Termination proof of /tmp/tmp74QhqE/Mod.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 360 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 120 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 330 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 IDP

↳11 IDPNonInfProof (⇒, 60 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: Mod

public class Mod {
  public static void main(String[] args) {
    int x = args[0].length();
    int y = args[1].length();
    mod(x, y);
  }
  public static int mod(int x, int y) {
   
    while (x >= y && y > 0) {
      x = minus(x,y);
     
    }
    return x;
  }

  public static int minus(int x, int y) {
    while (y != 0) {
      if (y > 0)  {
        y--;
        x--;
      } else  {
        y++;
        x++;
      }
    }
    return x;
  }

}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Mod.main([Ljava/lang/String;)V: Graph of 178 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: Mod.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:
508_0_mod_Load(EOS(STATIC_508), i84, i83, i84, i83) → 510_0_mod_LT(EOS(STATIC_510), i84, i83, i84, i83, i84)
510_0_mod_LT(EOS(STATIC_510), i84, i83, i84, i83, i84) → 513_0_mod_LT(EOS(STATIC_513), i84, i83, i84, i83, i84)
513_0_mod_LT(EOS(STATIC_513), i84, i83, i84, i83, i84) → 517_0_mod_Load(EOS(STATIC_517), i84, i83, i84) | >=(i83, i84)
517_0_mod_Load(EOS(STATIC_517), i84, i83, i84) → 521_0_mod_LE(EOS(STATIC_521), i84, i83, i84, i84)
521_0_mod_LE(EOS(STATIC_521), i90, i83, i90, i90) → 526_0_mod_LE(EOS(STATIC_526), i90, i83, i90, i90)
526_0_mod_LE(EOS(STATIC_526), i90, i83, i90, i90) → 533_0_mod_Load(EOS(STATIC_533), i90, i83, i90) | >(i90, 0)
533_0_mod_Load(EOS(STATIC_533), i90, i83, i90) → 540_0_mod_Load(EOS(STATIC_540), i90, i90, i83)
540_0_mod_Load(EOS(STATIC_540), i90, i90, i83) → 546_0_mod_InvokeMethod(EOS(STATIC_546), i90, i90, i83, i90)
546_0_mod_InvokeMethod(EOS(STATIC_546), i90, i90, i83, i90) → 550_0_minus_Load(EOS(STATIC_550), i90, i90, i83, i90, i83, i90)
550_0_minus_Load(EOS(STATIC_550), i90, i90, i83, i90, i83, i90) → 576_0_minus_Load(EOS(STATIC_576), i90, i90, i83, i90, i83, i90)
576_0_minus_Load(EOS(STATIC_576), i90, i90, i83, i90, i98, i99) → 580_0_minus_EQ(EOS(STATIC_580), i90, i90, i83, i90, i98, i99, i99)
580_0_minus_EQ(EOS(STATIC_580), i90, i90, i83, i90, i98, i106, i106) → 582_0_minus_EQ(EOS(STATIC_582), i90, i90, i83, i90, i98, i106, i106)
580_0_minus_EQ(EOS(STATIC_580), i90, i90, i83, i90, i98, matching1, matching2) → 583_0_minus_EQ(EOS(STATIC_583), i90, i90, i83, i90, i98, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
582_0_minus_EQ(EOS(STATIC_582), i90, i90, i83, i90, i98, i106, i106) → 585_0_minus_Load(EOS(STATIC_585), i90, i90, i83, i90, i98, i106) | >(i106, 0)
585_0_minus_Load(EOS(STATIC_585), i90, i90, i83, i90, i98, i106) → 590_0_minus_LE(EOS(STATIC_590), i90, i90, i83, i90, i98, i106, i106)
590_0_minus_LE(EOS(STATIC_590), i90, i90, i83, i90, i98, i106, i106) → 594_0_minus_Inc(EOS(STATIC_594), i90, i90, i83, i90, i98, i106) | >(i106, 0)
594_0_minus_Inc(EOS(STATIC_594), i90, i90, i83, i90, i98, i106) → 598_0_minus_Inc(EOS(STATIC_598), i90, i90, i83, i90, i98, +(i106, -1)) | >(i106, 0)
598_0_minus_Inc(EOS(STATIC_598), i90, i90, i83, i90, i98, i107) → 603_0_minus_JMP(EOS(STATIC_603), i90, i90, i83, i90, +(i98, -1), i107)
603_0_minus_JMP(EOS(STATIC_603), i90, i90, i83, i90, i108, i107) → 623_0_minus_Load(EOS(STATIC_623), i90, i90, i83, i90, i108, i107)
623_0_minus_Load(EOS(STATIC_623), i90, i90, i83, i90, i108, i107) → 576_0_minus_Load(EOS(STATIC_576), i90, i90, i83, i90, i108, i107)
583_0_minus_EQ(EOS(STATIC_583), i90, i90, i83, i90, i98, matching1, matching2) → 588_0_minus_Load(EOS(STATIC_588), i90, i90, i83, i90, i98) | &&(=(matching1, 0), =(matching2, 0))
588_0_minus_Load(EOS(STATIC_588), i90, i90, i83, i90, i98) → 592_0_minus_Return(EOS(STATIC_592), i90, i90, i83, i90, i98)
592_0_minus_Return(EOS(STATIC_592), i90, i90, i83, i90, i98) → 596_0_mod_Store(EOS(STATIC_596), i90, i90, i98)
596_0_mod_Store(EOS(STATIC_596), i90, i90, i98) → 601_0_mod_JMP(EOS(STATIC_601), i90, i98, i90)
601_0_mod_JMP(EOS(STATIC_601), i90, i98, i90) → 618_0_mod_Load(EOS(STATIC_618), i90, i98, i90)
618_0_mod_Load(EOS(STATIC_618), i90, i98, i90) → 505_0_mod_Load(EOS(STATIC_505), i90, i98, i90)
505_0_mod_Load(EOS(STATIC_505), i84, i83, i84) → 508_0_mod_Load(EOS(STATIC_508), i84, i83, i84, i83)
R rules:

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

P rules:
580_0_minus_EQ(EOS(STATIC_580), x0, x0, x1, x0, x2, x3, x3) → 580_0_minus_EQ(EOS(STATIC_580), x0, x0, x1, x0, +(x2, -1), +(x3, -1), +(x3, -1)) | >(x3, 0)
580_0_minus_EQ(EOS(STATIC_580), x0, x0, x1, x0, x2, 0, 0) → 580_0_minus_EQ(EOS(STATIC_580), x0, x0, x2, x0, x2, x0, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

580_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → 580_0_minus_EQ(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_580_0_minus_EQ1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_580_0_minus_EQ1(x1, x3, x4, x5, x6, x7)
Cond_580_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_580_0_minus_EQ(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

580_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7) → 580_0_minus_EQ(x3, x4, x5, x7)
Cond_580_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_580_0_minus_EQ(x1, x4, x5, x6, x8)
Cond_580_0_minus_EQ1(x1, x2, x3, x4, x5, x6) → Cond_580_0_minus_EQ1(x1, x4, x5, x6)

Filtered unneeded arguments:

Cond_580_0_minus_EQ(x1, x2, x3, x4, x5) → Cond_580_0_minus_EQ(x1, x3, x4, x5)
580_0_minus_EQ(x1, x2, x3, x4) → 580_0_minus_EQ(x2, x3, x4)
Cond_580_0_minus_EQ1(x1, x2, x3, x4) → Cond_580_0_minus_EQ1(x1, x3, x4)

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

P rules:
580_0_minus_EQ(x0, x2, x3) → 580_0_minus_EQ(x0, +(x2, -1), +(x3, -1)) | >(x3, 0)
580_0_minus_EQ(x0, x2, 0) → 580_0_minus_EQ(x0, x2, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

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

P rules:
580_0_MINUS_EQ(x0, x2, x3) → COND_580_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
COND_580_0_MINUS_EQ(TRUE, x0, x2, x3) → 580_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
580_0_MINUS_EQ(x0, x2, 0) → COND_580_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
COND_580_0_MINUS_EQ1(TRUE, x0, x2, 0) → 580_0_MINUS_EQ(x0, x2, 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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(2): 580_0_MINUS_EQ(x0[2], x2[2], 0) → COND_580_0_MINUS_EQ1(x2[2] >= x0[2] && x0[2] > 0, x0[2], x2[2], 0)
(3): COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 580_0_MINUS_EQ(x0[3], x2[3], x0[3])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(1) -> (2), if (x0[1] →^* x0[2]∧x2[1] + -1 →^* x2[2]∧x3[1] + -1 →^* 0)

(2) -> (3), if (x2[2] >= x0[2] && x0[2] > 0 ∧x0[2] →^* x0[3]∧x2[2] →^* x2[3])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x2[0]∧x0[3] →^* x3[0])

(3) -> (2), if (x0[3] →^* x0[2]∧x2[3] →^* x2[2]∧x0[3] →^* 0)

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@2f2bbce3 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 580_0_MINUS_EQ(x0, x2, x3) → COND_580_0_MINUS_EQ(>(x3, 0), x0, x2, x3) the following chains were created:

We consider the chain 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
(1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x3[0], 0)=TRUE ⇒ 580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧0 = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair COND_580_0_MINUS_EQ(TRUE, x0, x2, x3) → 580_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1)) the following chains were created:

We consider the chain 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9)    (>(x3[0], 0)=TRUE ⇒ COND_580_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_580_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥580_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x3[0] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
We consider the chain 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 580_0_MINUS_EQ(x0[2], x2[2], 0) → COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0) which results in the following constraint:
(15)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[2]∧+(x2[1], -1)=x2[2]∧+(x3[1], -1)=0 ⇒ COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
(16)    (>(x3[0], 0)=TRUE∧+(x3[0], -1)=0 ⇒ COND_580_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_580_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥580_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair 580_0_MINUS_EQ(x0, x2, 0) → COND_580_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0) the following chains were created:

We consider the chain 580_0_MINUS_EQ(x0[2], x2[2], 0) → COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0), COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 580_0_MINUS_EQ(x0[3], x2[3], x0[3]) which results in the following constraint:
(22)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x0[2]=x0[3]∧x2[2]=x2[3] ⇒ 580_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧580_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥))

We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ 580_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧580_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)

For Pair COND_580_0_MINUS_EQ1(TRUE, x0, x2, 0) → 580_0_MINUS_EQ(x0, x2, x0) the following chains were created:

We consider the chain COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 580_0_MINUS_EQ(x0[3], x2[3], x0[3]), 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(29)    (x0[3]=x0[0]∧x2[3]=x2[0]∧x0[3]=x3[0] ⇒ COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥580_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (29) using rule (IV) which results in the following new constraint:
(30)    (COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥580_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)
We consider the chain COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 580_0_MINUS_EQ(x0[3], x2[3], x0[3]), 580_0_MINUS_EQ(x0[2], x2[2], 0) → COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0) which results in the following constraint:
(35)    (x0[3]=x0[2]∧x2[3]=x2[2]∧x0[3]=0 ⇒ COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥580_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:
(36)    (COND_580_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥NonInfC∧COND_580_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥580_0_MINUS_EQ(0, x2[3], 0)∧(U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[(-1)bso_27] ≥ 0)

We simplified constraint (39) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(40)    ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

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

580_0_MINUS_EQ(x0, x2, x3) → COND_580_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
COND_580_0_MINUS_EQ(TRUE, x0, x2, x3) → 580_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
- (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
580_0_MINUS_EQ(x0, x2, 0) → COND_580_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)
COND_580_0_MINUS_EQ1(TRUE, x0, x2, 0) → 580_0_MINUS_EQ(x0, x2, x0)
- ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)
- ((U^Increasing(580_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧[(-1)bso_27] ≥ 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) = [3]
POL(580_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(COND_580_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_580_0_MINUS_EQ1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]

The following pairs are in P_>:

580_0_MINUS_EQ(x0[2], x2[2], 0) → COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

The following pairs are in P_bound:

580_0_MINUS_EQ(x0[2], x2[2], 0) → COND_580_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

The following pairs are in P_≥:

580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 580_0_MINUS_EQ(x0[3], x2[3], x0[3])

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

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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(3): COND_580_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 580_0_MINUS_EQ(x0[3], x2[3], x0[3])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x2[0]∧x0[3] →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(0): 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

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@2f2bbce3 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_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) the following chains were created:

We consider the chain 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>(x3[0], 0)=TRUE ⇒ COND_580_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_580_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥580_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x3[0] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

For Pair 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) the following chains were created:

We consider the chain 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
(8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (8) using rule (IV) which results in the following new constraint:
(9)    (>(x3[0], 0)=TRUE ⇒ 580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧580_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)
580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_580_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + x₄
POL(580_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + x₃
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))

The following pairs are in P_bound:

COND_580_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 580_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

The following pairs are in P_≥:

580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

There are no usable rules.

(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:
(0): 580_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_580_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

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