Termination proof of /tmp/tmpcQgp26/Mod.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 130 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 90 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 330 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 IDP

        ↳12 IDPtoQDPProof (⇒, 20 ms)

        ↳13 QDP

        ↳14 UsableRulesProof (⇔, 0 ms)

        ↳15 QDP

        ↳16 Instantiation (⇔, 0 ms)

        ↳17 QDP

        ↳18 UsableRulesProof (⇔, 0 ms)

        ↳19 QDP

        ↳20 Rewriting (⇔, 0 ms)

        ↳21 QDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 QDP

        ↳24 QReductionProof (⇔, 0 ms)

        ↳25 QDP

        ↳26 Narrowing (⇔, 0 ms)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔, 0 ms)

        ↳29 TRUE

    ↳30 IDP

        ↳31 IDependencyGraphProof (⇔, 0 ms)

        ↳32 IDP

        ↳33 IDPNonInfProof (⇒, 130 ms)

        ↳34 IDP

        ↳35 IDependencyGraphProof (⇔, 0 ms)

        ↳36 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:
547_0_mod_Load(EOS(STATIC_547), i91, i90, i91, i90) → 549_0_mod_LT(EOS(STATIC_549), i91, i90, i91, i90, i91)
549_0_mod_LT(EOS(STATIC_549), i91, i90, i91, i90, i91) → 552_0_mod_LT(EOS(STATIC_552), i91, i90, i91, i90, i91)
552_0_mod_LT(EOS(STATIC_552), i91, i90, i91, i90, i91) → 556_0_mod_Load(EOS(STATIC_556), i91, i90, i91) | >=(i90, i91)
556_0_mod_Load(EOS(STATIC_556), i91, i90, i91) → 561_0_mod_LE(EOS(STATIC_561), i91, i90, i91, i91)
561_0_mod_LE(EOS(STATIC_561), i97, i90, i97, i97) → 566_0_mod_LE(EOS(STATIC_566), i97, i90, i97, i97)
566_0_mod_LE(EOS(STATIC_566), i97, i90, i97, i97) → 573_0_mod_Load(EOS(STATIC_573), i97, i90, i97) | >(i97, 0)
573_0_mod_Load(EOS(STATIC_573), i97, i90, i97) → 580_0_mod_Load(EOS(STATIC_580), i97, i97, i90)
580_0_mod_Load(EOS(STATIC_580), i97, i97, i90) → 587_0_mod_InvokeMethod(EOS(STATIC_587), i97, i97, i90, i97)
587_0_mod_InvokeMethod(EOS(STATIC_587), i97, i97, i90, i97) → 592_0_minus_Load(EOS(STATIC_592), i97, i97, i90, i97, i90, i97)
592_0_minus_Load(EOS(STATIC_592), i97, i97, i90, i97, i90, i97) → 622_0_minus_Load(EOS(STATIC_622), i97, i97, i90, i97, i90, i97)
622_0_minus_Load(EOS(STATIC_622), i97, i97, i90, i97, i102, i103) → 626_0_minus_EQ(EOS(STATIC_626), i97, i97, i90, i97, i102, i103, i103)
626_0_minus_EQ(EOS(STATIC_626), i97, i97, i90, i97, i102, i113, i113) → 629_0_minus_EQ(EOS(STATIC_629), i97, i97, i90, i97, i102, i113, i113)
626_0_minus_EQ(EOS(STATIC_626), i97, i97, i90, i97, i102, matching1, matching2) → 630_0_minus_EQ(EOS(STATIC_630), i97, i97, i90, i97, i102, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
629_0_minus_EQ(EOS(STATIC_629), i97, i97, i90, i97, i102, i113, i113) → 632_0_minus_Load(EOS(STATIC_632), i97, i97, i90, i97, i102, i113) | >(i113, 0)
632_0_minus_Load(EOS(STATIC_632), i97, i97, i90, i97, i102, i113) → 636_0_minus_LE(EOS(STATIC_636), i97, i97, i90, i97, i102, i113, i113)
636_0_minus_LE(EOS(STATIC_636), i97, i97, i90, i97, i102, i113, i113) → 640_0_minus_Inc(EOS(STATIC_640), i97, i97, i90, i97, i102, i113) | >(i113, 0)
640_0_minus_Inc(EOS(STATIC_640), i97, i97, i90, i97, i102, i113) → 644_0_minus_Inc(EOS(STATIC_644), i97, i97, i90, i97, i102, +(i113, -1)) | >(i113, 0)
644_0_minus_Inc(EOS(STATIC_644), i97, i97, i90, i97, i102, i115) → 647_0_minus_JMP(EOS(STATIC_647), i97, i97, i90, i97, +(i102, -1), i115)
647_0_minus_JMP(EOS(STATIC_647), i97, i97, i90, i97, i116, i115) → 653_0_minus_Load(EOS(STATIC_653), i97, i97, i90, i97, i116, i115)
653_0_minus_Load(EOS(STATIC_653), i97, i97, i90, i97, i116, i115) → 622_0_minus_Load(EOS(STATIC_622), i97, i97, i90, i97, i116, i115)
630_0_minus_EQ(EOS(STATIC_630), i97, i97, i90, i97, i102, matching1, matching2) → 635_0_minus_Load(EOS(STATIC_635), i97, i97, i90, i97, i102) | &&(=(matching1, 0), =(matching2, 0))
635_0_minus_Load(EOS(STATIC_635), i97, i97, i90, i97, i102) → 638_0_minus_Return(EOS(STATIC_638), i97, i97, i90, i97, i102)
638_0_minus_Return(EOS(STATIC_638), i97, i97, i90, i97, i102) → 642_0_mod_Store(EOS(STATIC_642), i97, i97, i102)
642_0_mod_Store(EOS(STATIC_642), i97, i97, i102) → 645_0_mod_JMP(EOS(STATIC_645), i97, i102, i97)
645_0_mod_JMP(EOS(STATIC_645), i97, i102, i97) → 650_0_mod_Load(EOS(STATIC_650), i97, i102, i97)
650_0_mod_Load(EOS(STATIC_650), i97, i102, i97) → 542_0_mod_Load(EOS(STATIC_542), i97, i102, i97)
542_0_mod_Load(EOS(STATIC_542), i91, i90, i91) → 547_0_mod_Load(EOS(STATIC_547), i91, i90, i91, i90)
R rules:

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

P rules:
626_0_minus_EQ(EOS(STATIC_626), x0, x0, x1, x0, x2, x3, x3) → 626_0_minus_EQ(EOS(STATIC_626), x0, x0, x1, x0, +(x2, -1), +(x3, -1), +(x3, -1)) | >(x3, 0)
626_0_minus_EQ(EOS(STATIC_626), x0, x0, x1, x0, x2, 0, 0) → 626_0_minus_EQ(EOS(STATIC_626), x0, x0, x2, x0, x2, x0, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

626_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → 626_0_minus_EQ(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_626_0_minus_EQ1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_626_0_minus_EQ1(x1, x3, x4, x5, x6, x7)
Cond_626_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_626_0_minus_EQ(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

626_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7) → 626_0_minus_EQ(x3, x4, x5, x7)
Cond_626_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_626_0_minus_EQ(x1, x4, x5, x6, x8)
Cond_626_0_minus_EQ1(x1, x2, x3, x4, x5, x6) → Cond_626_0_minus_EQ1(x1, x4, x5, x6)

Filtered unneeded arguments:

Cond_626_0_minus_EQ(x1, x2, x3, x4, x5) → Cond_626_0_minus_EQ(x1, x3, x4, x5)
626_0_minus_EQ(x1, x2, x3, x4) → 626_0_minus_EQ(x2, x3, x4)
Cond_626_0_minus_EQ1(x1, x2, x3, x4) → Cond_626_0_minus_EQ1(x1, x3, x4)

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

P rules:
626_0_minus_EQ(x0, x2, x3) → 626_0_minus_EQ(x0, +(x2, -1), +(x3, -1)) | >(x3, 0)
626_0_minus_EQ(x0, x2, 0) → 626_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:
626_0_MINUS_EQ(x0, x2, x3) → COND_626_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
COND_626_0_MINUS_EQ(TRUE, x0, x2, x3) → 626_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
626_0_MINUS_EQ(x0, x2, 0) → COND_626_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
COND_626_0_MINUS_EQ1(TRUE, x0, x2, 0) → 626_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): 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(2): 626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_0_MINUS_EQ1(x2[2] >= x0[2] && x0[2] > 0, x0[2], x2[2], 0)
(3): COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_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@475170bc 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 626_0_MINUS_EQ(x0, x2, x3) → COND_626_0_MINUS_EQ(>(x3, 0), x0, x2, x3) the following chains were created:

We consider the chain 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_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] ⇒ 626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_626_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 ⇒ 626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_626_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_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x2[0] + [(-1)bni_20]x0[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_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x2[0] + [(-1)bni_20]x0[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_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x2[0] + [(-1)bni_20]x0[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_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧[(-1)bni_20] = 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 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_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧[(-1)bni_20] = 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair COND_626_0_MINUS_EQ(TRUE, x0, x2, x3) → 626_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1)) the following chains were created:

We consider the chain 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(626_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_626_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_626_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥626_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(626_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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x2[0] + [(-1)bni_22]x0[0] ≥ 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x2[0] + [(-1)bni_22]x0[0] ≥ 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x2[0] + [(-1)bni_22]x0[0] ≥ 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 = 0∧0 = 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)
We consider the chain 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(626_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_626_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_626_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥626_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(626_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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x2[0] + [(-1)bni_22]x0[0] ≥ 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x2[0] + [(-1)bni_22]x0[0] ≥ 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x2[0] + [(-1)bni_22]x0[0] ≥ 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 = 0∧0 = 0∧[1 + (-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(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

For Pair 626_0_MINUS_EQ(x0, x2, 0) → COND_626_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0) the following chains were created:

We consider the chain 626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0), COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_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] ⇒ 626_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧626_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_626_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 ⇒ 626_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧626_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_626_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_626_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] + [(-1)bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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_626_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] + [(-1)bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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_626_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] + [(-1)bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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_626_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] ≥ 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_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)Bound*bni_24 + (-1)bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_626_0_MINUS_EQ1(TRUE, x0, x2, 0) → 626_0_MINUS_EQ(x0, x2, x0) the following chains were created:

We consider the chain COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3]), 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_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_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥626_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(626_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_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥626_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(626_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(626_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(626_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(626_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(626_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_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3]), 626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_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_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥626_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(626_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_626_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥NonInfC∧COND_626_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥626_0_MINUS_EQ(0, x2[3], 0)∧(U^Increasing(626_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(626_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(626_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(626_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(626_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.

626_0_MINUS_EQ(x0, x2, x3) → COND_626_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧[(-1)bni_20] = 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
COND_626_0_MINUS_EQ(TRUE, x0, x2, x3) → 626_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)
- (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_23] ≥ 0)
626_0_MINUS_EQ(x0, x2, 0) → COND_626_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)Bound*bni_24 + (-1)bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)
COND_626_0_MINUS_EQ1(TRUE, x0, x2, 0) → 626_0_MINUS_EQ(x0, x2, x0)
- ((U^Increasing(626_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)
- ((U^Increasing(626_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) = 0
POL(626_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + x₂ + [-1]x₁
POL(COND_626_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + x₃ + [-1]x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_626_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_>:

COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))

The following pairs are in P_bound:

626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

The following pairs are in P_≥:

626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)
COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_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) 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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(2): 626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_0_MINUS_EQ1(x2[2] >= x0[2] && x0[2] > 0, x0[2], x2[2], 0)
(3): COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_0_MINUS_EQ(x0[3], x2[3], x0[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)

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

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
(2): 626_0_MINUS_EQ(x0[2], x2[2], 0) → COND_626_0_MINUS_EQ1(x2[2] >= x0[2] && x0[2] > 0, x0[2], x2[2], 0)

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

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

The set Q is empty.

(12) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(13) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(x0[2], x2[2], pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(x2[2], x0[2]), greater_int(x0[2], pos(01))), x0[2], x2[2], pos(01))

The TRS R consists of the following rules:

and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true
greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(01), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(01), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(01)) → false
greatereq_int(neg(s(x)), neg(01)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(01), pos(01)) → false
greater_int(pos(01), neg(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(neg(01), neg(01)) → false
greater_int(pos(01), pos(s(y))) → false
greater_int(neg(01), pos(s(y))) → false
greater_int(pos(01), neg(s(y))) → true
greater_int(neg(01), neg(s(y))) → true
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
greater_int(pos(s(x)), neg(01)) → true
greater_int(neg(s(x)), neg(01)) → false
greater_int(pos(s(x)), neg(s(y))) → true
greater_int(neg(s(x)), pos(s(y))) → false
greater_int(pos(s(x)), pos(s(y))) → greater_int(pos(x), pos(y))
greater_int(neg(s(x)), neg(s(y))) → greater_int(neg(x), neg(y))

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(14) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(15) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(x0[2], x2[2], pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(x2[2], x0[2]), greater_int(x0[2], pos(01))), x0[2], x2[2], pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(01), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(01), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(01)) → false
greatereq_int(neg(s(x)), neg(01)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(01), pos(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(16) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 626_0_MINUS_EQ(x0[2], x2[2], pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(x2[2], x0[2]), greater_int(x0[2], pos(01))), x0[2], x2[2], pos(01)) we obtained the following new rules [LPAR04]:

626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), greater_int(pos(01), pos(01))), pos(01), z1, pos(01))

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), greater_int(pos(01), pos(01))), pos(01), z1, pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(01), neg(y)) → true
greatereq_int(pos(x), neg(y)) → true
greatereq_int(pos(01), pos(s(y))) → false
greatereq_int(neg(x), pos(s(y))) → false
greatereq_int(neg(s(x)), pos(01)) → false
greatereq_int(neg(s(x)), neg(01)) → false
greatereq_int(pos(s(x)), pos(s(y))) → greatereq_int(pos(x), pos(y))
greatereq_int(neg(s(x)), neg(s(y))) → greatereq_int(neg(x), neg(y))
greater_int(pos(01), pos(01)) → false
greater_int(neg(01), pos(01)) → false
greater_int(pos(s(x)), pos(01)) → true
greater_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(18) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(19) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), greater_int(pos(01), pos(01))), pos(01), z1, pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(s(x)), pos(01)) → false
greater_int(pos(01), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(20) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule 626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), greater_int(pos(01), pos(01))), pos(01), z1, pos(01)) at position [0,1] we obtained the following new rules [LPAR04]:

626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), false), pos(01), z1, pos(01))

(21) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), false), pos(01), z1, pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(s(x)), pos(01)) → false
greater_int(pos(01), pos(01)) → false
and(false, false) → false
and(false, true) → false
and(true, false) → false
and(true, true) → true

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(22) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(23) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), false), pos(01), z1, pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(true, false) → false

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))
greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(24) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

greater_int(pos(01), pos(01))
greater_int(pos(01), neg(01))
greater_int(neg(01), pos(01))
greater_int(neg(01), neg(01))
greater_int(pos(01), pos(s(x0)))
greater_int(neg(01), pos(s(x0)))
greater_int(pos(01), neg(s(x0)))
greater_int(neg(01), neg(s(x0)))
greater_int(pos(s(x0)), pos(01))
greater_int(neg(s(x0)), pos(01))
greater_int(pos(s(x0)), neg(01))
greater_int(neg(s(x0)), neg(01))
greater_int(pos(s(x0)), neg(s(x1)))
greater_int(neg(s(x0)), pos(s(x1)))
greater_int(pos(s(x0)), pos(s(x1)))
greater_int(neg(s(x0)), neg(s(x1)))

(25) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), false), pos(01), z1, pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(true, false) → false

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(26) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 626_0_MINUS_EQ(pos(01), z1, pos(01)) → COND_626_0_MINUS_EQ1(and(greatereq_int(z1, pos(01)), false), pos(01), z1, pos(01)) at position [0] we obtained the following new rules [LPAR04]:

626_0_MINUS_EQ(pos(01), pos(x0), pos(01)) → COND_626_0_MINUS_EQ1(and(true, false), pos(01), pos(x0), pos(01))
626_0_MINUS_EQ(pos(01), neg(01), pos(01)) → COND_626_0_MINUS_EQ1(and(true, false), pos(01), neg(01), pos(01))
626_0_MINUS_EQ(pos(01), neg(s(x0)), pos(01)) → COND_626_0_MINUS_EQ1(and(false, false), pos(01), neg(s(x0)), pos(01))

(27) Obligation:

Q DP problem:
The TRS P consists of the following rules:

COND_626_0_MINUS_EQ1(true, x0[3], x2[3], pos(01)) → 626_0_MINUS_EQ(x0[3], x2[3], x0[3])
626_0_MINUS_EQ(pos(01), pos(x0), pos(01)) → COND_626_0_MINUS_EQ1(and(true, false), pos(01), pos(x0), pos(01))
626_0_MINUS_EQ(pos(01), neg(01), pos(01)) → COND_626_0_MINUS_EQ1(and(true, false), pos(01), neg(01), pos(01))
626_0_MINUS_EQ(pos(01), neg(s(x0)), pos(01)) → COND_626_0_MINUS_EQ1(and(false, false), pos(01), neg(s(x0)), pos(01))

The TRS R consists of the following rules:

greatereq_int(pos(x), pos(01)) → true
greatereq_int(neg(01), pos(01)) → true
greatereq_int(neg(s(x)), pos(01)) → false
and(false, false) → false
and(true, false) → false

The set Q consists of the following terms:

and(false, false)
and(false, true)
and(true, false)
and(true, true)
greatereq_int(pos(x0), pos(01))
greatereq_int(neg(01), pos(01))
greatereq_int(neg(01), neg(x0))
greatereq_int(pos(x0), neg(x1))
greatereq_int(pos(01), pos(s(x0)))
greatereq_int(neg(x0), pos(s(x1)))
greatereq_int(neg(s(x0)), pos(01))
greatereq_int(neg(s(x0)), neg(01))
greatereq_int(pos(s(x0)), pos(s(x1)))
greatereq_int(neg(s(x0)), neg(s(x1)))

We have to consider all minimal (P,Q,R)-chains.

(28) DependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE

(30) 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): 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(3): COND_626_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 626_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.

(31) IDependencyGraphProof (EQUIVALENT transformation)

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

(32) 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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(0): 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_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.

(33) 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@475170bc 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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) the following chains were created:

We consider the chain 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(626_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_626_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_626_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥626_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(626_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(626_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(626_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(626_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(626_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(626_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 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) the following chains were created:

We consider the chain 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_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] ⇒ 626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_626_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 ⇒ 626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧626_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_626_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_626_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_626_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_626_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_626_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_626_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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(626_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)
626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_626_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_626_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + x₄
POL(626_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_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))

The following pairs are in P_bound:

COND_626_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 626_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

The following pairs are in P_≥:

626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

There are no usable rules.

(34) 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): 626_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_626_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

The set Q is empty.

(35) 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) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) Obligation:

(12) IDPtoQDPProof (SOUND transformation)

(13) Obligation:

(14) UsableRulesProof (EQUIVALENT transformation)

(15) Obligation:

(16) Instantiation (EQUIVALENT transformation)

(17) Obligation:

(18) UsableRulesProof (EQUIVALENT transformation)

(19) Obligation:

(20) Rewriting (EQUIVALENT transformation)

(21) Obligation:

(22) UsableRulesProof (EQUIVALENT transformation)

(23) Obligation:

(24) QReductionProof (EQUIVALENT transformation)

(25) Obligation:

(26) Narrowing (EQUIVALENT transformation)

(27) Obligation:

(28) DependencyGraphProof (EQUIVALENT transformation)

(29) TRUE

(30) Obligation:

(31) IDependencyGraphProof (EQUIVALENT transformation)

(32) Obligation:

(33) IDPNonInfProof (SOUND transformation)

(34) Obligation:

(35) IDependencyGraphProof (EQUIVALENT transformation)

(36) TRUE