Termination proof of /tmp/tmpu9rugJ/CountUpRound.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 110 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 110 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 290 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 TRUE

    ↳12 IDP

        ↳13 IDependencyGraphProof (⇔, 0 ms)

        ↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: CountUpRound

public class CountUpRound{
  public static int round (int x) {

    if (x % 2 == 0) return x;
    else return x+1;
  }


  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();



    while (x > y) {

      y = round(y+1);

    }


  }

}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
CountUpRound.main([Ljava/lang/String;)V: Graph of 193 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: CountUpRound.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 27 rules for P and 0 rules for R.

P rules:
313_0_main_Load(EOS(STATIC_313), i18, i47, i18) → 319_0_main_LE(EOS(STATIC_319), i18, i47, i18, i47)
319_0_main_LE(EOS(STATIC_319), i18, i47, i18, i47) → 333_0_main_LE(EOS(STATIC_333), i18, i47, i18, i47)
333_0_main_LE(EOS(STATIC_333), i18, i47, i18, i47) → 344_0_main_Load(EOS(STATIC_344), i18, i47) | >(i18, i47)
344_0_main_Load(EOS(STATIC_344), i18, i47) → 352_0_main_ConstantStackPush(EOS(STATIC_352), i18, i47)
352_0_main_ConstantStackPush(EOS(STATIC_352), i18, i47) → 362_0_main_IntArithmetic(EOS(STATIC_362), i18, i47, 1)
362_0_main_IntArithmetic(EOS(STATIC_362), i18, i47, matching1) → 371_0_main_InvokeMethod(EOS(STATIC_371), i18, +(i47, 1)) | &&(>=(i47, 0), =(matching1, 1))
371_0_main_InvokeMethod(EOS(STATIC_371), i18, i53) → 379_0_round_Load(EOS(STATIC_379), i18, i53, i53)
379_0_round_Load(EOS(STATIC_379), i18, i53, i53) → 398_0_round_ConstantStackPush(EOS(STATIC_398), i18, i53, i53, i53)
398_0_round_ConstantStackPush(EOS(STATIC_398), i18, i53, i53, i53) → 408_0_round_IntArithmetic(EOS(STATIC_408), i18, i53, i53, i53, 2)
408_0_round_IntArithmetic(EOS(STATIC_408), i18, i53, i53, i53, matching1) → 418_0_round_NE(EOS(STATIC_418), i18, i53, i53, %(i53, 2)) | =(matching1, 2)
418_0_round_NE(EOS(STATIC_418), i18, i53, i53, matching1) → 426_0_round_NE(EOS(STATIC_426), i18, i53, i53, 1) | =(matching1, 1)
418_0_round_NE(EOS(STATIC_418), i18, i53, i53, matching1) → 427_0_round_NE(EOS(STATIC_427), i18, i53, i53, 0) | =(matching1, 0)
426_0_round_NE(EOS(STATIC_426), i18, i53, i53, matching1) → 434_0_round_Load(EOS(STATIC_434), i18, i53, i53) | &&(>(1, 0), =(matching1, 1))
434_0_round_Load(EOS(STATIC_434), i18, i53, i53) → 442_0_round_ConstantStackPush(EOS(STATIC_442), i18, i53, i53)
442_0_round_ConstantStackPush(EOS(STATIC_442), i18, i53, i53) → 450_0_round_IntArithmetic(EOS(STATIC_450), i18, i53, i53, 1)
450_0_round_IntArithmetic(EOS(STATIC_450), i18, i53, i53, matching1) → 461_0_round_Return(EOS(STATIC_461), i18, i53, +(i53, 1)) | &&(>(i53, 0), =(matching1, 1))
461_0_round_Return(EOS(STATIC_461), i18, i53, i59) → 470_0_main_Store(EOS(STATIC_470), i18, i59)
470_0_main_Store(EOS(STATIC_470), i18, i59) → 482_0_main_JMP(EOS(STATIC_482), i18, i59)
482_0_main_JMP(EOS(STATIC_482), i18, i59) → 486_0_main_Load(EOS(STATIC_486), i18, i59)
486_0_main_Load(EOS(STATIC_486), i18, i59) → 306_0_main_Load(EOS(STATIC_306), i18, i59)
306_0_main_Load(EOS(STATIC_306), i18, i47) → 313_0_main_Load(EOS(STATIC_313), i18, i47, i18)
427_0_round_NE(EOS(STATIC_427), i18, i53, i53, matching1) → 436_0_round_Load(EOS(STATIC_436), i18, i53, i53) | =(matching1, 0)
436_0_round_Load(EOS(STATIC_436), i18, i53, i53) → 445_0_round_Return(EOS(STATIC_445), i18, i53, i53, i53)
445_0_round_Return(EOS(STATIC_445), i18, i53, i53, i53) → 453_0_main_Store(EOS(STATIC_453), i18, i53)
453_0_main_Store(EOS(STATIC_453), i18, i53) → 463_0_main_JMP(EOS(STATIC_463), i18, i53)
463_0_main_JMP(EOS(STATIC_463), i18, i53) → 477_0_main_Load(EOS(STATIC_477), i18, i53)
477_0_main_Load(EOS(STATIC_477), i18, i53) → 306_0_main_Load(EOS(STATIC_306), i18, i53)
R rules:

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

P rules:
313_0_main_Load(EOS(STATIC_313), x0, x1, x0) → 313_0_main_Load(EOS(STATIC_313), x0, +(x1, 2), x0) | &&(&&(>(+(x1, 1), 0), <(x1, x0)), =(1, %(+(x1, 1), 2)))
313_0_main_Load(EOS(STATIC_313), x0, x1, x0) → 313_0_main_Load(EOS(STATIC_313), x0, +(x1, 1), x0) | &&(&&(>(+(x1, 1), 0), <(x1, x0)), =(0, %(+(x1, 1), 2)))
R rules:

Filtered ground terms:

313_0_main_Load(x1, x2, x3, x4) → 313_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_313_0_main_Load1(x1, x2, x3, x4, x5) → Cond_313_0_main_Load1(x1, x3, x4, x5)
Cond_313_0_main_Load(x1, x2, x3, x4, x5) → Cond_313_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

313_0_main_Load(x1, x2, x3) → 313_0_main_Load(x2, x3)
Cond_313_0_main_Load(x1, x2, x3, x4) → Cond_313_0_main_Load(x1, x3, x4)
Cond_313_0_main_Load1(x1, x2, x3, x4) → Cond_313_0_main_Load1(x1, x3, x4)

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

P rules:
313_0_main_Load(x1, x0) → 313_0_main_Load(+(x1, 2), x0) | &&(&&(>(x1, -1), <(x1, x0)), =(1, %(+(x1, 1), 2)))
313_0_main_Load(x1, x0) → 313_0_main_Load(+(x1, 1), x0) | &&(&&(>(x1, -1), <(x1, x0)), =(0, %(+(x1, 1), 2)))
R rules:

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

P rules:
313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), =(1, %(+(x1, 1), 2))), x1, x0)
COND_313_0_MAIN_LOAD(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 2), x0)
313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD1(&&(&&(>(x1, -1), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0)
COND_313_0_MAIN_LOAD1(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 1), x0)
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0] && 1 = x1[0] + 1 % 2, x1[0], x0[0])
(1): COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(x1[1] + 2, x0[1])
(2): 313_0_MAIN_LOAD(x1[2], x0[2]) → COND_313_0_MAIN_LOAD1(x1[2] > -1 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2, x1[2], x0[2])
(3): COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 313_0_MAIN_LOAD(x1[3] + 1, x0[3])

(0) -> (1), if (x1[0] > -1 && x1[0] < x0[0] && 1 = x1[0] + 1 % 2 ∧x1[0] →^* x1[1]∧x0[0] →^* x0[1])

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

(1) -> (2), if (x1[1] + 2 →^* x1[2]∧x0[1] →^* x0[2])

(2) -> (3), if (x1[2] > -1 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2 ∧x1[2] →^* x1[3]∧x0[2] →^* x0[3])

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

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

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@7a0e046e Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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

For Pair 313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), =(1, %(+(x1, 1), 2))), x1, x0) the following chains were created:

We consider the chain 313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0]), COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(+(x1[1], 2), x0[1]) which results in the following constraint:
(1)    (&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2)))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 313_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧313_0_MAIN_LOAD(x1[0], x0[0])≥COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])∧(U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], -1)=TRUE∧<(x1[0], x0[0])=TRUE∧>=(1, %(+(x1[0], 1), 2))=TRUE∧<=(1, %(+(x1[0], 1), 2))=TRUE ⇒ 313_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧313_0_MAIN_LOAD(x1[0], x0[0])≥COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])∧(U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [(-1)bni_12]x1[0] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [(-1)bni_12]x1[0] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x0[0] + [(-1)bni_12]x1[0] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥)∧[(4)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] + [(2)bni_12]x0[0] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (6) using rule (IDP_POLY_GCD) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥)∧[(4)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] + [(2)bni_12]x0[0] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

For Pair COND_313_0_MAIN_LOAD(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 2), x0) the following chains were created:

We consider the chain COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(+(x1[1], 2), x0[1]) which results in the following constraint:
(8)    (COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥313_0_MAIN_LOAD(+(x1[1], 2), x0[1])∧(U^Increasing(313_0_MAIN_LOAD(+(x1[1], 2), x0[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[1], 2), x0[1])), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[1], 2), x0[1])), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[1], 2), x0[1])), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[1], 2), x0[1])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair 313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD1(&&(&&(>(x1, -1), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0) the following chains were created:

We consider the chain 313_0_MAIN_LOAD(x1[2], x0[2]) → COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2]), COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 313_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:
(13)    (&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2)))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 313_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧313_0_MAIN_LOAD(x1[2], x0[2])≥COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])∧(U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥))

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14)    (>(x1[2], -1)=TRUE∧<(x1[2], x0[2])=TRUE∧>=(0, %(+(x1[2], 1), 2))=TRUE∧<=(0, %(+(x1[2], 1), 2))=TRUE ⇒ 313_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧313_0_MAIN_LOAD(x1[2], x0[2])≥COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])∧(U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[2] + [(-1)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[2] + [(-1)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x0[2] + [(-1)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(4)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(2)bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (18) using rule (IDP_POLY_GCD) which results in the following new constraint:
(19)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(4)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(2)bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_313_0_MAIN_LOAD1(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 1), x0) the following chains were created:

We consider the chain COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 313_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:
(20)    (COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥NonInfC∧COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥313_0_MAIN_LOAD(+(x1[3], 1), x0[3])∧(U^Increasing(313_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    ((U^Increasing(313_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), =(1, %(+(x1, 1), 2))), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])), ≥)∧[(4)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] + [(2)bni_12]x0[0] ≥ 0∧[1 + (-1)bso_13] ≥ 0)
COND_313_0_MAIN_LOAD(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 2), x0)
- ((U^Increasing(313_0_MAIN_LOAD(+(x1[1], 2), x0[1])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)
313_0_MAIN_LOAD(x1, x0) → COND_313_0_MAIN_LOAD1(&&(&&(>(x1, -1), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0)
- (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(4)bni_16 + (-1)Bound*bni_16] + [bni_16]x1[2] + [(2)bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)
COND_313_0_MAIN_LOAD1(TRUE, x1, x0) → 313_0_MAIN_LOAD(+(x1, 1), x0)
- ((U^Increasing(313_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(313_0_MAIN_LOAD(x₁, x₂)) = [2] + [2]x₂ + [-1]x₁
POL(COND_313_0_MAIN_LOAD(x₁, x₂, x₃)) = [1] + [2]x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<(x₁, x₂)) = [-1]
POL(=(x₁, x₂)) = [-1]
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(2) = [2]
POL(COND_313_0_MAIN_LOAD1(x₁, x₂, x₃)) = [2] + [2]x₃ + [-1]x₂
POL(0) = 0

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(%(x₁, 2)^-1 @ {}) = min{x₂, [-1]x₂}
POL(%(x₁, 2)¹ @ {}) = max{x₂, [-1]x₂}

The following pairs are in P_>:

313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])
COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(+(x1[1], 2), x0[1])
COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 313_0_MAIN_LOAD(+(x1[3], 1), x0[3])

The following pairs are in P_bound:

313_0_MAIN_LOAD(x1[0], x0[0]) → COND_313_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), =(1, %(+(x1[0], 1), 2))), x1[0], x0[0])
313_0_MAIN_LOAD(x1[2], x0[2]) → COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])

The following pairs are in P_≥:

313_0_MAIN_LOAD(x1[2], x0[2]) → COND_313_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])

There are no usable rules.

(8) Complex Obligation (AND)

(9) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 313_0_MAIN_LOAD(x1[2], x0[2]) → COND_313_0_MAIN_LOAD1(x1[2] > -1 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2, x1[2], x0[2])

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_313_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 313_0_MAIN_LOAD(x1[1] + 2, x0[1])
(3): COND_313_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 313_0_MAIN_LOAD(x1[3] + 1, x0[3])

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE