Termination proof of /tmp/tmp297x

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPtoQDPProof (⇒)

        ↳7 QDP

        ↳8 DependencyGraphProof (⇔)

        ↳9 QDP

        ↳10 UsableRulesProof (⇔)

        ↳11 QDP

        ↳12 QReductionProof (⇔)

        ↳13 QDP

        ↳14 Narrowing (⇔)

        ↳15 QDP

        ↳16 Narrowing (⇔)

        ↳17 QDP

        ↳18 Narrowing (⇔)

        ↳19 QDP

        ↳20 Narrowing (⇔)

        ↳21 QDP

        ↳22 Narrowing (⇔)

        ↳23 QDP

        ↳24 Narrowing (⇔)

        ↳25 QDP

        ↳26 Narrowing (⇔)

        ↳27 QDP

        ↳28 Narrowing (⇔)

        ↳29 QDP

        ↳30 UsableRulesProof (⇔)

        ↳31 QDP

        ↳32 QReductionProof (⇔)

        ↳33 QDP

        ↳34 Narrowing (⇔)

        ↳35 QDP

        ↳36 DependencyGraphProof (⇔)

        ↳37 QDP

        ↳38 Narrowing (⇔)

        ↳39 QDP

        ↳40 DependencyGraphProof (⇔)

        ↳41 QDP

        ↳42 Narrowing (⇔)

        ↳43 QDP

        ↳44 DependencyGraphProof (⇔)

        ↳45 QDP

        ↳46 Narrowing (⇔)

        ↳47 QDP

        ↳48 DependencyGraphProof (⇔)

        ↳49 QDP

        ↳50 Narrowing (⇔)

        ↳51 QDP

        ↳52 DependencyGraphProof (⇔)

        ↳53 QDP

        ↳54 Narrowing (⇔)

        ↳55 QDP

        ↳56 Narrowing (⇔)

        ↳57 QDP

        ↳58 DependencyGraphProof (⇔)

        ↳59 QDP

        ↳60 Narrowing (⇔)

        ↳61 QDP

        ↳62 DependencyGraphProof (⇔)

        ↳63 QDP

        ↳64 UsableRulesProof (⇔)

        ↳65 QDP

        ↳66 QReductionProof (⇔)

        ↳67 QDP

        ↳68 Instantiation (⇔)

        ↳69 QDP

        ↳70 Instantiation (⇔)

        ↳71 QDP

        ↳72 Instantiation (⇔)

        ↳73 QDP

        ↳74 QDPSizeChangeProof (⇔)

        ↳75 YES

    ↳76 IDP

        ↳77 IDPNonInfProof (⇒)

        ↳78 AND

            ↳79 IDP

                ↳80 IDependencyGraphProof (⇔)

                ↳81 TRUE

            ↳82 IDP

                ↳83 IDependencyGraphProof (⇔)

                ↳84 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_14 (Sun Microsystems Inc.) Main-Class: DivTernary2/DivTernary2

package DivTernary2;
public class DivTernary2 {
    // by Thomas Kolbe
    public static Nats div(final Nats x, final Nats y, final Nats z) {
        if (z.isZero()) {
            return Nats.zero();
        }
        if (y.isZero()) {
            return Nats.succ(div(x, z, z));
        }
        if (x.isZero()) {
            return Nats.zero();
        }
        return div(x.pred, y.pred, z);
    }

    public static void main(final String[] args) {

        final Nats x = new Nats(args[0].length());
        final Nats y = new Nats(args[1].length());

        div(x, y, y);
    }
}


package DivTernary2;

public class Nats {
    Nats pred;

    public Nats() {
    }

    public Nats(final int n) {
        if (n <= 0) {
            this.pred = null;
        } else {
            this.pred = new Nats(n - 1);
        }
    }

    public int toInt() {
        if (this.pred == null) {
            return 0;
        }
        return this.pred.toInt() + 1;
    }

    @Override
    public String toString() {
        return (String.valueOf(toInt()));
    }

    public boolean isZero() {
        return (this.pred == null);
    }

    public static Nats zero() {
        return new Nats();
    }

    public static Nats succ(final Nats x) {
        final Nats y = new Nats();
        y.pred = x;
        return y;
    }

    /**
     * @return
     */
    public Nats copy() {
        if (this.pred == null) {
            return new Nats();
        }
        final Nats predCopy = this.pred.copy();
        return succ(predCopy);
    }

}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
DivTernary2.DivTernary2.main([Ljava/lang/String;)V: Graph of 157 nodes with 0 SCCs.

DivTernary2.Nats.<init>(I)V: Graph of 27 nodes with 0 SCCs.

DivTernary2.DivTernary2.div(LDivTernary2/Nats;LDivTernary2/Nats;LDivTernary2/Nats;)LDivTernary2/Nats;: Graph of 179 nodes with 0 SCCs.

DivTernary2.Nats.isZero()Z: Graph of 39 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 72 rules for P and 154 rules for R.

Combined rules. Obtained 20 rules for P and 21 rules for R.

Filtered ground terms:

2255_0_isZero_Load(x1, x2) → 2255_0_isZero_Load(x2)
DivTernary2.Nats(x1, x2) → DivTernary2.Nats(x2)
2963_0_isZero_Return(x1, x2) → 2963_0_isZero_Return
2941_0_isZero_Return(x1, x2) → 2941_0_isZero_Return
2922_0_isZero_Return(x1, x2) → 2922_0_isZero_Return
3240_0_isZero_Load(x1, x2) → 3240_0_isZero_Load(x2)
3191_0_div_EQ(x1, x2, x3, x4, x5) → 3191_0_div_EQ(x2, x3, x4, x5)
3101_0_isZero_Return(x1, x2) → 3101_0_isZero_Return
3063_0_isZero_Return(x1, x2) → 3063_0_isZero_Return
2998_0_isZero_Return(x1, x2) → 2998_0_isZero_Return
2681_0_isZero_Load(x1, x2) → 2681_0_isZero_Load(x2)
2711_0_isZero_FieldAccess(x1, x2) → 2711_0_isZero_FieldAccess(x2)
3817_0_div_InvokeMethod(x1, x2, x3, x4, x5) → 3817_0_div_InvokeMethod(x3, x4, x5)
java.lang.NullPointerException(x1) → java.lang.NullPointerException
java.lang.RuntimeException(x1) → java.lang.RuntimeException
java.lang.Exception(x1) → java.lang.Exception
java.lang.Throwable(x1) → java.lang.Throwable
3789_0_div_InvokeMethod(x1, x2, x3, x4, x5) → 3789_0_div_InvokeMethod(x4, x5)
3746_0_div_InvokeMethod(x1, x2, x3, x4, x5, x6) → 3746_0_div_InvokeMethod(x4, x5)
3725_0_div_Return(x1, x2) → 3725_0_div_Return(x2)
3694_0_div_Return(x1, x2) → 3694_0_div_Return
3656_0_div_Return(x1, x2, x3, x4, x5) → 3656_0_div_Return(x2, x3, x4)
3547_0_div_Return(x1, x2) → 3547_0_div_Return
3561_0_div_Return(x1, x2, x3, x4, x5) → 3561_0_div_Return(x2, x3, x4)
3550_0_div_InvokeMethod(x1, x2, x3, x4, x5) → 3550_0_div_InvokeMethod(x3, x5)
3422_0_div_InvokeMethod(x1, x2, x3, x4, x5, x6) → 3422_0_div_InvokeMethod(x3, x5)
2760_0_div_Return(x1, x2, x3, x4, x5) → 2760_0_div_Return(x2, x3, x4)

Filtered duplicate args:

2255_1_div_InvokeMethod(x1, x2, x3, x4, x5) → 2255_1_div_InvokeMethod(x1, x2, x3, x5)
3240_1_div_InvokeMethod(x1, x2, x3, x4, x5) → 3240_1_div_InvokeMethod(x1, x3, x4, x5)
3336_1_div_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → 3336_1_div_InvokeMethod(x1, x3, x5, x7)
2681_1_div_InvokeMethod(x1, x2, x3, x4, x5) → 2681_1_div_InvokeMethod(x1, x2, x4, x5)

Finished conversion. Obtained 14 rules for P and 21 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 15 rules for P and 12 rules for R.

Combined rules. Obtained 1 rules for P and 2 rules for R.

Filtered ground terms:

87_0_<init>_InvokeMethod(x1, x2) → 87_0_<init>_InvokeMethod(x2)
Cond_87_0_<init>_InvokeMethod(x1, x2, x3) → Cond_87_0_<init>_InvokeMethod(x1, x3)
339_0_<init>_Return(x1) → 339_0_<init>_Return
211_0_<init>_Return(x1) → 211_0_<init>_Return

Combined rules. Obtained 1 rules for P and 2 rules for R.

Finished conversion. Obtained 1 rules for P and 2 rules for R. System has predefined symbols.

(4) Complex Obligation (AND)

(5) 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:
none

The ITRS R consists of the following rules:
2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1)) → 3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x2)), java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2)) → 3547_0_div_Return
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1)) → 3550_0_div_InvokeMethod(x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(NULL)))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3547_0_div_Return
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3694_0_div_Return
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(NULL)))))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)) → 3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The integer pair graph contains the following rules and edges:
(0): 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0]))
(1): 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
(2): 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
(3): 2681_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, x1[3], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[3], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)), 1)
(4): 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), 1)
(5): 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), 1)
(6): 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), 0)
(7): 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), 0)
(8): 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), 0)
(9): 3191_0_DIV_EQ(x0[9], java.lang.Object(DivTernary2.Nats(x1[9])), java.lang.Object(x2[9]), 1) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[9])), x0[9], java.lang.Object(x2[9]), java.lang.Object(x2[9]))
(10): 3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), 0) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
(11): 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
(12): 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
(13): 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))

(0) -> (3), if ((2255_0_isZero_Load(java.lang.Object(x2[0])) →^* 2998_0_isZero_Return)∧(x1[0] →^* x1[3])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))) →^* java.lang.Object(DivTernary2.Nats(NULL)))∧(java.lang.Object(x2[0]) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(0) -> (4), if ((2255_0_isZero_Load(java.lang.Object(x2[0])) →^* 3063_0_isZero_Return)∧(x1[0] →^* x1[4])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))) →^* java.lang.Object(x2[4]))∧(java.lang.Object(x2[0]) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(0) -> (5), if ((2255_0_isZero_Load(java.lang.Object(x2[0])) →^* 3101_0_isZero_Return)∧(x1[0] →^* x1[5])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))) →^* java.lang.Object(x2[5]))∧(java.lang.Object(x2[0]) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(0) -> (6), if ((2255_0_isZero_Load(java.lang.Object(x2[0])) →^* 2922_0_isZero_Return)∧(x1[0] →^* x1[6])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))))∧(java.lang.Object(x2[0]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))))

(0) -> (7), if ((2255_0_isZero_Load(java.lang.Object(x2[0])) →^* 2941_0_isZero_Return)∧(x1[0] →^* x1[7])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))) →^* java.lang.Object(x3[7]))∧(java.lang.Object(x2[0]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))))

(0) -> (8), if ((2255_0_isZero_Load(java.lang.Object(x2[0])) →^* 2963_0_isZero_Return)∧(x1[0] →^* x1[8])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))) →^* java.lang.Object(x3[8]))∧(java.lang.Object(x2[0]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))))

(1) -> (3), if ((2255_0_isZero_Load(java.lang.Object(x2[1])) →^* 2998_0_isZero_Return)∧(x1[1] →^* x1[3])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))) →^* java.lang.Object(DivTernary2.Nats(NULL)))∧(java.lang.Object(x2[1]) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(1) -> (4), if ((2255_0_isZero_Load(java.lang.Object(x2[1])) →^* 3063_0_isZero_Return)∧(x1[1] →^* x1[4])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))) →^* java.lang.Object(x2[4]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(1) -> (5), if ((2255_0_isZero_Load(java.lang.Object(x2[1])) →^* 3101_0_isZero_Return)∧(x1[1] →^* x1[5])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))) →^* java.lang.Object(x2[5]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(1) -> (6), if ((2255_0_isZero_Load(java.lang.Object(x2[1])) →^* 2922_0_isZero_Return)∧(x1[1] →^* x1[6])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))))∧(java.lang.Object(x2[1]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))))

(1) -> (7), if ((2255_0_isZero_Load(java.lang.Object(x2[1])) →^* 2941_0_isZero_Return)∧(x1[1] →^* x1[7])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))) →^* java.lang.Object(x3[7]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))))

(1) -> (8), if ((2255_0_isZero_Load(java.lang.Object(x2[1])) →^* 2963_0_isZero_Return)∧(x1[1] →^* x1[8])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))) →^* java.lang.Object(x3[8]))∧(java.lang.Object(x2[1]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))))

(2) -> (3), if ((2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) →^* 2998_0_isZero_Return)∧(x1[2] →^* x1[3])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(2) -> (4), if ((2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) →^* 3063_0_isZero_Return)∧(x1[2] →^* x1[4])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(x2[4]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(2) -> (5), if ((2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) →^* 3101_0_isZero_Return)∧(x1[2] →^* x1[5])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(x2[5]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(DivTernary2.Nats(NULL))))

(2) -> (6), if ((2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) →^* 2922_0_isZero_Return)∧(x1[2] →^* x1[6])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))))

(2) -> (7), if ((2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) →^* 2941_0_isZero_Return)∧(x1[2] →^* x1[7])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(x3[7]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))))

(2) -> (8), if ((2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) →^* 2963_0_isZero_Return)∧(x1[2] →^* x1[8])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(x3[8]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))))

(3) -> (9), if ((x1[3] →^* x0[9])∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(DivTernary2.Nats(x1[9])))∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(x2[9])))

(3) -> (10), if ((x1[3] →^* java.lang.Object(x0[10]))∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(DivTernary2.Nats(x1[10])))∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(x2[10]))∧(1 →^* 0))

(4) -> (9), if ((x1[4] →^* x0[9])∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(DivTernary2.Nats(x1[9])))∧(java.lang.Object(x2[4]) →^* java.lang.Object(x2[9])))

(4) -> (10), if ((x1[4] →^* java.lang.Object(x0[10]))∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(DivTernary2.Nats(x1[10])))∧(java.lang.Object(x2[4]) →^* java.lang.Object(x2[10]))∧(1 →^* 0))

(5) -> (9), if ((x1[5] →^* x0[9])∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(DivTernary2.Nats(x1[9])))∧(java.lang.Object(x2[5]) →^* java.lang.Object(x2[9])))

(5) -> (10), if ((x1[5] →^* java.lang.Object(x0[10]))∧(java.lang.Object(DivTernary2.Nats(NULL)) →^* java.lang.Object(DivTernary2.Nats(x1[10])))∧(java.lang.Object(x2[5]) →^* java.lang.Object(x2[10]))∧(1 →^* 0))

(6) -> (9), if ((x1[6] →^* x0[9])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))) →^* java.lang.Object(DivTernary2.Nats(x1[9])))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))) →^* java.lang.Object(x2[9]))∧(0 →^* 1))

(6) -> (10), if ((x1[6] →^* java.lang.Object(x0[10]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))) →^* java.lang.Object(DivTernary2.Nats(x1[10])))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))) →^* java.lang.Object(x2[10])))

(7) -> (9), if ((x1[7] →^* x0[9])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))) →^* java.lang.Object(DivTernary2.Nats(x1[9])))∧(java.lang.Object(x3[7]) →^* java.lang.Object(x2[9]))∧(0 →^* 1))

(7) -> (10), if ((x1[7] →^* java.lang.Object(x0[10]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))) →^* java.lang.Object(DivTernary2.Nats(x1[10])))∧(java.lang.Object(x3[7]) →^* java.lang.Object(x2[10])))

(8) -> (9), if ((x1[8] →^* x0[9])∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))) →^* java.lang.Object(DivTernary2.Nats(x1[9])))∧(java.lang.Object(x3[8]) →^* java.lang.Object(x2[9]))∧(0 →^* 1))

(8) -> (10), if ((x1[8] →^* java.lang.Object(x0[10]))∧(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))) →^* java.lang.Object(DivTernary2.Nats(x1[10])))∧(java.lang.Object(x3[8]) →^* java.lang.Object(x2[10])))

(9) -> (0), if ((2255_0_isZero_Load(java.lang.Object(x2[9])) →^* 2922_0_isZero_Return)∧(x0[9] →^* x1[0])∧(java.lang.Object(x2[9]) →^* java.lang.Object(x2[0]))∧(java.lang.Object(x2[9]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))))

(9) -> (1), if ((2255_0_isZero_Load(java.lang.Object(x2[9])) →^* 2941_0_isZero_Return)∧(x0[9] →^* x1[1])∧(java.lang.Object(x2[9]) →^* java.lang.Object(x2[1]))∧(java.lang.Object(x2[9]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))))

(9) -> (2), if ((2255_0_isZero_Load(java.lang.Object(x2[9])) →^* 2963_0_isZero_Return)∧(x0[9] →^* x1[2])∧(java.lang.Object(x2[9]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))))

(10) -> (11), if ((2255_0_isZero_Load(java.lang.Object(x0[10])) →^* 2922_0_isZero_Return)∧(java.lang.Object(DivTernary2.Nats(x1[10])) →^* java.lang.Object(DivTernary2.Nats(x2[11])))∧(java.lang.Object(x2[10]) →^* java.lang.Object(x3[11]))∧(java.lang.Object(x0[10]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))))

(10) -> (12), if ((2255_0_isZero_Load(java.lang.Object(x0[10])) →^* 2941_0_isZero_Return)∧(java.lang.Object(DivTernary2.Nats(x1[10])) →^* java.lang.Object(DivTernary2.Nats(x2[12])))∧(java.lang.Object(x2[10]) →^* java.lang.Object(x3[12]))∧(java.lang.Object(x0[10]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))))

(10) -> (13), if ((2255_0_isZero_Load(java.lang.Object(x0[10])) →^* 2963_0_isZero_Return)∧(java.lang.Object(DivTernary2.Nats(x1[10])) →^* java.lang.Object(DivTernary2.Nats(x2[13])))∧(java.lang.Object(x2[10]) →^* java.lang.Object(x3[13]))∧(java.lang.Object(x0[10]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))))

(11) -> (0), if ((2255_0_isZero_Load(java.lang.Object(x3[11])) →^* 2922_0_isZero_Return)∧(java.lang.Object(x1[11]) →^* x1[0])∧(x2[11] →^* java.lang.Object(x2[0]))∧(java.lang.Object(x3[11]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))))

(11) -> (1), if ((2255_0_isZero_Load(java.lang.Object(x3[11])) →^* 2941_0_isZero_Return)∧(java.lang.Object(x1[11]) →^* x1[1])∧(x2[11] →^* java.lang.Object(x2[1]))∧(java.lang.Object(x3[11]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))))

(11) -> (2), if ((2255_0_isZero_Load(java.lang.Object(x3[11])) →^* 2963_0_isZero_Return)∧(java.lang.Object(x1[11]) →^* x1[2])∧(x2[11] →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))∧(java.lang.Object(x3[11]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))))

(12) -> (0), if ((2255_0_isZero_Load(java.lang.Object(x3[12])) →^* 2922_0_isZero_Return)∧(java.lang.Object(x1[12]) →^* x1[0])∧(x2[12] →^* java.lang.Object(x2[0]))∧(java.lang.Object(x3[12]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))))

(12) -> (1), if ((2255_0_isZero_Load(java.lang.Object(x3[12])) →^* 2941_0_isZero_Return)∧(java.lang.Object(x1[12]) →^* x1[1])∧(x2[12] →^* java.lang.Object(x2[1]))∧(java.lang.Object(x3[12]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))))

(12) -> (2), if ((2255_0_isZero_Load(java.lang.Object(x3[12])) →^* 2963_0_isZero_Return)∧(java.lang.Object(x1[12]) →^* x1[2])∧(x2[12] →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))∧(java.lang.Object(x3[12]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))))

(13) -> (0), if ((2255_0_isZero_Load(java.lang.Object(x3[13])) →^* 2922_0_isZero_Return)∧(java.lang.Object(x1[13]) →^* x1[0])∧(x2[13] →^* java.lang.Object(x2[0]))∧(java.lang.Object(x3[13]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))))

(13) -> (1), if ((2255_0_isZero_Load(java.lang.Object(x3[13])) →^* 2941_0_isZero_Return)∧(java.lang.Object(x1[13]) →^* x1[1])∧(x2[13] →^* java.lang.Object(x2[1]))∧(java.lang.Object(x3[13]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))))

(13) -> (2), if ((2255_0_isZero_Load(java.lang.Object(x3[13])) →^* 2963_0_isZero_Return)∧(java.lang.Object(x1[13]) →^* x1[2])∧(x2[13] →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))∧(java.lang.Object(x3[13]) →^* java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))))

The set Q consists of the following terms:
2255_0_isZero_Load(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2))
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
2681_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, x1[3], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[3], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(x0[9], java.lang.Object(DivTernary2.Nats(x1[9])), java.lang.Object(x2[9]), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[9])), x0[9], java.lang.Object(x2[9]), java.lang.Object(x2[9]))
3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1)) → 3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x2)), java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2)) → 3547_0_div_Return
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1)) → 3550_0_div_InvokeMethod(x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(NULL)))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3547_0_div_Return
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3694_0_div_Return
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(NULL)))))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)) → 3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2))
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(8) DependencyGraphProof (EQUIVALENT transformation)

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

(9) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(x0[9], java.lang.Object(DivTernary2.Nats(x1[9])), java.lang.Object(x2[9]), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[9])), x0[9], java.lang.Object(x2[9]), java.lang.Object(x2[9]))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0]))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1)) → 3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x2)), java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2)) → 3547_0_div_Return
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1)) → 3550_0_div_InvokeMethod(x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(NULL)))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3547_0_div_Return
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)) → 3694_0_div_Return
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(NULL)))))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3)) → 3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)) → 3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2)) → 3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2))
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(10) 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.

(11) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(x0[9], java.lang.Object(DivTernary2.Nats(x1[9])), java.lang.Object(x2[9]), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[9])), x0[9], java.lang.Object(x2[9]), java.lang.Object(x2[9]))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0]))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2))
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(12) 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].

3336_1_div_InvokeMethod(2760_0_div_Return(x0, java.lang.Object(x1), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x2)), x0, java.lang.Object(x1))
3509_1_div_InvokeMethod(2760_0_div_Return(x0, x1, java.lang.Object(x2)), x0, x1, java.lang.Object(x2))
3509_1_div_InvokeMethod(3422_0_div_InvokeMethod(x0, java.lang.Object(x1)), x0, NULL, java.lang.Object(x1))
3509_1_div_InvokeMethod(3547_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3550_0_div_InvokeMethod(x0, java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3561_0_div_Return(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3656_0_div_Return(x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), x0, java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2))
3509_1_div_InvokeMethod(3694_0_div_Return, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))
3509_1_div_InvokeMethod(3725_0_div_Return(java.lang.Object(DivTernary2.Nats(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x2)))), java.lang.Object(x3))
3509_1_div_InvokeMethod(3746_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), NULL, java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1))
3509_1_div_InvokeMethod(3789_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(x1)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(x1))
3509_1_div_InvokeMethod(3817_0_div_InvokeMethod(java.lang.Object(DivTernary2.Nats(x0)), java.lang.Object(DivTernary2.Nats(x1)), java.lang.Object(x2)), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x0)))), java.lang.Object(DivTernary2.Nats(java.lang.Object(DivTernary2.Nats(x1)))), java.lang.Object(x2))

(13) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(x0[9], java.lang.Object(DivTernary2.Nats(x1[9])), java.lang.Object(x2[9]), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[9])), x0[9], java.lang.Object(x2[9]), java.lang.Object(x2[9]))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0]))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(14) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3191_0_DIV_EQ(x0[9], java.lang.Object(DivTernary2.Nats(x1[9])), java.lang.Object(x2[9]), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[9])), x0[9], java.lang.Object(x2[9]), java.lang.Object(x2[9])) at position [0] we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))

(15) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0]))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(16) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[0], java.lang.Object(x2[0]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[0])), x1[0], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[0]))), java.lang.Object(x2[0])) at position [0] we obtained the following new rules [LPAR04]:

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))

(17) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10]))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(18) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3191_0_DIV_EQ(java.lang.Object(x0[10]), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x0[10])), java.lang.Object(DivTernary2.Nats(x1[10])), java.lang.Object(x2[10]), java.lang.Object(x0[10])) at position [0] we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))

(19) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11]))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(20) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[11])), java.lang.Object(x3[11]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[11])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[11])), java.lang.Object(x1[11]), x2[11], java.lang.Object(x3[11])) at position [0] we obtained the following new rules [LPAR04]:

3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

(21) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1]))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(22) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[1], java.lang.Object(x2[1]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x2[1])), x1[1], java.lang.Object(DivTernary2.Nats(java.lang.Object(x3[1]))), java.lang.Object(x2[1])) at position [0] we obtained the following new rules [LPAR04]:

2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))

(23) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(24) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) → 2681_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))), x1[2], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[2])))) at position [0] we obtained the following new rules [LPAR04]:

2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))

(25) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12]))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(26) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[12])), java.lang.Object(x3[12]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[12])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[12])), java.lang.Object(x1[12]), x2[12], java.lang.Object(x3[12])) at position [0] we obtained the following new rules [LPAR04]:

3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

(27) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13]))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(28) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(x2[13])), java.lang.Object(x3[13]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x1[13])))) → 2255_1_DIV_INVOKEMETHOD(2255_0_isZero_Load(java.lang.Object(x3[13])), java.lang.Object(x1[13]), x2[13], java.lang.Object(x3[13])) at position [0] we obtained the following new rules [LPAR04]:

3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

(29) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2255_0_isZero_Load(java.lang.Object(x0)) → 2711_0_isZero_FieldAccess(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(30) 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.

(31) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2255_0_isZero_Load(java.lang.Object(x0))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(32) 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].

2255_0_isZero_Load(java.lang.Object(x0))

(33) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(34) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(x0), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(x0), java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)))

(35) Obligation:

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

2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(NULL)), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(36) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(37) Obligation:

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

3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(38) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))

(39) Obligation:

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

3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(40) DependencyGraphProof (EQUIVALENT transformation)

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

(41) Obligation:

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

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(42) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3191_0_DIV_EQ(java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(NULL)))

(43) Obligation:

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

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(NULL)))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(44) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(45) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(46) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))

(47) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(49) Obligation:

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

3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(50) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))

(51) Obligation:

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

3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(52) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(53) Obligation:

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

3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(54) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y1))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y1)))) at position [0] we obtained the following new rules [LPAR04]:

2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

(55) Obligation:

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

3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(56) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), java.lang.Object(y2), y0, java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))

(57) Obligation:

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

3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(58) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(59) Obligation:

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

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(60) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(x0), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2711_0_isZero_FieldAccess(java.lang.Object(x0)), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(x0)) at position [0] we obtained the following new rules [LPAR04]:

2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))

(61) Obligation:

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

2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2998_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(62) DependencyGraphProof (EQUIVALENT transformation)

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

(63) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

The TRS R consists of the following rules:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2922_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 2998_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2963_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3101_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2941_0_isZero_Return
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL))) → 3063_0_isZero_Return

The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(64) 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.

(65) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

R is empty.
The set Q consists of the following terms:

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

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

(66) 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].

2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2711_0_isZero_FieldAccess(java.lang.Object(DivTernary2.Nats(NULL)))

(67) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))

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

(68) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))

(69) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))

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

(70) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))

(71) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))

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

(72) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule 3191_0_DIV_EQ(y0, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) we obtained the following new rules [LPAR04]:

3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))

(73) Obligation:

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

2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))

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

(74) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 3 >= 3, 1 >= 4
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 1 >= 1, 4 > 2, 2 > 3, 3 >= 4
2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6])))) → 3191_0_DIV_EQ(x1[6], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[6]))), pos(01))
The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 2, 3 >= 3, 4 >= 3
2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, x1[8], java.lang.Object(x3[8]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8])))) → 3191_0_DIV_EQ(x1[8], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[8]))), java.lang.Object(x3[8]), pos(01))
The graph contains the following edges 2 >= 1, 4 >= 2, 3 >= 3
2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, x1[7], java.lang.Object(x3[7]), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7])))) → 3191_0_DIV_EQ(x1[7], java.lang.Object(DivTernary2.Nats(java.lang.Object(x2[7]))), java.lang.Object(x3[7]), pos(01))
The graph contains the following edges 2 >= 1, 4 >= 2, 3 >= 3
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 4 > 2, 2 > 3, 3 >= 4
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 3 >= 3, 1 >= 4
3191_0_DIV_EQ(java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), pos(01)) → 3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y1)), java.lang.Object(y2), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 3 >= 3, 1 >= 4
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 4 > 2, 2 > 3, 3 >= 4
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))
The graph contains the following edges 1 >= 2, 3 >= 3, 3 >= 4
2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, x1[5], java.lang.Object(x2[5]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[5], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[5]), pos(s(01)))
The graph contains the following edges 2 >= 1, 4 >= 2, 3 >= 3
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3101_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
2255_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, x1[4], java.lang.Object(x2[4]), java.lang.Object(DivTernary2.Nats(NULL))) → 3191_0_DIV_EQ(x1[4], java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(x2[4]), pos(s(01)))
The graph contains the following edges 2 >= 1, 4 >= 2, 3 >= 3
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(3063_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(NULL)))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 4 > 2, 2 > 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 4 > 2, 2 > 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 1 >= 1, 4 > 2, 2 > 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 4 > 2, 2 > 3, 3 >= 4
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))
The graph contains the following edges 1 >= 2, 3 >= 3, 3 >= 4
3191_0_DIV_EQ(z0, java.lang.Object(DivTernary2.Nats(NULL)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), pos(s(01))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, z0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x2))))
The graph contains the following edges 1 >= 2, 3 >= 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 4 > 2, 2 > 3, 3 >= 4
3240_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(DivTernary2.Nats(y0)), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, java.lang.Object(y2), y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 1 >= 1, 4 > 2, 2 > 3, 3 >= 4
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2922_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 4 >= 3, 3 >= 4
2255_1_DIV_INVOKEMETHOD(2963_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 2 >= 2, 3 >= 3, 4 >= 3, 3 >= 4, 4 >= 4
2255_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))), java.lang.Object(DivTernary2.Nats(java.lang.Object(y2)))) → 2681_1_DIV_INVOKEMETHOD(2941_0_isZero_Return, y0, java.lang.Object(DivTernary2.Nats(java.lang.Object(y2))), java.lang.Object(DivTernary2.Nats(java.lang.Object(x0))))
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3, 3 >= 4

(75) YES

(76) 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

The ITRS R consists of the following rules:
258_1_<init>_InvokeMethod(211_0_<init>_Return, 0) → 339_0_<init>_Return
258_1_<init>_InvokeMethod(339_0_<init>_Return, x0) → 339_0_<init>_Return

The integer pair graph contains the following rules and edges:
(0): 87_0_<INIT>_INVOKEMETHOD(x0[0]) → COND_87_0_<INIT>_INVOKEMETHOD(x0[0] > 0, x0[0])
(1): COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1]) → 87_0_<INIT>_INVOKEMETHOD(x0[1] - 1)

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

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

The set Q consists of the following terms:
258_1_<init>_InvokeMethod(211_0_<init>_Return, 0)
258_1_<init>_InvokeMethod(339_0_<init>_Return, x0)

(77) IDPNonInfProof (SOUND transformation)

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 87_0_<INIT>_INVOKEMETHOD(x0) → COND_87_0_<INIT>_INVOKEMETHOD(>(x0, 0), x0) the following chains were created:

We consider the chain 87_0_<INIT>_INVOKEMETHOD(x0[0]) → COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0]), COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1]) → 87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1)) which results in the following constraint:
(1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 87_0_<INIT>_INVOKEMETHOD(x0[0])≥NonInfC∧87_0_<INIT>_INVOKEMETHOD(x0[0])≥COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])∧(U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE ⇒ 87_0_<INIT>_INVOKEMETHOD(x0[0])≥NonInfC∧87_0_<INIT>_INVOKEMETHOD(x0[0])≥COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])∧(U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11 + (2)bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)

For Pair COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0) → 87_0_<INIT>_INVOKEMETHOD(-(x0, 1)) the following chains were created:

We consider the chain COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1]) → 87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1)) which results in the following constraint:
(7)    (COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1])≥NonInfC∧COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1])≥87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))∧(U^Increasing(87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))), ≥)∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))), ≥)∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))), ≥)∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_14] ≥ 0)

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

87_0_<INIT>_INVOKEMETHOD(x0) → COND_87_0_<INIT>_INVOKEMETHOD(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11 + (2)bni_11] + [(2)bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0) → 87_0_<INIT>_INVOKEMETHOD(-(x0, 1))
- ((U^Increasing(87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_14] ≥ 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(258_1_<init>_InvokeMethod(x₁, x₂)) = [-1]
POL(211_0_<init>_Return) = [-1]
POL(0) = 0
POL(339_0_<init>_Return) = [-1]
POL(87_0_<INIT>_INVOKEMETHOD(x₁)) = [2]x₁
POL(COND_87_0_<INIT>_INVOKEMETHOD(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1]) → 87_0_<INIT>_INVOKEMETHOD(-(x0[1], 1))

The following pairs are in P_bound:

87_0_<INIT>_INVOKEMETHOD(x0[0]) → COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])

The following pairs are in P_≥:

87_0_<INIT>_INVOKEMETHOD(x0[0]) → COND_87_0_<INIT>_INVOKEMETHOD(>(x0[0], 0), x0[0])

There are no usable rules.

(78) Complex Obligation (AND)

(79) 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

The ITRS R consists of the following rules:
258_1_<init>_InvokeMethod(211_0_<init>_Return, 0) → 339_0_<init>_Return
258_1_<init>_InvokeMethod(339_0_<init>_Return, x0) → 339_0_<init>_Return

The integer pair graph contains the following rules and edges:
(0): 87_0_<INIT>_INVOKEMETHOD(x0[0]) → COND_87_0_<INIT>_INVOKEMETHOD(x0[0] > 0, x0[0])

The set Q consists of the following terms:
258_1_<init>_InvokeMethod(211_0_<init>_Return, 0)
258_1_<init>_InvokeMethod(339_0_<init>_Return, x0)

(80) IDependencyGraphProof (EQUIVALENT transformation)

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

(81) TRUE

(82) 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

The ITRS R consists of the following rules:
258_1_<init>_InvokeMethod(211_0_<init>_Return, 0) → 339_0_<init>_Return
258_1_<init>_InvokeMethod(339_0_<init>_Return, x0) → 339_0_<init>_Return

The integer pair graph contains the following rules and edges:
(1): COND_87_0_<INIT>_INVOKEMETHOD(TRUE, x0[1]) → 87_0_<INIT>_INVOKEMETHOD(x0[1] - 1)

The set Q consists of the following terms:
258_1_<init>_InvokeMethod(211_0_<init>_Return, 0)
258_1_<init>_InvokeMethod(339_0_<init>_Return, x0)

(83) IDependencyGraphProof (EQUIVALENT transformation)

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