Termination proof of /tmp/tmpbrFzQy/PastaC2.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 (⇒, 100 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 430 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 IDP

↳11 IDPNonInfProof (⇒, 70 ms)

↳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: PastaC2

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaC2 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        while (x >= 0) {
            x = x+1;
            int y = 1;
            while (x >= y) {
                y++;
            }
            x = x-2;
        }
    }
}


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:
PastaC2.main([Ljava/lang/String;)V: Graph of 128 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: PastaC2.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 28 rules for P and 0 rules for R.

P rules:
394_0_main_LT(EOS(STATIC_394), i45, i45) → 398_0_main_LT(EOS(STATIC_398), i45, i45)
398_0_main_LT(EOS(STATIC_398), i45, i45) → 402_0_main_Load(EOS(STATIC_402), i45) | >=(i45, 0)
402_0_main_Load(EOS(STATIC_402), i45) → 406_0_main_ConstantStackPush(EOS(STATIC_406), i45)
406_0_main_ConstantStackPush(EOS(STATIC_406), i45) → 408_0_main_IntArithmetic(EOS(STATIC_408), i45, 1)
408_0_main_IntArithmetic(EOS(STATIC_408), i45, matching1) → 410_0_main_Store(EOS(STATIC_410), +(i45, 1)) | &&(>=(i45, 0), =(matching1, 1))
410_0_main_Store(EOS(STATIC_410), i46) → 412_0_main_ConstantStackPush(EOS(STATIC_412), i46)
412_0_main_ConstantStackPush(EOS(STATIC_412), i46) → 414_0_main_Store(EOS(STATIC_414), i46, 1)
414_0_main_Store(EOS(STATIC_414), i46, matching1) → 416_0_main_Load(EOS(STATIC_416), i46, 1) | =(matching1, 1)
416_0_main_Load(EOS(STATIC_416), i46, matching1) → 431_0_main_Load(EOS(STATIC_431), i46, 1) | =(matching1, 1)
431_0_main_Load(EOS(STATIC_431), i46, i47) → 459_0_main_Load(EOS(STATIC_459), i46, i47)
459_0_main_Load(EOS(STATIC_459), i46, i51) → 487_0_main_Load(EOS(STATIC_487), i46, i51)
487_0_main_Load(EOS(STATIC_487), i46, i54) → 515_0_main_Load(EOS(STATIC_515), i46, i54)
515_0_main_Load(EOS(STATIC_515), i46, i58) → 518_0_main_Load(EOS(STATIC_518), i46, i58, i46)
518_0_main_Load(EOS(STATIC_518), i46, i58, i46) → 521_0_main_LT(EOS(STATIC_521), i46, i58, i46, i58)
521_0_main_LT(EOS(STATIC_521), i46, i58, i46, i58) → 523_0_main_LT(EOS(STATIC_523), i46, i58, i46, i58)
521_0_main_LT(EOS(STATIC_521), i46, i58, i46, i58) → 524_0_main_LT(EOS(STATIC_524), i46, i58, i46, i58)
523_0_main_LT(EOS(STATIC_523), i46, i58, i46, i58) → 526_0_main_Load(EOS(STATIC_526), i46) | <(i46, i58)
526_0_main_Load(EOS(STATIC_526), i46) → 530_0_main_ConstantStackPush(EOS(STATIC_530), i46)
530_0_main_ConstantStackPush(EOS(STATIC_530), i46) → 535_0_main_IntArithmetic(EOS(STATIC_535), i46, 2)
535_0_main_IntArithmetic(EOS(STATIC_535), i46, matching1) → 541_0_main_Store(EOS(STATIC_541), -(i46, 2)) | &&(>(i46, 0), =(matching1, 2))
541_0_main_Store(EOS(STATIC_541), i62) → 543_0_main_JMP(EOS(STATIC_543), i62)
543_0_main_JMP(EOS(STATIC_543), i62) → 546_0_main_Load(EOS(STATIC_546), i62)
546_0_main_Load(EOS(STATIC_546), i62) → 391_0_main_Load(EOS(STATIC_391), i62)
391_0_main_Load(EOS(STATIC_391), i42) → 394_0_main_LT(EOS(STATIC_394), i42, i42)
524_0_main_LT(EOS(STATIC_524), i46, i58, i46, i58) → 528_0_main_Inc(EOS(STATIC_528), i46, i58) | >=(i46, i58)
528_0_main_Inc(EOS(STATIC_528), i46, i58) → 532_0_main_JMP(EOS(STATIC_532), i46, +(i58, 1)) | >(i58, 0)
532_0_main_JMP(EOS(STATIC_532), i46, i60) → 539_0_main_Load(EOS(STATIC_539), i46, i60)
539_0_main_Load(EOS(STATIC_539), i46, i60) → 515_0_main_Load(EOS(STATIC_515), i46, i60)
R rules:

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

P rules:
521_0_main_LT(EOS(STATIC_521), x0, x1, x0, x1) → 521_0_main_LT(EOS(STATIC_521), +(-(x0, 2), 1), 1, +(-(x0, 2), 1), 1) | &&(>(x1, x0), >(+(x0, 1), 2))
521_0_main_LT(EOS(STATIC_521), x0, x1, x0, x1) → 521_0_main_LT(EOS(STATIC_521), x0, +(x1, 1), x0, +(x1, 1)) | &&(>(x1, 0), <=(x1, x0))
R rules:

Filtered ground terms:

521_0_main_LT(x1, x2, x3, x4, x5) → 521_0_main_LT(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_521_0_main_LT1(x1, x2, x3, x4, x5, x6) → Cond_521_0_main_LT1(x1, x3, x4, x5, x6)
Cond_521_0_main_LT(x1, x2, x3, x4, x5, x6) → Cond_521_0_main_LT(x1, x3, x4, x5, x6)

Filtered duplicate args:

521_0_main_LT(x1, x2, x3, x4) → 521_0_main_LT(x3, x4)
Cond_521_0_main_LT(x1, x2, x3, x4, x5) → Cond_521_0_main_LT(x1, x4, x5)
Cond_521_0_main_LT1(x1, x2, x3, x4, x5) → Cond_521_0_main_LT1(x1, x4, x5)

Filtered unneeded arguments:

Cond_521_0_main_LT(x1, x2, x3) → Cond_521_0_main_LT(x1, x2)

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

P rules:
521_0_main_LT(x0, x1) → 521_0_main_LT(+(-(x0, 2), 1), 1) | &&(>(x1, x0), >(x0, 1))
521_0_main_LT(x0, x1) → 521_0_main_LT(x0, +(x1, 1)) | &&(>(x1, 0), <=(x1, x0))
R rules:

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

P rules:
521_0_MAIN_LT(x0, x1) → COND_521_0_MAIN_LT(&&(>(x1, x0), >(x0, 1)), x0, x1)
COND_521_0_MAIN_LT(TRUE, x0, x1) → 521_0_MAIN_LT(+(-(x0, 2), 1), 1)
521_0_MAIN_LT(x0, x1) → COND_521_0_MAIN_LT1(&&(>(x1, 0), <=(x1, x0)), x0, x1)
COND_521_0_MAIN_LT1(TRUE, x0, x1) → 521_0_MAIN_LT(x0, +(x1, 1))
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): 521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(x1[0] > x0[0] && x0[0] > 1, x0[0], x1[0])
(1): COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(x0[1] - 2 + 1, 1)
(2): 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])
(3): COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], x1[3] + 1)

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

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

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

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

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5c66d27c 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 521_0_MAIN_LT(x0, x1) → COND_521_0_MAIN_LT(&&(>(x1, x0), >(x0, 1)), x0, x1) the following chains were created:

We consider the chain 521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(+(-(x0[1], 2), 1), 1) which results in the following constraint:
(1)    (&&(>(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 521_0_MAIN_LT(x0[0], x1[0])≥NonInfC∧521_0_MAIN_LT(x0[0], x1[0])≥COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])∧(U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ 521_0_MAIN_LT(x0[0], x1[0])≥NonInfC∧521_0_MAIN_LT(x0[0], x1[0])≥COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])∧(U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

For Pair COND_521_0_MAIN_LT(TRUE, x0, x1) → 521_0_MAIN_LT(+(-(x0, 2), 1), 1) the following chains were created:

We consider the chain 521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(+(-(x0[1], 2), 1), 1), 521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]) which results in the following constraint:
(8)    (&&(>(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧+(-(x0[1], 2), 1)=x0[0]1∧1=x1[0]1 ⇒ COND_521_0_MAIN_LT(TRUE, x0[1], x1[1])≥NonInfC∧COND_521_0_MAIN_LT(TRUE, x0[1], x1[1])≥521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)∧(U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ COND_521_0_MAIN_LT(TRUE, x0[0], x1[0])≥NonInfC∧COND_521_0_MAIN_LT(TRUE, x0[0], x1[0])≥521_0_MAIN_LT(+(-(x0[0], 2), 1), 1)∧(U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(3)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)
We consider the chain 521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(+(-(x0[1], 2), 1), 1), 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:
(15)    (&&(>(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧+(-(x0[1], 2), 1)=x0[2]∧1=x1[2] ⇒ COND_521_0_MAIN_LT(TRUE, x0[1], x1[1])≥NonInfC∧COND_521_0_MAIN_LT(TRUE, x0[1], x1[1])≥521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)∧(U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))

We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(16)    (>(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ COND_521_0_MAIN_LT(TRUE, x0[0], x1[0])≥NonInfC∧COND_521_0_MAIN_LT(TRUE, x0[0], x1[0])≥521_0_MAIN_LT(+(-(x0[0], 2), 1), 1)∧(U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(3)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair 521_0_MAIN_LT(x0, x1) → COND_521_0_MAIN_LT1(&&(>(x1, 0), <=(x1, x0)), x0, x1) the following chains were created:

We consider the chain 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1)) which results in the following constraint:
(22)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 521_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧521_0_MAIN_LT(x0[2], x1[2])≥COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))

We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ 521_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧521_0_MAIN_LT(x0[2], x1[2])≥COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_521_0_MAIN_LT1(TRUE, x0, x1) → 521_0_MAIN_LT(x0, +(x1, 1)) the following chains were created:

We consider the chain 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1)), 521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]) which results in the following constraint:
(29)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[0]∧+(x1[3], 1)=x1[0] ⇒ COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3])≥521_0_MAIN_LT(x0[3], +(x1[3], 1))∧(U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ COND_521_0_MAIN_LT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_521_0_MAIN_LT1(TRUE, x0[2], x1[2])≥521_0_MAIN_LT(x0[2], +(x1[2], 1))∧(U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)
We consider the chain 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1)), 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:
(36)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[2]1∧+(x1[3], 1)=x1[2]1 ⇒ COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3])≥521_0_MAIN_LT(x0[3], +(x1[3], 1))∧(U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))

We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(37)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ COND_521_0_MAIN_LT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_521_0_MAIN_LT1(TRUE, x0[2], x1[2])≥521_0_MAIN_LT(x0[2], +(x1[2], 1))∧(U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

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

521_0_MAIN_LT(x0, x1) → COND_521_0_MAIN_LT(&&(>(x1, x0), >(x0, 1)), x0, x1)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
COND_521_0_MAIN_LT(TRUE, x0, x1) → 521_0_MAIN_LT(+(-(x0, 2), 1), 1)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(3)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(3)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)
521_0_MAIN_LT(x0, x1) → COND_521_0_MAIN_LT1(&&(>(x1, 0), <=(x1, x0)), x0, x1)
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)
COND_521_0_MAIN_LT1(TRUE, x0, x1) → 521_0_MAIN_LT(x0, +(x1, 1))
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 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(521_0_MAIN_LT(x₁, x₂)) = [2] + x₁
POL(COND_521_0_MAIN_LT(x₁, x₂, x₃)) = [1] + x₂
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(2) = [2]
POL(COND_521_0_MAIN_LT1(x₁, x₂, x₃)) = [2] + x₂ + [-1]x₁
POL(0) = 0
POL(<=(x₁, x₂)) = [-1]

The following pairs are in P_>:

521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])

The following pairs are in P_bound:

521_0_MAIN_LT(x0[0], x1[0]) → COND_521_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])
COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)
521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])
COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1))

The following pairs are in P_≥:

COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(+(-(x0[1], 2), 1), 1)
521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])
COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1))

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

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

(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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_521_0_MAIN_LT(TRUE, x0[1], x1[1]) → 521_0_MAIN_LT(x0[1] - 2 + 1, 1)
(2): 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])
(3): COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], x1[3] + 1)

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

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

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

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

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

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], x1[3] + 1)
(2): 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])

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

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

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@5c66d27c Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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

For Pair COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1)) the following chains were created:

We consider the chain 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1)), 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:
(1)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[2]1∧+(x1[3], 1)=x1[2]1 ⇒ COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3])≥521_0_MAIN_LT(x0[3], +(x1[3], 1))∧(U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ COND_521_0_MAIN_LT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_521_0_MAIN_LT1(TRUE, x0[2], x1[2])≥521_0_MAIN_LT(x0[2], +(x1[2], 1))∧(U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

For Pair 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) the following chains were created:

We consider the chain 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1)) which results in the following constraint:
(8)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 521_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧521_0_MAIN_LT(x0[2], x1[2])≥COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))

We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ 521_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧521_0_MAIN_LT(x0[2], x1[2])≥COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1))
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(521_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)
521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

POL(TRUE) = [3]
POL(FALSE) = 0
POL(COND_521_0_MAIN_LT1(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(521_0_MAIN_LT(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(<=(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1))

The following pairs are in P_bound:

COND_521_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 521_0_MAIN_LT(x0[3], +(x1[3], 1))
521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])

The following pairs are in P_≥:

521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])

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

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

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 521_0_MAIN_LT(x0[2], x1[2]) → COND_521_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(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) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) IDPNonInfProof (SOUND transformation)

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE