Termination proof of /tmp/tmpvRfNtN/Gopher.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPNonInfProof (⇒)

        ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDPtoQDPProof (⇒)

        ↳12 QDP

        ↳13 UsableRulesProof (⇔)

        ↳14 QDP

        ↳15 QReductionProof (⇔)

        ↳16 QDP

        ↳17 MRRProof (⇔)

        ↳18 QDP

        ↳19 PisEmptyProof (⇔)

        ↳20 YES

    ↳21 IDP

        ↳22 IDPNonInfProof (⇒)

        ↳23 AND

            ↳24 IDP

                ↳25 IDependencyGraphProof (⇔)

                ↳26 TRUE

            ↳27 IDP

                ↳28 IDependencyGraphProof (⇔)

                ↳29 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: Gopher/Gopher

package Gopher;
public class Gopher {
    // by Boyer & Moore (1979)

  public static Tree gopher(Tree x) {
    if (x == null) return null;
    if (x.left == null) return x;
    else return gopher(new Tree(x.left.left, new Tree(x.left.right, x.right)));
  }

  public static void main(String[] args) {
    Random.args = args;
    Tree x = Tree.createTree(Random.random());
    Tree y = gopher(x);
  }
}


package Gopher;

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

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


package Gopher;

public class Tree {
    Tree left;
    Tree right;
    int value;

    public static Tree createTree(final int numElements) {
        final Tree t = new Tree(Random.random());

        for (int i = 0; i < numElements; i++) {
            t.insert(Random.random());
        }

        return t;
    }

    public Tree(final Tree l, final Tree r) {
        this.left = l;
        this.right = r;
    }

    public Tree(final int val) {
        this.value = val;
    }

    public void insert(final int v) {
        if (v <= this.value) {
            if (this.left == null) {
                this.left = new Tree(v);
            } else {
                this.left.insert(v);
            }
        } else {
            if (this.right == null) {
                this.right = new Tree(v);
            } else {
                this.right.insert(v);
            }
        }
    }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Gopher.Gopher.main([Ljava/lang/String;)V: Graph of 110 nodes with 0 SCCs.

Gopher.Tree.createTree(I)LGopher/Tree;: Graph of 193 nodes with 1 SCC.

Gopher.Gopher.gopher(LGopher/Tree;)LGopher/Tree;: Graph of 56 nodes with 0 SCCs.

Gopher.Tree.insert(I)V: Graph of 72 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 30 rules for P and 51 rules for R.

Combined rules. Obtained 2 rules for P and 8 rules for R.

Filtered ground terms:

3430_0_insert_Load(x1, x2, x3, x4) → 3430_0_insert_Load(x2, x3, x4)
Cond_3430_0_insert_Load1(x1, x2, x3, x4, x5) → Cond_3430_0_insert_Load1(x1, x3, x4, x5)
Gopher.Tree(x1, x2, x3, x4) → Gopher.Tree(x2, x3, x4)
Cond_3430_0_insert_Load(x1, x2, x3, x4, x5) → Cond_3430_0_insert_Load(x1, x3, x4, x5)
4020_0_insert_Return(x1) → 4020_0_insert_Return
3964_0_insert_Return(x1) → 3964_0_insert_Return
3880_0_insert_Return(x1) → 3880_0_insert_Return
3877_0_insert_Return(x1) → 3877_0_insert_Return

Filtered duplicate args:

3430_0_insert_Load(x1, x2, x3) → 3430_0_insert_Load(x1, x3)
Cond_3430_0_insert_Load1(x1, x2, x3, x4) → Cond_3430_0_insert_Load1(x1, x2, x4)
Cond_3430_0_insert_Load(x1, x2, x3, x4) → Cond_3430_0_insert_Load(x1, x2, x4)

Filtered unneeded arguments:

3730_1_insert_InvokeMethod(x1, x2, x3) → 3730_1_insert_InvokeMethod(x1, x2)
3737_1_insert_InvokeMethod(x1, x2, x3) → 3737_1_insert_InvokeMethod(x1, x2)

Combined rules. Obtained 2 rules for P and 8 rules for R.

Finished conversion. Obtained 2 rules for P and 8 rules for R. System has predefined symbols.

Log for SCC 1:

Generated 43 rules for P and 12 rules for R.

Combined rules. Obtained 1 rules for P and 3 rules for R.

Filtered ground terms:

Gopher.Tree(x1, x2, x3) → Gopher.Tree(x2, x3)
3105_0_gopher_NONNULL(x1, x2, x3) → 3105_0_gopher_NONNULL(x2, x3)
4668_0_gopher_Return(x1, x2) → 4668_0_gopher_Return(x2)
4022_0_gopher_Return(x1, x2) → 4022_0_gopher_Return(x2)
3173_0_gopher_Return(x1, x2, x3) → 3173_0_gopher_Return(x2, x3)

Filtered duplicate args:

3105_0_gopher_NONNULL(x1, x2) → 3105_0_gopher_NONNULL(x2)
3173_0_gopher_Return(x1, x2) → 3173_0_gopher_Return(x2)

Finished conversion. Obtained 1 rules for P and 3 rules for R. System has no predefined symbols.

Log for SCC 2:

Generated 37 rules for P and 157 rules for R.

Combined rules. Obtained 6 rules for P and 13 rules for R.

Filtered ground terms:

4773_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 4773_1_createTree_InvokeMethod(x1, x2, x4)
Gopher.Tree(x1) → Gopher.Tree
4773_0_random_ArrayAccess(x1, x2, x3) → 4773_0_random_ArrayAccess(x2, x3)
Cond_4980_1_createTree_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod3(x1, x3, x5, x7)
4020_0_insert_Return(x1) → 4020_0_insert_Return
4980_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → 4980_1_createTree_InvokeMethod(x1, x2, x4, x6)
Cond_4980_1_createTree_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod2(x1, x3, x5, x7)
3964_0_insert_Return(x1) → 3964_0_insert_Return
Cond_4980_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod1(x1, x3, x5, x7)
3880_0_insert_Return(x1) → 3880_0_insert_Return
Cond_4980_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_4980_1_createTree_InvokeMethod(x1, x3, x5, x7)
3877_0_insert_Return(x1) → 3877_0_insert_Return
4980_0_insert_Load(x1, x2, x3) → 4980_0_insert_Load(x3)
Cond_4808_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_4808_1_createTree_InvokeMethod(x1, x2, x3, x5)
4808_0_random_IntArithmetic(x1, x2, x3, x4) → 4808_0_random_IntArithmetic(x2, x3)
4808_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 4808_1_createTree_InvokeMethod(x1, x2, x4)
Cond_4773_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_4773_1_createTree_InvokeMethod(x1, x2, x3, x5)
3517_0_insert_GT(x1, x2, x3, x4, x5) → 3517_0_insert_GT(x3, x4, x5)
Cond_3517_0_insert_GT3(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT3(x1, x4, x5, x6)
Cond_3517_0_insert_GT2(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT2(x1, x4, x5, x6)
Cond_3517_0_insert_GT1(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT1(x1, x4, x5, x6)
Cond_3517_0_insert_GT(x1, x2, x3, x4, x5, x6) → Cond_3517_0_insert_GT(x1, x4, x5, x6)

Filtered duplicate args:

3517_0_insert_GT(x1, x2, x3) → 3517_0_insert_GT(x2, x3)
Cond_3517_0_insert_GT3(x1, x2, x3, x4) → Cond_3517_0_insert_GT3(x1, x3, x4)
Cond_3517_0_insert_GT2(x1, x2, x3, x4) → Cond_3517_0_insert_GT2(x1, x3, x4)
Cond_3517_0_insert_GT1(x1, x2, x3, x4) → Cond_3517_0_insert_GT1(x1, x3, x4)
Cond_3517_0_insert_GT(x1, x2, x3, x4) → Cond_3517_0_insert_GT(x1, x3, x4)

Filtered unneeded arguments:

4980_1_createTree_InvokeMethod(x1, x2, x3, x4) → 4980_1_createTree_InvokeMethod(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod1(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod1(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod2(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod2(x1, x2, x3)
Cond_4980_1_createTree_InvokeMethod3(x1, x2, x3, x4) → Cond_4980_1_createTree_InvokeMethod3(x1, x2, x3)
Cond_3517_0_insert_GT(x1, x2, x3) → Cond_3517_0_insert_GT(x1)
Cond_3517_0_insert_GT1(x1, x2, x3) → Cond_3517_0_insert_GT1(x1, x2)
3737_1_insert_InvokeMethod(x1, x2, x3) → 3737_1_insert_InvokeMethod(x1, x2)
Cond_3517_0_insert_GT2(x1, x2, x3) → Cond_3517_0_insert_GT2(x1)
Cond_3517_0_insert_GT3(x1, x2, x3) → Cond_3517_0_insert_GT3(x1, x2)
3730_1_insert_InvokeMethod(x1, x2, x3) → 3730_1_insert_InvokeMethod(x1, x2)

Filtered all non-integer terms:

4808_0_random_IntArithmetic(x1, x2) → 4808_0_random_IntArithmetic(x2)
3737_1_insert_InvokeMethod(x1, x2) → 3737_1_insert_InvokeMethod(x1)
3730_1_insert_InvokeMethod(x1, x2) → 3730_1_insert_InvokeMethod(x1)

Filtered all free variables:

4808_1_createTree_InvokeMethod(x1, x2, x3) → 4808_1_createTree_InvokeMethod(x2, x3)
Cond_4808_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_4808_1_createTree_InvokeMethod(x1, x3, x4)
4980_1_createTree_InvokeMethod(x1, x2, x3) → 4980_1_createTree_InvokeMethod(x2, x3)
4773_1_createTree_InvokeMethod(x1, x2, x3) → 4773_1_createTree_InvokeMethod(x2, x3)
Cond_4773_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_4773_1_createTree_InvokeMethod(x1, x3, x4)
3517_0_insert_GT(x1, x2) → 3517_0_insert_GT(x1)

Combined rules. Obtained 1 rules for P and 13 rules for R.

Finished conversion. Obtained 1 rules for P and 13 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:

Integer

The ITRS R consists of the following rules:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(1): COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])
(2): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])
(3): COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])

(0) -> (1), if ((x3[0] > x0[0] →^* TRUE)∧(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])) →^* java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])))∧(x3[0] →^* x3[1]))

(1) -> (0), if ((java.lang.Object(x1[1]) →^* java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])))∧(x3[1] →^* x3[0]))

(1) -> (2), if ((java.lang.Object(x1[1]) →^* java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))))∧(x3[1] →^* x3[2]))

(2) -> (3), if ((x3[2] <= x0[2] →^* TRUE)∧(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))) →^* java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))))∧(x3[2] →^* x3[3]))

(3) -> (0), if ((java.lang.Object(x2[3]) →^* java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])))∧(x3[3] →^* x3[0]))

(3) -> (2), if ((java.lang.Object(x2[3]) →^* java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))))∧(x3[3] →^* x3[2]))

The set Q consists of the following terms:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))

(6) 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 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_3430_0_INSERT_LOAD(>(x3, x0), java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) the following chains were created:

We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]) which results in the following constraint:
(1)    (>(x3[0], x0[0])=TRUE∧java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1] ⇒ 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥NonInfC∧3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])∧(U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
(2)    (>(x3[0], x0[0])=TRUE ⇒ 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥NonInfC∧3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])∧(U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(2)bni_28]x2[0] + [bni_28]x1[0] + [bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(2)bni_28]x2[0] + [bni_28]x1[0] + [bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(2)bni_28]x2[0] + [bni_28]x1[0] + [bni_28]x0[0] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧0 ≥ 0∧[(2)bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28 + (-1)Bound*bni_28] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_29] ≥ 0)

For Pair COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → 3430_0_INSERT_LOAD(java.lang.Object(x1), x3) the following chains were created:

We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) which results in the following constraint:
(7)    (>(x3[0], x0[0])=TRUE∧java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1]∧java.lang.Object(x1[1])=java.lang.Object(Gopher.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1))∧x3[1]=x3[0]1 ⇒ COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥NonInfC∧COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥))

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:
(8)    (>(x3[0], x0[0])=TRUE ⇒ COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1)), x2[0])), x3[0])≥NonInfC∧COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1)), x2[0])), x3[0])≥3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1)), x3[0])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[0]1 + [bni_30]x1[0]1 + [bni_30]x0[0]1 + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[0]1 + [bni_30]x1[0]1 + [bni_30]x0[0]1 + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(2)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[0]1 + [bni_30]x1[0]1 + [bni_30]x0[0]1 + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(12)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(2)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)
We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) which results in the following constraint:
(13)    (>(x3[0], x0[0])=TRUE∧java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1]∧java.lang.Object(x1[1])=java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2])))∧x3[1]=x3[2] ⇒ COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥NonInfC∧COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥))

We simplified constraint (13) using rules (I), (II), (III), (IV) which results in the following new constraint:
(14)    (>(x3[0], x0[0])=TRUE ⇒ COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x2[0])), x3[0])≥NonInfC∧COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x2[0])), x3[0])≥3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[0])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[2] + [bni_30]x1[2] + [bni_30]x0[2] + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[2] + [bni_30]x1[2] + [bni_30]x0[2] + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧[(3)bni_30 + (-1)Bound*bni_30] + [(2)bni_30]x2[0] + [(2)bni_30]x2[2] + [bni_30]x1[2] + [bni_30]x0[2] + [bni_30]x0[0] ≥ 0∧[1 + (-1)bso_31] + [2]x2[0] + x0[0] ≥ 0)

We simplified constraint (17) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(18)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(3)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)

For Pair 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_3430_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) the following chains were created:

We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]) which results in the following constraint:
(19)    (<=(x3[2], x0[2])=TRUE∧java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3] ⇒ 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥NonInfC∧3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])∧(U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥))

We simplified constraint (19) using rules (I), (II), (IV) which results in the following new constraint:
(20)    (<=(x3[2], x0[2])=TRUE ⇒ 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥NonInfC∧3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])∧(U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧[(2)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]x2[2] + [bni_32]x1[2] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧[(2)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]x2[2] + [bni_32]x1[2] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧[(2)bni_32 + (-1)Bound*bni_32] + [(2)bni_32]x2[2] + [bni_32]x1[2] + [bni_32]x0[2] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧0 ≥ 0∧[(2)bni_32] ≥ 0∧[bni_32] ≥ 0∧[bni_32] ≥ 0∧[(2)bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

For Pair COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → 3430_0_INSERT_LOAD(java.lang.Object(x2), x3) the following chains were created:

We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) which results in the following constraint:
(25)    (<=(x3[2], x0[2])=TRUE∧java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3]∧java.lang.Object(x2[3])=java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))∧x3[3]=x3[0] ⇒ COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥NonInfC∧COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥))

We simplified constraint (25) using rules (I), (II), (III), (IV) which results in the following new constraint:
(26)    (<=(x3[2], x0[2])=TRUE ⇒ COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])))), x3[2])≥NonInfC∧COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])))), x3[2])≥3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[2])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(4)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[0] + [(2)bni_34]x1[0] + [(2)bni_34]x0[0] + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[3 + (-1)bso_35] + [2]x2[0] + x1[0] + x0[0] + x1[2] + x0[2] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(4)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[0] + [(2)bni_34]x1[0] + [(2)bni_34]x0[0] + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[3 + (-1)bso_35] + [2]x2[0] + x1[0] + x0[0] + x1[2] + x0[2] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(4)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[0] + [(2)bni_34]x1[0] + [(2)bni_34]x0[0] + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[3 + (-1)bso_35] + [2]x2[0] + x1[0] + x0[0] + x1[2] + x0[2] ≥ 0)

We simplified constraint (29) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(30)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(4)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[3 + (-1)bso_35] ≥ 0∧[1] ≥ 0)
We consider the chain 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) which results in the following constraint:
(31)    (<=(x3[2], x0[2])=TRUE∧java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3]∧java.lang.Object(x2[3])=java.lang.Object(Gopher.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1)))∧x3[3]=x3[2]1 ⇒ COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥NonInfC∧COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥))

We simplified constraint (31) using rules (I), (II), (III), (IV) which results in the following new constraint:
(32)    (<=(x3[2], x0[2])=TRUE ⇒ COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(Gopher.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1))))), x3[2])≥NonInfC∧COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(Gopher.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1))))), x3[2])≥3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1))), x3[2])∧(U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥))

We simplified constraint (32) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(33)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(6)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[2]1 + [(2)bni_34]x1[2]1 + [(2)bni_34]x0[2]1 + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[4 + (-1)bso_35] + [2]x2[2]1 + x1[2]1 + x0[2]1 + x1[2] + x0[2] ≥ 0)

We simplified constraint (33) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(34)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(6)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[2]1 + [(2)bni_34]x1[2]1 + [(2)bni_34]x0[2]1 + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[4 + (-1)bso_35] + [2]x2[2]1 + x1[2]1 + x0[2]1 + x1[2] + x0[2] ≥ 0)

We simplified constraint (34) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(35)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧[(6)bni_34 + (-1)Bound*bni_34] + [(4)bni_34]x2[2]1 + [(2)bni_34]x1[2]1 + [(2)bni_34]x0[2]1 + [bni_34]x1[2] + [bni_34]x0[2] ≥ 0∧[4 + (-1)bso_35] + [2]x2[2]1 + x1[2]1 + x0[2]1 + x1[2] + x0[2] ≥ 0)

We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(36)    (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(6)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[4 + (-1)bso_35] ≥ 0∧[1] ≥ 0)

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

3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_3430_0_INSERT_LOAD(>(x3, x0), java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3)
- (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])), ≥)∧0 ≥ 0∧[(2)bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28] ≥ 0∧[bni_28 + (-1)Bound*bni_28] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_29] ≥ 0)
COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0, java.lang.Object(x1), x2)), x3) → 3430_0_INSERT_LOAD(java.lang.Object(x1), x3)
- (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(2)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])), ≥)∧0 ≥ 0∧[(2)bni_30] ≥ 0∧[(2)bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[bni_30] ≥ 0∧[(3)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_31] ≥ 0∧[1] ≥ 0)
3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_3430_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3)
- (0 ≥ 0 ⇒ (U^Increasing(COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])), ≥)∧0 ≥ 0∧[(2)bni_32] ≥ 0∧[bni_32] ≥ 0∧[bni_32] ≥ 0∧[(2)bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)
COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0, x1, java.lang.Object(x2))), x3) → 3430_0_INSERT_LOAD(java.lang.Object(x2), x3)
- (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(4)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[3 + (-1)bso_35] ≥ 0∧[1] ≥ 0)
- (0 ≥ 0 ⇒ (U^Increasing(3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])), ≥)∧0 ≥ 0∧[(4)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[(2)bni_34] ≥ 0∧[bni_34] ≥ 0∧[bni_34] ≥ 0∧[(6)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[4 + (-1)bso_35] ≥ 0∧[1] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(3730_1_insert_InvokeMethod(x₁, x₂)) = 0
POL(3877_0_insert_Return) = 0
POL(java.lang.Object(x₁)) = [1] + x₁
POL(3964_0_insert_Return) = 0
POL(3880_0_insert_Return) = 0
POL(4020_0_insert_Return) = 0
POL(3737_1_insert_InvokeMethod(x₁, x₂)) = 0
POL(3430_0_INSERT_LOAD(x₁, x₂)) = [-1] + x₁
POL(Gopher.Tree(x₁, x₂, x₃)) = [2]x₃ + x₂ + x₁
POL(COND_3430_0_INSERT_LOAD(x₁, x₂, x₃)) = [-1] + x₂
POL(>(x₁, x₂)) = 0
POL(COND_3430_0_INSERT_LOAD1(x₁, x₂, x₃)) = [-1] + x₂
POL(<=(x₁, x₂)) = 0

The following pairs are in P_>:

COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])
COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])

The following pairs are in P_bound:

3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
COND_3430_0_INSERT_LOAD(TRUE, java.lang.Object(Gopher.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 3430_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])
3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])
COND_3430_0_INSERT_LOAD1(TRUE, java.lang.Object(Gopher.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 3430_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])

The following pairs are in P_≥:

3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])

There are no usable rules.

(7) 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:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 3964_0_insert_Return
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0)) → 4020_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_3430_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(2): 3430_0_INSERT_LOAD(java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_3430_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(Gopher.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])

The set Q consists of the following terms:
3730_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3730_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3880_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(3964_0_insert_Return, java.lang.Object(x0))
3737_1_insert_InvokeMethod(4020_0_insert_Return, java.lang.Object(x0))

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

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

The following domains are used:
none

The ITRS R consists of the following rules:
3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))) → 4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))

The integer pair graph contains the following rules and edges:
(0): 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

(0) -> (0), if ((java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))) →^* java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0]', x1[0]')), x2[0]'))))

The set Q consists of the following terms:
3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

(11) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(12) Obligation:

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

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

The TRS R consists of the following rules:

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))) → 4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7))))) → 4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2)))))))

The set Q consists of the following terms:

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

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

(13) 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.

(14) Obligation:

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

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

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

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

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

(15) 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].

3968_1_gopher_InvokeMethod(3173_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1))))), java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, x1)))))
3968_1_gopher_InvokeMethod(4022_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(NULL, x0)), java.lang.Object(Gopher.Tree(x1, x2)))))
3968_1_gopher_InvokeMethod(4668_0_gopher_Return(java.lang.Object(Gopher.Tree(NULL, java.lang.Object(Gopher.Tree(x0, java.lang.Object(Gopher.Tree(x1, x2))))))), java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x3, x4)), x5)), java.lang.Object(Gopher.Tree(x6, x7)))))

(16) Obligation:

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

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

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

(17) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(java.lang.Object(Gopher.Tree(x0[0], x1[0])), x2[0]))) → 3105_0_GOPHER_NONNULL(java.lang.Object(Gopher.Tree(x0[0], java.lang.Object(Gopher.Tree(x1[0], x2[0])))))

Used ordering: Polynomial interpretation [POLO]:

POL(3105_0_GOPHER_NONNULL(x₁)) = 2·x₁
POL(Gopher.Tree(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(java.lang.Object(x₁)) = 2 + x₁

(18) Obligation:

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

(19) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(20) YES

(21) Obligation:

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

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
4980_0_insert_Load(x0) → 3517_0_insert_GT(x0)
3517_0_insert_GT(x0) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3737_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3964_0_insert_Return
3517_0_insert_GT(x0) → 3730_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3730_1_insert_InvokeMethod(3877_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return) → 3964_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])
(1): COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], x1[1] + 1)

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

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

The set Q consists of the following terms:
4980_0_insert_Load(x0)
3517_0_insert_GT(x0)
3737_1_insert_InvokeMethod(3877_0_insert_Return)
3737_1_insert_InvokeMethod(3880_0_insert_Return)
3737_1_insert_InvokeMethod(3964_0_insert_Return)
3737_1_insert_InvokeMethod(4020_0_insert_Return)
3730_1_insert_InvokeMethod(3877_0_insert_Return)
3730_1_insert_InvokeMethod(3880_0_insert_Return)
3730_1_insert_InvokeMethod(3964_0_insert_Return)
3730_1_insert_InvokeMethod(4020_0_insert_Return)

(22) 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 4773_1_CREATETREE_INVOKEMETHOD(x0, x1) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1) the following chains were created:

We consider the chain 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0]), COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1)) which results in the following constraint:
(1)    (&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1)))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x1[0], 0)=TRUE∧>(x0[0], +(x1[0], 1))=TRUE ⇒ 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(-1)bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0, x1) → 4773_1_CREATETREE_INVOKEMETHOD(x0, +(x1, 1)) the following chains were created:

We consider the chain COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1)) which results in the following constraint:
(7)    (COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1])≥NonInfC∧COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1])≥4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))∧(U^Increasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

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

4773_1_CREATETREE_INVOKEMETHOD(x0, x1) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])), ≥)∧[(3)bni_18 + (-1)Bound*bni_18] + [bni_18]x1[0] + [(2)bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)
COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0, x1) → 4773_1_CREATETREE_INVOKEMETHOD(x0, +(x1, 1))
- ((U^Increasing(4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(4980_0_insert_Load(x₁)) = [-1]
POL(3517_0_insert_GT(x₁)) = [-1]
POL(4020_0_insert_Return) = [-1]
POL(3737_1_insert_InvokeMethod(x₁)) = [-1]
POL(3877_0_insert_Return) = [-1]
POL(3880_0_insert_Return) = [-1]
POL(3964_0_insert_Return) = [-1]
POL(3730_1_insert_InvokeMethod(x₁)) = [-1]
POL(4773_1_CREATETREE_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂ + [2]x₁
POL(COND_4773_1_CREATETREE_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(>(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))

The following pairs are in P_bound:

4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

The following pairs are in P_≥:

4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

There are no usable rules.

(23) Complex Obligation (AND)

(24) Obligation:

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

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
4980_0_insert_Load(x0) → 3517_0_insert_GT(x0)
3517_0_insert_GT(x0) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3737_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3964_0_insert_Return
3517_0_insert_GT(x0) → 3730_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3730_1_insert_InvokeMethod(3877_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return) → 3964_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 4773_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_4773_1_CREATETREE_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])

The set Q consists of the following terms:
4980_0_insert_Load(x0)
3517_0_insert_GT(x0)
3737_1_insert_InvokeMethod(3877_0_insert_Return)
3737_1_insert_InvokeMethod(3880_0_insert_Return)
3737_1_insert_InvokeMethod(3964_0_insert_Return)
3737_1_insert_InvokeMethod(4020_0_insert_Return)
3730_1_insert_InvokeMethod(3877_0_insert_Return)
3730_1_insert_InvokeMethod(3880_0_insert_Return)
3730_1_insert_InvokeMethod(3964_0_insert_Return)
3730_1_insert_InvokeMethod(4020_0_insert_Return)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE

(27) 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:
4980_0_insert_Load(x0) → 3517_0_insert_GT(x0)
3517_0_insert_GT(x0) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3737_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3737_1_insert_InvokeMethod(3877_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3880_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(3964_0_insert_Return) → 4020_0_insert_Return
3737_1_insert_InvokeMethod(4020_0_insert_Return) → 4020_0_insert_Return
3517_0_insert_GT(x0) → 3964_0_insert_Return
3517_0_insert_GT(x0) → 3730_1_insert_InvokeMethod(3517_0_insert_GT(x0))
3730_1_insert_InvokeMethod(3877_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3880_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(3964_0_insert_Return) → 3964_0_insert_Return
3730_1_insert_InvokeMethod(4020_0_insert_Return) → 3964_0_insert_Return

The integer pair graph contains the following rules and edges:
(1): COND_4773_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 4773_1_CREATETREE_INVOKEMETHOD(x0[1], x1[1] + 1)

The set Q consists of the following terms:
4980_0_insert_Load(x0)
3517_0_insert_GT(x0)
3737_1_insert_InvokeMethod(3877_0_insert_Return)
3737_1_insert_InvokeMethod(3880_0_insert_Return)
3737_1_insert_InvokeMethod(3964_0_insert_Return)
3737_1_insert_InvokeMethod(4020_0_insert_Return)
3730_1_insert_InvokeMethod(3877_0_insert_Return)
3730_1_insert_InvokeMethod(3880_0_insert_Return)
3730_1_insert_InvokeMethod(3964_0_insert_Return)
3730_1_insert_InvokeMethod(4020_0_insert_Return)

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Obligation:

(8) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDPtoQDPProof (SOUND transformation)

(12) Obligation:

(13) UsableRulesProof (EQUIVALENT transformation)

(14) Obligation:

(15) QReductionProof (EQUIVALENT transformation)

(16) Obligation:

(17) MRRProof (EQUIVALENT transformation)

(18) Obligation:

(19) PisEmptyProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) IDPNonInfProof (SOUND transformation)

(23) Complex Obligation (AND)

(24) Obligation:

(25) IDependencyGraphProof (EQUIVALENT transformation)

(26) TRUE

(27) Obligation:

(28) IDependencyGraphProof (EQUIVALENT transformation)

(29) TRUE