0[7], x1[7] - 1)

(0) -> (1), if (x1[0] > 1 && !(x0[0] = 2) ∧x0[0]* x0[1]x1[0]* x1[1])

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

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

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

(1) -> (6), if (x0[1]* x0[6]x1[1] / 2* x1[6])

(2) -> (3), if (x1[2] > 0 && !(x1[2] = 1) && !(x0[2] = 2) ∧x0[2]* x0[3]x1[2]* x1[3])

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

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

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

(3) -> (6), if (x0[3]* x0[6]x1[3] - 1* x1[6])

(4) -> (5), if (x1[4] > 1 && !(x0[4] = 2) ∧x0[4]* x0[5]x1[4]* x1[5])

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

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

(5) -> (4), if (x0[5]* x0[4]x1[5] / 2* x1[4])

(5) -> (6), if (x0[5]* x0[6]x1[5] / 2* x1[6])

(6) -> (7), if (x1[6] > 0 && !(x1[6] = 1) && !(x0[6] = 2) ∧x0[6]* x0[7]x1[6]* x1[7])

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

(7) -> (2), if (x0[7]* x0[2]x1[7] - 1* x1[2])

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

(7) -> (6), if (x0[7]* x0[6]x1[7] - 1* x1[6])

The set Q consists of the following terms:
342_0_power_GT(x0, 0)
Cond_554_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
Cond_543_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
550_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1)
Cond_550_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
554_1_power_InvokeMethod(671_0_power_Return, x0, x1)
Cond_554_1_power_InvokeMethod1(TRUE, 671_0_power_Return, x0, x1)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1)
Cond_554_1_power_InvokeMethod3(TRUE, 351_0_power_Return(x0), x0, x1)
537_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1)
Cond_537_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
543_1_power_InvokeMethod(671_0_power_Return, x0, x1)
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1)
537_1_power_InvokeMethod(862_0_power_Return(x0, x1), x0, x2)
Cond_537_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x1), x0, x2)
550_1_power_InvokeMethod(862_0_power_Return(x0, x1), x0, x2)
Cond_550_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x1), x0, x2)
537_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2)
Cond_537_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x2)
550_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2)
Cond_550_1_power_InvokeMethod2(TRUE, 924_0_power_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.IdpDefaultShapeHeuristic@2949784b Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT(&&(>(x1, 1), !(=(x0, 2))), x0, x1) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[0], x1[0]) → COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0]), COND_342_0_POWER_GT(TRUE, x0[1], x1[1]) → 342_0_POWER_GT(x0[1], /(x1[1], 2)) which results in the following constraint:

(1)    (&&(>(x1[0], 1), !(=(x0[0], 2)))=TRUEx0[0]=x0[1]x1[0]=x1[1]342_0_POWER_GT(x0[0], x1[0])≥NonInfC∧342_0_POWER_GT(x0[0], x1[0])≥COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])∧(UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥))

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

(2)    (>(x1[0], 1)=TRUE<(x0[0], 2)=TRUE342_0_POWER_GT(x0[0], x1[0])≥NonInfC∧342_0_POWER_GT(x0[0], x1[0])≥COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])∧(UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥))

(3)    (>(x1[0], 1)=TRUE>(x0[0], 2)=TRUE342_0_POWER_GT(x0[0], x1[0])≥NonInfC∧342_0_POWER_GT(x0[0], x1[0])≥COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])∧(UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥))

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

(4)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[(-1)bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(5)    (x1[0] + [-2] ≥ 0∧x0[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[(-1)bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(6)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[(-1)bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(7)    (x1[0] + [-2] ≥ 0∧x0[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[(-1)bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(8)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[(-1)bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(9)    (x1[0] + [-2] ≥ 0∧x0[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[(-1)bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(10)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(11)    (x1[0] ≥ 0∧x0[0] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(12)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

(13)    (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

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

(14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

For Pair COND_342_0_POWER_GT(TRUE, x0, x1) → 342_0_POWER_GT(x0, /(x1, 2)) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[0], x1[0]) → COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0]), COND_342_0_POWER_GT(TRUE, x0[1], x1[1]) → 342_0_POWER_GT(x0[1], /(x1[1], 2)) which results in the following constraint:

(15)    (&&(>(x1[0], 1), !(=(x0[0], 2)))=TRUEx0[0]=x0[1]x1[0]=x1[1]COND_342_0_POWER_GT(TRUE, x0[1], x1[1])≥NonInfC∧COND_342_0_POWER_GT(TRUE, x0[1], x1[1])≥342_0_POWER_GT(x0[1], /(x1[1], 2))∧(UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥))

We simplified constraint (15) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

(16)    (>(x1[0], 1)=TRUE<(x0[0], 2)=TRUECOND_342_0_POWER_GT(TRUE, x0[0], x1[0])≥NonInfC∧COND_342_0_POWER_GT(TRUE, x0[0], x1[0])≥342_0_POWER_GT(x0[0], /(x1[0], 2))∧(UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥))

(17)    (>(x1[0], 1)=TRUE>(x0[0], 2)=TRUECOND_342_0_POWER_GT(TRUE, x0[0], x1[0])≥NonInfC∧COND_342_0_POWER_GT(TRUE, x0[0], x1[0])≥342_0_POWER_GT(x0[0], /(x1[0], 2))∧(UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥))

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

(18)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[(-1)bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] + x1[0] + [-1]max{x1[0], [-1]x1[0]} ≥ 0)

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

(19)    (x1[0] + [-2] ≥ 0∧x0[0] + [-3] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[(-1)bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] + x1[0] + [-1]max{x1[0], [-1]x1[0]} ≥ 0)

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

(20)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[(-1)bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] + x1[0] + [-1]max{x1[0], [-1]x1[0]} ≥ 0)

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

(21)    (x1[0] + [-2] ≥ 0∧x0[0] + [-3] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[(-1)bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] + x1[0] + [-1]max{x1[0], [-1]x1[0]} ≥ 0)

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

(22)    (x1[0] + [-2] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[2]x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[(-1)bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(23)    (x1[0] + [-2] ≥ 0∧x0[0] + [-3] ≥ 0∧[2]x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[(-1)bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(24)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(25)    (x1[0] ≥ 0∧x0[0] + [-3] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(26)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

(27)    (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(28)    (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(29)    (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(30)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[4] + [2]x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

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

(31)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

For Pair 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT1(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), x0, x1) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[2], x1[2]) → COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2]), COND_342_0_POWER_GT1(TRUE, x0[3], x1[3]) → 342_0_POWER_GT(x0[3], -(x1[3], 1)) which results in the following constraint:

(32)    (&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2)))=TRUEx0[2]=x0[3]x1[2]=x1[3]342_0_POWER_GT(x0[2], x1[2])≥NonInfC∧342_0_POWER_GT(x0[2], x1[2])≥COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])∧(UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥))

We simplified constraint (32) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

(33)    (>(x1[2], 0)=TRUE<(x0[2], 2)=TRUE<(x1[2], 1)=TRUE342_0_POWER_GT(x0[2], x1[2])≥NonInfC∧342_0_POWER_GT(x0[2], x1[2])≥COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])∧(UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥))

(34)    (>(x1[2], 0)=TRUE<(x0[2], 2)=TRUE>(x1[2], 1)=TRUE342_0_POWER_GT(x0[2], x1[2])≥NonInfC∧342_0_POWER_GT(x0[2], x1[2])≥COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])∧(UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥))

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

(35)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(36)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(37)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(38)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(42)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(43)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(44)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(45)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(46)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

We solved constraint (43) using rule (IDP_SMT_SPLIT).We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(47)    (x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

We solved constraint (45) using rule (IDP_SMT_SPLIT).We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(48)    (x1[2] ≥ 0∧x0[2] + [-3] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(49)    ([1] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(50)    ([1] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

(51)    ([1] + x1[2] ≥ 0∧[1] + x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

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

(52)    ([1] + x1[2] ≥ 0∧x0[2] + [-3] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

We simplified constraint (52) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(53)    ([1] + x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

For Pair COND_342_0_POWER_GT1(TRUE, x0, x1) → 342_0_POWER_GT(x0, -(x1, 1)) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[2], x1[2]) → COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2]), COND_342_0_POWER_GT1(TRUE, x0[3], x1[3]) → 342_0_POWER_GT(x0[3], -(x1[3], 1)) which results in the following constraint:

(54)    (&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2)))=TRUEx0[2]=x0[3]x1[2]=x1[3]COND_342_0_POWER_GT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_342_0_POWER_GT1(TRUE, x0[3], x1[3])≥342_0_POWER_GT(x0[3], -(x1[3], 1))∧(UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥))

We simplified constraint (54) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

(55)    (>(x1[2], 0)=TRUE<(x0[2], 2)=TRUE<(x1[2], 1)=TRUECOND_342_0_POWER_GT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_342_0_POWER_GT1(TRUE, x0[2], x1[2])≥342_0_POWER_GT(x0[2], -(x1[2], 1))∧(UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥))

(56)    (>(x1[2], 0)=TRUE<(x0[2], 2)=TRUE>(x1[2], 1)=TRUECOND_342_0_POWER_GT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_342_0_POWER_GT1(TRUE, x0[2], x1[2])≥342_0_POWER_GT(x0[2], -(x1[2], 1))∧(UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥))

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

(57)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(58)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(59)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

We simplified constraint (58) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(60)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(62)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(64)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

We simplified constraint (59) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(65)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(66)    (x1[2] + [-1] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(67)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧[-1]x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(68)    (x1[2] + [-1] ≥ 0∧x0[2] + [-3] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

We solved constraint (65) using rule (IDP_SMT_SPLIT).We simplified constraint (66) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(69)    (x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

We solved constraint (67) using rule (IDP_SMT_SPLIT).We simplified constraint (68) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(70)    (x1[2] ≥ 0∧x0[2] + [-3] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[(-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(71)    ([1] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(72)    ([1] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

(73)    ([1] + x1[2] ≥ 0∧[1] + x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

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

(74)    ([1] + x1[2] ≥ 0∧x0[2] + [-3] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

We simplified constraint (74) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(75)    ([1] + x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

For Pair 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT2(&&(>(x1, 1), !(=(x0, 2))), x0, x1) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[4], x1[4]) → COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4]), COND_342_0_POWER_GT2(TRUE, x0[5], x1[5]) → 342_0_POWER_GT(x0[5], /(x1[5], 2)) which results in the following constraint:

(76)    (&&(>(x1[4], 1), !(=(x0[4], 2)))=TRUEx0[4]=x0[5]x1[4]=x1[5]342_0_POWER_GT(x0[4], x1[4])≥NonInfC∧342_0_POWER_GT(x0[4], x1[4])≥COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])∧(UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥))

We simplified constraint (76) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

(77)    (>(x1[4], 1)=TRUE<(x0[4], 2)=TRUE342_0_POWER_GT(x0[4], x1[4])≥NonInfC∧342_0_POWER_GT(x0[4], x1[4])≥COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])∧(UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥))

(78)    (>(x1[4], 1)=TRUE>(x0[4], 2)=TRUE342_0_POWER_GT(x0[4], x1[4])≥NonInfC∧342_0_POWER_GT(x0[4], x1[4])≥COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])∧(UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥))

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

(79)    (x1[4] + [-2] ≥ 0∧[1] + [-1]x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[(-1)bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

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

(80)    (x1[4] + [-2] ≥ 0∧x0[4] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[(-1)bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

We simplified constraint (79) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(81)    (x1[4] + [-2] ≥ 0∧[1] + [-1]x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[(-1)bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

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

(82)    (x1[4] + [-2] ≥ 0∧x0[4] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[(-1)bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

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

(83)    (x1[4] + [-2] ≥ 0∧[1] + [-1]x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[(-1)bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

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

(84)    (x1[4] + [-2] ≥ 0∧x0[4] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[(-1)bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

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

(85)    (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

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

(86)    (x1[4] ≥ 0∧x0[4] + [-3] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

We simplified constraint (85) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(87)    (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

(88)    (x1[4] ≥ 0∧[1] + x0[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

We simplified constraint (86) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(89)    (x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

For Pair COND_342_0_POWER_GT2(TRUE, x0, x1) → 342_0_POWER_GT(x0, /(x1, 2)) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[4], x1[4]) → COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4]), COND_342_0_POWER_GT2(TRUE, x0[5], x1[5]) → 342_0_POWER_GT(x0[5], /(x1[5], 2)) which results in the following constraint:

(90)    (&&(>(x1[4], 1), !(=(x0[4], 2)))=TRUEx0[4]=x0[5]x1[4]=x1[5]COND_342_0_POWER_GT2(TRUE, x0[5], x1[5])≥NonInfC∧COND_342_0_POWER_GT2(TRUE, x0[5], x1[5])≥342_0_POWER_GT(x0[5], /(x1[5], 2))∧(UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥))

We simplified constraint (90) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

(91)    (>(x1[4], 1)=TRUE<(x0[4], 2)=TRUECOND_342_0_POWER_GT2(TRUE, x0[4], x1[4])≥NonInfC∧COND_342_0_POWER_GT2(TRUE, x0[4], x1[4])≥342_0_POWER_GT(x0[4], /(x1[4], 2))∧(UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥))

(92)    (>(x1[4], 1)=TRUE>(x0[4], 2)=TRUECOND_342_0_POWER_GT2(TRUE, x0[4], x1[4])≥NonInfC∧COND_342_0_POWER_GT2(TRUE, x0[4], x1[4])≥342_0_POWER_GT(x0[4], /(x1[4], 2))∧(UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥))

We simplified constraint (91) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(93)    (x1[4] + [-2] ≥ 0∧[1] + [-1]x0[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[(-1)bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] + x1[4] + [-1]max{x1[4], [-1]x1[4]} ≥ 0)

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

(94)    (x1[4] + [-2] ≥ 0∧x0[4] + [-3] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[(-1)bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] + x1[4] + [-1]max{x1[4], [-1]x1[4]} ≥ 0)

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

(95)    (x1[4] + [-2] ≥ 0∧[1] + [-1]x0[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[(-1)bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] + x1[4] + [-1]max{x1[4], [-1]x1[4]} ≥ 0)

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

(96)    (x1[4] + [-2] ≥ 0∧x0[4] + [-3] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[(-1)bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] + x1[4] + [-1]max{x1[4], [-1]x1[4]} ≥ 0)

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

(97)    (x1[4] + [-2] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧[2]x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[(-1)bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

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

(98)    (x1[4] + [-2] ≥ 0∧x0[4] + [-3] ≥ 0∧[2]x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[(-1)bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

We simplified constraint (97) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(99)    (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧[4] + [2]x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

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

(100)    (x1[4] ≥ 0∧x0[4] + [-3] ≥ 0∧[4] + [2]x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

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

(101)    (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧[4] + [2]x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

(102)    (x1[4] ≥ 0∧[1] + x0[4] ≥ 0∧[4] + [2]x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

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

(103)    (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧x0[4] ≥ 0∧[2] + x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

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

(104)    (x1[4] ≥ 0∧[1] + x0[4] ≥ 0∧x0[4] ≥ 0∧[2] + x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

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

(105)    (x1[4] ≥ 0∧x0[4] ≥ 0∧[4] + [2]x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

We simplified constraint (105) using rule (IDP_POLY_GCD) which results in the following new constraint:

(106)    (x1[4] ≥ 0∧x0[4] ≥ 0∧[2] + x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

For Pair 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT3(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), x0, x1) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[6], x1[6]) → COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6]), COND_342_0_POWER_GT3(TRUE, x0[7], x1[7]) → 342_0_POWER_GT(x0[7], -(x1[7], 1)) which results in the following constraint:

(107)    (&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2)))=TRUEx0[6]=x0[7]x1[6]=x1[7]342_0_POWER_GT(x0[6], x1[6])≥NonInfC∧342_0_POWER_GT(x0[6], x1[6])≥COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])∧(UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥))

We simplified constraint (107) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

(108)    (>(x1[6], 0)=TRUE<(x0[6], 2)=TRUE<(x1[6], 1)=TRUE342_0_POWER_GT(x0[6], x1[6])≥NonInfC∧342_0_POWER_GT(x0[6], x1[6])≥COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])∧(UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥))

(109)    (>(x1[6], 0)=TRUE<(x0[6], 2)=TRUE>(x1[6], 1)=TRUE342_0_POWER_GT(x0[6], x1[6])≥NonInfC∧342_0_POWER_GT(x0[6], x1[6])≥COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])∧(UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥))

We simplified constraint (108) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(110)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (109) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(111)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (110) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(112)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (111) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(113)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (114) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(115)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (116) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(117)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (112) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(118)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (113) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(119)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (115) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(120)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (117) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(121)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We solved constraint (118) using rule (IDP_SMT_SPLIT).We simplified constraint (119) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(122)    (x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1] + x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We solved constraint (120) using rule (IDP_SMT_SPLIT).We simplified constraint (121) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(123)    (x1[6] ≥ 0∧x0[6] + [-3] ≥ 0∧[-1] + x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[(-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (122) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(124)    ([1] + x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (124) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(125)    ([1] + x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

(126)    ([1] + x1[6] ≥ 0∧[1] + x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (123) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(127)    ([1] + x1[6] ≥ 0∧x0[6] + [-3] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

We simplified constraint (127) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(128)    ([1] + x1[6] ≥ 0∧x0[6] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

For Pair COND_342_0_POWER_GT3(TRUE, x0, x1) → 342_0_POWER_GT(x0, -(x1, 1)) the following chains were created:
• We consider the chain 342_0_POWER_GT(x0[6], x1[6]) → COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6]), COND_342_0_POWER_GT3(TRUE, x0[7], x1[7]) → 342_0_POWER_GT(x0[7], -(x1[7], 1)) which results in the following constraint:

(129)    (&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2)))=TRUEx0[6]=x0[7]x1[6]=x1[7]COND_342_0_POWER_GT3(TRUE, x0[7], x1[7])≥NonInfC∧COND_342_0_POWER_GT3(TRUE, x0[7], x1[7])≥342_0_POWER_GT(x0[7], -(x1[7], 1))∧(UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥))

We simplified constraint (129) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

(130)    (>(x1[6], 0)=TRUE<(x0[6], 2)=TRUE<(x1[6], 1)=TRUECOND_342_0_POWER_GT3(TRUE, x0[6], x1[6])≥NonInfC∧COND_342_0_POWER_GT3(TRUE, x0[6], x1[6])≥342_0_POWER_GT(x0[6], -(x1[6], 1))∧(UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥))

(131)    (>(x1[6], 0)=TRUE<(x0[6], 2)=TRUE>(x1[6], 1)=TRUECOND_342_0_POWER_GT3(TRUE, x0[6], x1[6])≥NonInfC∧COND_342_0_POWER_GT3(TRUE, x0[6], x1[6])≥342_0_POWER_GT(x0[6], -(x1[6], 1))∧(UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥))

We simplified constraint (130) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(132)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (131) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(133)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (132) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(134)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (133) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(135)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (136) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(137)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (138) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(139)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (134) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(140)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (135) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(141)    (x1[6] + [-1] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (137) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(142)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧[-1]x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (139) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(143)    (x1[6] + [-1] ≥ 0∧x0[6] + [-3] ≥ 0∧x1[6] + [-2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We solved constraint (140) using rule (IDP_SMT_SPLIT).We simplified constraint (141) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(144)    (x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧[-1] + x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We solved constraint (142) using rule (IDP_SMT_SPLIT).We simplified constraint (143) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(145)    (x1[6] ≥ 0∧x0[6] + [-3] ≥ 0∧[-1] + x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[(-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (144) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(146)    ([1] + x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (146) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(147)    ([1] + x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

(148)    ([1] + x1[6] ≥ 0∧[1] + x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (145) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(149)    ([1] + x1[6] ≥ 0∧x0[6] + [-3] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

We simplified constraint (149) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(150)    ([1] + x1[6] ≥ 0∧x0[6] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT(&&(>(x1, 1), !(=(x0, 2))), x0, x1)
• (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)
• (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])), ≥)∧[bni_104 + (-1)Bound*bni_104] + [bni_104]x1[0] ≥ 0∧[(-1)bso_105] ≥ 0)

• COND_342_0_POWER_GT(TRUE, x0, x1) → 342_0_POWER_GT(x0, /(x1, 2))
• (x1[0] ≥ 0∧[1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)
• (x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)
• (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + x1[0] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[1], /(x1[1], 2))), ≥)∧[bni_106 + (-1)Bound*bni_106] + [bni_106]x1[0] ≥ 0∧[1 + (-1)bso_110] ≥ 0)

• 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT1(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), x0, x1)
• ([1] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)
• ([1] + x1[2] ≥ 0∧[1] + x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)
• ([1] + x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])), ≥)∧[bni_111 + (-1)Bound*bni_111] + [bni_111]x1[2] ≥ 0∧[(-1)bso_112] ≥ 0)

• COND_342_0_POWER_GT1(TRUE, x0, x1) → 342_0_POWER_GT(x0, -(x1, 1))
• ([1] + x1[2] ≥ 0∧[1] + [-1]x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)
• ([1] + x1[2] ≥ 0∧[1] + x0[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)
• ([1] + x1[2] ≥ 0∧x0[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[3], -(x1[3], 1))), ≥)∧[bni_113 + (-1)Bound*bni_113] + [bni_113]x1[2] ≥ 0∧[1 + (-1)bso_114] ≥ 0)

• 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT2(&&(>(x1, 1), !(=(x0, 2))), x0, x1)
• (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)
• (x1[4] ≥ 0∧[1] + x0[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)
• (x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])), ≥)∧[bni_115 + (-1)Bound*bni_115] + [bni_115]x1[4] ≥ 0∧[(-1)bso_116] ≥ 0)

• COND_342_0_POWER_GT2(TRUE, x0, x1) → 342_0_POWER_GT(x0, /(x1, 2))
• (x1[4] ≥ 0∧[1] + [-1]x0[4] ≥ 0∧x0[4] ≥ 0∧[2] + x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)
• (x1[4] ≥ 0∧[1] + x0[4] ≥ 0∧x0[4] ≥ 0∧[2] + x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)
• (x1[4] ≥ 0∧x0[4] ≥ 0∧[2] + x1[4] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[5], /(x1[5], 2))), ≥)∧[bni_117 + (-1)Bound*bni_117] + [bni_117]x1[4] ≥ 0∧[1 + (-1)bso_118] ≥ 0)

• 342_0_POWER_GT(x0, x1) → COND_342_0_POWER_GT3(&&(&&(>(x1, 0), !(=(x1, 1))), !(=(x0, 2))), x0, x1)
• ([1] + x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)
• ([1] + x1[6] ≥ 0∧[1] + x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)
• ([1] + x1[6] ≥ 0∧x0[6] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])), ≥)∧[bni_119 + (-1)Bound*bni_119] + [bni_119]x1[6] ≥ 0∧[(-1)bso_120] ≥ 0)

• COND_342_0_POWER_GT3(TRUE, x0, x1) → 342_0_POWER_GT(x0, -(x1, 1))
• ([1] + x1[6] ≥ 0∧[1] + [-1]x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)
• ([1] + x1[6] ≥ 0∧[1] + x0[6] ≥ 0∧x1[6] ≥ 0∧x0[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 0)
• ([1] + x1[6] ≥ 0∧x0[6] ≥ 0∧x1[6] ≥ 0 ⇒ (UIncreasing(342_0_POWER_GT(x0[7], -(x1[7], 1))), ≥)∧[bni_121 + (-1)Bound*bni_121] + [bni_121]x1[6] ≥ 0∧[1 + (-1)bso_122] ≥ 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(342_0_power_GT(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(0) = 0
POL(351_0_power_Return(x1)) = [-1] + [-1]x1
POL(554_1_power_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(1) = [1]
POL(Cond_554_1_power_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(>(x1, x2)) = [-1]
POL(543_1_power_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(Cond_543_1_power_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(671_0_power_Return) = [-1]
POL(550_1_power_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(Cond_550_1_power_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(<=(x1, x2)) = [-1]
POL(862_0_power_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(*(x1, x2)) = x1·x2
POL(Cond_554_1_power_InvokeMethod1(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(Cond_554_1_power_InvokeMethod3(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(537_1_power_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(Cond_537_1_power_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(924_0_power_Return(x1)) = [-1] + [-1]x1
POL(Cond_537_1_power_InvokeMethod1(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(Cond_550_1_power_InvokeMethod1(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(Cond_537_1_power_InvokeMethod2(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(Cond_550_1_power_InvokeMethod2(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(342_0_POWER_GT(x1, x2)) = [-1] + x2
POL(COND_342_0_POWER_GT(x1, x2, x3)) = [-1] + x3
POL(!(x1)) = [-1]
POL(=(x1, x2)) = [-1]
POL(2) = [2]
POL(COND_342_0_POWER_GT1(x1, x2, x3)) = [-1] + x3 + [-1]x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_342_0_POWER_GT2(x1, x2, x3)) = [-1] + x3 + [-1]x1
POL(COND_342_0_POWER_GT3(x1, x2, x3)) = [-1] + x3 + [-1]x1

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {342_0_POWER_GT_2/1}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

COND_342_0_POWER_GT(TRUE, x0[1], x1[1]) → 342_0_POWER_GT(x0[1], /(x1[1], 2))
COND_342_0_POWER_GT1(TRUE, x0[3], x1[3]) → 342_0_POWER_GT(x0[3], -(x1[3], 1))
COND_342_0_POWER_GT2(TRUE, x0[5], x1[5]) → 342_0_POWER_GT(x0[5], /(x1[5], 2))
COND_342_0_POWER_GT3(TRUE, x0[7], x1[7]) → 342_0_POWER_GT(x0[7], -(x1[7], 1))

The following pairs are in Pbound:

342_0_POWER_GT(x0[0], x1[0]) → COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])
COND_342_0_POWER_GT(TRUE, x0[1], x1[1]) → 342_0_POWER_GT(x0[1], /(x1[1], 2))
342_0_POWER_GT(x0[2], x1[2]) → COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])
COND_342_0_POWER_GT1(TRUE, x0[3], x1[3]) → 342_0_POWER_GT(x0[3], -(x1[3], 1))
342_0_POWER_GT(x0[4], x1[4]) → COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])
COND_342_0_POWER_GT2(TRUE, x0[5], x1[5]) → 342_0_POWER_GT(x0[5], /(x1[5], 2))
342_0_POWER_GT(x0[6], x1[6]) → COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])
COND_342_0_POWER_GT3(TRUE, x0[7], x1[7]) → 342_0_POWER_GT(x0[7], -(x1[7], 1))

The following pairs are in P:

342_0_POWER_GT(x0[0], x1[0]) → COND_342_0_POWER_GT(&&(>(x1[0], 1), !(=(x0[0], 2))), x0[0], x1[0])
342_0_POWER_GT(x0[2], x1[2]) → COND_342_0_POWER_GT1(&&(&&(>(x1[2], 0), !(=(x1[2], 1))), !(=(x0[2], 2))), x0[2], x1[2])
342_0_POWER_GT(x0[4], x1[4]) → COND_342_0_POWER_GT2(&&(>(x1[4], 1), !(=(x0[4], 2))), x0[4], x1[4])
342_0_POWER_GT(x0[6], x1[6]) → COND_342_0_POWER_GT3(&&(&&(>(x1[6], 0), !(=(x1[6], 1))), !(=(x0[6], 2))), x0[6], x1[6])

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

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

### (8) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
342_0_power_GT(x0, 0) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_554_1_power_InvokeMethod(x0 > 1, 351_0_power_Return(x0), x0, 1)
Cond_554_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 351_0_power_Return(x0)
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_543_1_power_InvokeMethod(x0 > 1, 351_0_power_Return(x0), x0, 1)
Cond_543_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 671_0_power_Return
550_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_550_1_power_InvokeMethod(x0 <= 1, 351_0_power_Return(x0), x0, 1)
Cond_550_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 862_0_power_Return(x0, x0 * x0)
554_1_power_InvokeMethod(671_0_power_Return, x0, x2) → Cond_554_1_power_InvokeMethod1(x0 > 1, 671_0_power_Return, x0, x2)
Cond_554_1_power_InvokeMethod1(TRUE, 671_0_power_Return, x0, x2) → 351_0_power_Return(x0)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1) → Cond_554_1_power_InvokeMethod3(x0 > 1, 351_0_power_Return(x0), x0, x1)
Cond_554_1_power_InvokeMethod3(TRUE, 351_0_power_Return(x0), x0, x1) → 351_0_power_Return(x0)
537_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1) → Cond_537_1_power_InvokeMethod(x0 <= 1, 351_0_power_Return(x0), x0, 1)
Cond_537_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1) → 924_0_power_Return(x0 * x0)
543_1_power_InvokeMethod(671_0_power_Return, x0, x1) → 671_0_power_Return
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1) → 671_0_power_Return
537_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → Cond_537_1_power_InvokeMethod1(x2 <= 1, 862_0_power_Return(x0, x2), x0, x1)
Cond_537_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x2), x0, x1) → 924_0_power_Return(x2 * x2)
550_1_power_InvokeMethod(862_0_power_Return(x0, x2), x0, x1) → Cond_550_1_power_InvokeMethod1(x2 <= 1 && x0 <= 1, 862_0_power_Return(x0, x2), x0, x1)
Cond_550_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x2), x0, x1) → 862_0_power_Return(x0, x0 * x2)
537_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2) → Cond_537_1_power_InvokeMethod2(x0 <= 1, 924_0_power_Return(x0), x1, x2)
Cond_537_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x2) → 924_0_power_Return(x0 * x0)
550_1_power_InvokeMethod(924_0_power_Return(x0), x1, x3) → Cond_550_1_power_InvokeMethod2(x1 <= 1 && x0 <= 1, 924_0_power_Return(x0), x1, x3)
Cond_550_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x3) → 862_0_power_Return(x1, x1 * x0)

The integer pair graph contains the following rules and edges:
(0): 342_0_POWER_GT(x0[0], x1[0]) → COND_342_0_POWER_GT(x1[0] > 1 && !(x0[0] = 2), x0[0], x1[0])
(2): 342_0_POWER_GT(x0[2], x1[2]) → COND_342_0_POWER_GT1(x1[2] > 0 && !(x1[2] = 1) && !(x0[2] = 2), x0[2], x1[2])
(4): 342_0_POWER_GT(x0[4], x1[4]) → COND_342_0_POWER_GT2(x1[4] > 1 && !(x0[4] = 2), x0[4], x1[4])
(6): 342_0_POWER_GT(x0[6], x1[6]) → COND_342_0_POWER_GT3(x1[6] > 0 && !(x1[6] = 1) && !(x0[6] = 2), x0[6], x1[6])

The set Q consists of the following terms:
342_0_power_GT(x0, 0)
Cond_554_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
Cond_543_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
550_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1)
Cond_550_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
554_1_power_InvokeMethod(671_0_power_Return, x0, x1)
Cond_554_1_power_InvokeMethod1(TRUE, 671_0_power_Return, x0, x1)
554_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1)
Cond_554_1_power_InvokeMethod3(TRUE, 351_0_power_Return(x0), x0, x1)
537_1_power_InvokeMethod(351_0_power_Return(x0), x0, 1)
Cond_537_1_power_InvokeMethod(TRUE, 351_0_power_Return(x0), x0, 1)
543_1_power_InvokeMethod(671_0_power_Return, x0, x1)
543_1_power_InvokeMethod(351_0_power_Return(x0), x0, x1)
537_1_power_InvokeMethod(862_0_power_Return(x0, x1), x0, x2)
Cond_537_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x1), x0, x2)
550_1_power_InvokeMethod(862_0_power_Return(x0, x1), x0, x2)
Cond_550_1_power_InvokeMethod1(TRUE, 862_0_power_Return(x0, x1), x0, x2)
537_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2)
Cond_537_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x2)
550_1_power_InvokeMethod(924_0_power_Return(x0), x1, x2)
Cond_550_1_power_InvokeMethod2(TRUE, 924_0_power_Return(x0), x1, x2)

### (9) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.