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 DependencyGraphProof (⇔)
        ↳14 QDP
        ↳15 UsableRulesProof (⇔)
        ↳16 QDP
        ↳17 QReductionProof (⇔)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳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: BinTreeChanger/BinTreeChanger 
package BinTreeChanger;

public class BinTreeChanger {
  public static void main(final String[] args) {
    Random.args = args;

    final Tree t = Tree.createTree(Random.random());

    t.applyTreeChanger(new AnswerAdder());
  }
}

class AnswerAdder implements TreeChanger {
    public void change(final Tree t) {
        final Tree newLeft = new Tree(42);
        final Tree newRight = new Tree(42);
        newLeft.left = t.left;
        newRight.right = t.right;
        t.left = newLeft;
        t.right = newRight;
    }
}


package BinTreeChanger;

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

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


package BinTreeChanger;

public interface TreeChanger {
    public void change(Tree t);
}


package BinTreeChanger;

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 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);
            }
        }
    }

    public void applyTreeChanger(final TreeChanger c) {
        final Tree oldLeft = this.left;
        final Tree oldRight = this.right;
        c.change(this);
        if (oldLeft != null) {
            oldLeft.applyTreeChanger(c);
        }
        if (oldRight != null) {
            oldRight.applyTreeChanger(c);
        }
    }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
BinTreeChanger.BinTreeChanger.main([Ljava/lang/String;)V: Graph of 116 nodes with 0 SCCs.

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

BinTreeChanger.Tree.applyTreeChanger(LBinTreeChanger/TreeChanger;)V: Graph of 100 nodes with 0 SCCs.

BinTreeChanger.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:


2601_0_insert_Load(x1, x2, x3, x4) → 2601_0_insert_Load(x2, x3, x4)
Cond_2601_0_insert_Load1(x1, x2, x3, x4, x5) → Cond_2601_0_insert_Load1(x1, x3, x4, x5)
BinTreeChanger.Tree(x1, x2, x3, x4) → BinTreeChanger.Tree(x2, x3, x4)
Cond_2601_0_insert_Load(x1, x2, x3, x4, x5) → Cond_2601_0_insert_Load(x1, x3, x4, x5)
2874_0_insert_Return(x1) → 2874_0_insert_Return
2853_0_insert_Return(x1) → 2853_0_insert_Return
2816_0_insert_Return(x1) → 2816_0_insert_Return
2815_0_insert_Return(x1) → 2815_0_insert_Return

Filtered duplicate args:


2601_0_insert_Load(x1, x2, x3) → 2601_0_insert_Load(x1, x3)
Cond_2601_0_insert_Load1(x1, x2, x3, x4) → Cond_2601_0_insert_Load1(x1, x2, x4)
Cond_2601_0_insert_Load(x1, x2, x3, x4) → Cond_2601_0_insert_Load(x1, x2, x4)

Filtered unneeded arguments:


2749_1_insert_InvokeMethod(x1, x2, x3) → 2749_1_insert_InvokeMethod(x1, x2)
2754_1_insert_InvokeMethod(x1, x2, x3) → 2754_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 86 rules for P and 24 rules for R.


Combined rules. Obtained 6 rules for P and 8 rules for R.


Filtered ground terms:


2509_0_applyTreeChanger_FieldAccess(x1, x2, x3) → 2509_0_applyTreeChanger_FieldAccess(x2, x3)
BinTreeChanger.Tree(x1, x2, x3) → BinTreeChanger.Tree(x2, x3)
3028_0_applyTreeChanger_Return(x1) → 3028_0_applyTreeChanger_Return
4463_0_applyTreeChanger_Return(x1) → 4463_0_applyTreeChanger_Return
3924_0_applyTreeChanger_Return(x1) → 3924_0_applyTreeChanger_Return
3721_0_applyTreeChanger_Return(x1) → 3721_0_applyTreeChanger_Return

Filtered duplicate args:


2509_0_applyTreeChanger_FieldAccess(x1, x2) → 2509_0_applyTreeChanger_FieldAccess(x2)

Finished conversion. Obtained 6 rules for P and 8 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:


2935_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 2935_1_createTree_InvokeMethod(x1, x2, x4)
BinTreeChanger.Tree(x1) → BinTreeChanger.Tree
2935_0_random_ArrayAccess(x1, x2, x3) → 2935_0_random_ArrayAccess(x2, x3)
Cond_3247_1_createTree_InvokeMethod3(x1, x2, x3, x4, x5, x6, x7) → Cond_3247_1_createTree_InvokeMethod3(x1, x3, x5, x7)
2874_0_insert_Return(x1) → 2874_0_insert_Return
3247_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → 3247_1_createTree_InvokeMethod(x1, x2, x4, x6)
Cond_3247_1_createTree_InvokeMethod2(x1, x2, x3, x4, x5, x6, x7) → Cond_3247_1_createTree_InvokeMethod2(x1, x3, x5, x7)
2853_0_insert_Return(x1) → 2853_0_insert_Return
Cond_3247_1_createTree_InvokeMethod1(x1, x2, x3, x4, x5, x6, x7) → Cond_3247_1_createTree_InvokeMethod1(x1, x3, x5, x7)
2816_0_insert_Return(x1) → 2816_0_insert_Return
Cond_3247_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6, x7) → Cond_3247_1_createTree_InvokeMethod(x1, x3, x5, x7)
2815_0_insert_Return(x1) → 2815_0_insert_Return
3247_0_insert_Load(x1, x2, x3) → 3247_0_insert_Load(x3)
Cond_2967_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2967_1_createTree_InvokeMethod(x1, x2, x3, x5)
2967_0_random_IntArithmetic(x1, x2, x3, x4) → 2967_0_random_IntArithmetic(x2, x3)
2967_1_createTree_InvokeMethod(x1, x2, x3, x4, x5) → 2967_1_createTree_InvokeMethod(x1, x2, x4)
Cond_2935_1_createTree_InvokeMethod(x1, x2, x3, x4, x5, x6) → Cond_2935_1_createTree_InvokeMethod(x1, x2, x3, x5)
2638_0_insert_GT(x1, x2, x3, x4, x5) → 2638_0_insert_GT(x3, x4, x5)
Cond_2638_0_insert_GT3(x1, x2, x3, x4, x5, x6) → Cond_2638_0_insert_GT3(x1, x4, x5, x6)
Cond_2638_0_insert_GT2(x1, x2, x3, x4, x5, x6) → Cond_2638_0_insert_GT2(x1, x4, x5, x6)
Cond_2638_0_insert_GT1(x1, x2, x3, x4, x5, x6) → Cond_2638_0_insert_GT1(x1, x4, x5, x6)
Cond_2638_0_insert_GT(x1, x2, x3, x4, x5, x6) → Cond_2638_0_insert_GT(x1, x4, x5, x6)

Filtered duplicate args:


2638_0_insert_GT(x1, x2, x3) → 2638_0_insert_GT(x2, x3)
Cond_2638_0_insert_GT3(x1, x2, x3, x4) → Cond_2638_0_insert_GT3(x1, x3, x4)
Cond_2638_0_insert_GT2(x1, x2, x3, x4) → Cond_2638_0_insert_GT2(x1, x3, x4)
Cond_2638_0_insert_GT1(x1, x2, x3, x4) → Cond_2638_0_insert_GT1(x1, x3, x4)
Cond_2638_0_insert_GT(x1, x2, x3, x4) → Cond_2638_0_insert_GT(x1, x3, x4)

Filtered unneeded arguments:


3247_1_createTree_InvokeMethod(x1, x2, x3, x4) → 3247_1_createTree_InvokeMethod(x1, x2, x3)
Cond_3247_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_3247_1_createTree_InvokeMethod(x1, x2, x3)
Cond_3247_1_createTree_InvokeMethod1(x1, x2, x3, x4) → Cond_3247_1_createTree_InvokeMethod1(x1, x2, x3)
Cond_3247_1_createTree_InvokeMethod2(x1, x2, x3, x4) → Cond_3247_1_createTree_InvokeMethod2(x1, x2, x3)
Cond_3247_1_createTree_InvokeMethod3(x1, x2, x3, x4) → Cond_3247_1_createTree_InvokeMethod3(x1, x2, x3)
Cond_2638_0_insert_GT(x1, x2, x3) → Cond_2638_0_insert_GT(x1)
Cond_2638_0_insert_GT1(x1, x2, x3) → Cond_2638_0_insert_GT1(x1, x2)
2754_1_insert_InvokeMethod(x1, x2, x3) → 2754_1_insert_InvokeMethod(x1, x2)
Cond_2638_0_insert_GT2(x1, x2, x3) → Cond_2638_0_insert_GT2(x1)
Cond_2638_0_insert_GT3(x1, x2, x3) → Cond_2638_0_insert_GT3(x1, x2)
2749_1_insert_InvokeMethod(x1, x2, x3) → 2749_1_insert_InvokeMethod(x1, x2)

Filtered all non-integer terms:


2967_0_random_IntArithmetic(x1, x2) → 2967_0_random_IntArithmetic(x2)
2754_1_insert_InvokeMethod(x1, x2) → 2754_1_insert_InvokeMethod(x1)
2749_1_insert_InvokeMethod(x1, x2) → 2749_1_insert_InvokeMethod(x1)

Filtered all free variables:


2967_1_createTree_InvokeMethod(x1, x2, x3) → 2967_1_createTree_InvokeMethod(x2, x3)
Cond_2967_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_2967_1_createTree_InvokeMethod(x1, x3, x4)
3247_1_createTree_InvokeMethod(x1, x2, x3) → 3247_1_createTree_InvokeMethod(x2, x3)
2935_1_createTree_InvokeMethod(x1, x2, x3) → 2935_1_createTree_InvokeMethod(x2, x3)
Cond_2935_1_createTree_InvokeMethod(x1, x2, x3, x4) → Cond_2935_1_createTree_InvokeMethod(x1, x3, x4)
2638_0_insert_GT(x1, x2) → 2638_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:
2749_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2874_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2754_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2874_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(1): COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])
(2): 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])
(3): COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])

(0) -> (1), if ((x3[0] > x0[0] →* TRUE)∧(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])) →* java.lang.Object(BinTreeChanger.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(BinTreeChanger.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(BinTreeChanger.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(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))) →* java.lang.Object(BinTreeChanger.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(BinTreeChanger.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(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))))∧(x3[3] →* x3[2]))



The set Q consists of the following terms:
2749_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0))
2749_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0))
2749_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0))
2749_1_insert_InvokeMethod(2874_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2874_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 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_2601_0_INSERT_LOAD(>(x3, x0), java.lang.Object(BinTreeChanger.Tree(x0, java.lang.Object(x1), x2)), x3) the following chains were created:
  • We consider the chain 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 2601_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(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1] ⇒ 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥NonInfC∧2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])∧(UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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 ⇒ 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥NonInfC∧2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])≥COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])∧(UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0, java.lang.Object(x1), x2)), x3) → 2601_0_INSERT_LOAD(java.lang.Object(x1), x3) the following chains were created:
  • We consider the chain 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1]∧java.lang.Object(x1[1])=java.lang.Object(BinTreeChanger.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1))∧x3[1]=x3[0]1 ⇒ COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥NonInfC∧COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])∧(UIncreasing(2601_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_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(BinTreeChanger.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1)), x2[0])), x3[0])≥NonInfC∧COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(BinTreeChanger.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1)), x2[0])), x3[0])≥2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0]1, java.lang.Object(x1[0]1), x2[0]1)), x3[0])∧(UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]), COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1]), 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))=java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1]))∧x3[0]=x3[1]∧java.lang.Object(x1[1])=java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2])))∧x3[1]=x3[2] ⇒ COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥NonInfC∧COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1])≥2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])∧(UIncreasing(2601_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_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x2[0])), x3[0])≥NonInfC∧COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x2[0])), x3[0])≥2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[0])∧(UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_2601_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(BinTreeChanger.Tree(x0, x1, java.lang.Object(x2))), x3) the following chains were created:
  • We consider the chain 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 2601_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(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3] ⇒ 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥NonInfC∧2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])∧(UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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 ⇒ 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥NonInfC∧2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])≥COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])∧(UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0, x1, java.lang.Object(x2))), x3) → 2601_0_INSERT_LOAD(java.lang.Object(x2), x3) the following chains were created:
  • We consider the chain 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3]∧java.lang.Object(x2[3])=java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0]))∧x3[3]=x3[0] ⇒ COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥NonInfC∧COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])∧(UIncreasing(2601_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_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])))), x3[2])≥NonInfC∧COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])))), x3[2])≥2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[2])∧(UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]), COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3]), 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2])))=java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3])))∧x3[2]=x3[3]∧java.lang.Object(x2[3])=java.lang.Object(BinTreeChanger.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1)))∧x3[3]=x3[2]1 ⇒ COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥NonInfC∧COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3])≥2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])∧(UIncreasing(2601_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_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(BinTreeChanger.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1))))), x3[2])≥NonInfC∧COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(BinTreeChanger.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1))))), x3[2])≥2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2]1, x1[2]1, java.lang.Object(x2[2]1))), x3[2])∧(UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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.
  • 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0, java.lang.Object(x1), x2)), x3) → COND_2601_0_INSERT_LOAD(>(x3, x0), java.lang.Object(BinTreeChanger.Tree(x0, java.lang.Object(x1), x2)), x3)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.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_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0, java.lang.Object(x1), x2)), x3) → 2601_0_INSERT_LOAD(java.lang.Object(x1), x3)
    • (0 ≥ 0 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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)

  • 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0, x1, java.lang.Object(x2))), x3) → COND_2601_0_INSERT_LOAD1(<=(x3, x0), java.lang.Object(BinTreeChanger.Tree(x0, x1, java.lang.Object(x2))), x3)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0, x1, java.lang.Object(x2))), x3) → 2601_0_INSERT_LOAD(java.lang.Object(x2), x3)
    • (0 ≥ 0 ⇒ (UIncreasing(2601_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 ⇒ (UIncreasing(2601_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 Pbound 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 Pbound.
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(2749_1_insert_InvokeMethod(x1, x2)) = 0   
POL(2815_0_insert_Return) = 0   
POL(java.lang.Object(x1)) = [1] + x1   
POL(2853_0_insert_Return) = 0   
POL(2816_0_insert_Return) = 0   
POL(2874_0_insert_Return) = 0   
POL(2754_1_insert_InvokeMethod(x1, x2)) = 0   
POL(2601_0_INSERT_LOAD(x1, x2)) = [-1] + x1   
POL(BinTreeChanger.Tree(x1, x2, x3)) = [2]x3 + x2 + x1   
POL(COND_2601_0_INSERT_LOAD(x1, x2, x3)) = [-1] + x2   
POL(>(x1, x2)) = 0   
POL(COND_2601_0_INSERT_LOAD1(x1, x2, x3)) = [-1] + x2   
POL(<=(x1, x2)) = 0   

The following pairs are in P>:

COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])
COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])

The following pairs are in Pbound:

2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
COND_2601_0_INSERT_LOAD(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[1], java.lang.Object(x1[1]), x2[1])), x3[1]) → 2601_0_INSERT_LOAD(java.lang.Object(x1[1]), x3[1])
2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])
COND_2601_0_INSERT_LOAD1(TRUE, java.lang.Object(BinTreeChanger.Tree(x0[3], x1[3], java.lang.Object(x2[3]))), x3[3]) → 2601_0_INSERT_LOAD(java.lang.Object(x2[3]), x3[3])

The following pairs are in P≥:

2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(>(x3[0], x0[0]), java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(<=(x3[2], x0[2]), java.lang.Object(BinTreeChanger.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:
2749_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2874_0_insert_Return, java.lang.Object(x0)) → 2853_0_insert_Return
2754_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2874_0_insert_Return, java.lang.Object(x0)) → 2874_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0]) → COND_2601_0_INSERT_LOAD(x3[0] > x0[0], java.lang.Object(BinTreeChanger.Tree(x0[0], java.lang.Object(x1[0]), x2[0])), x3[0])
(2): 2601_0_INSERT_LOAD(java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2]) → COND_2601_0_INSERT_LOAD1(x3[2] <= x0[2], java.lang.Object(BinTreeChanger.Tree(x0[2], x1[2], java.lang.Object(x2[2]))), x3[2])


The set Q consists of the following terms:
2749_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0))
2749_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0))
2749_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0))
2749_1_insert_InvokeMethod(2874_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2815_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2816_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2853_0_insert_Return, java.lang.Object(x0))
2754_1_insert_InvokeMethod(2874_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:
4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return

The integer pair graph contains the following rules and edges:
(0): 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))) → 3021_1_APPLYTREECHANGER_INVOKEMETHOD(2509_0_applyTreeChanger_FieldAccess(java.lang.Object(x0[0])), x1[0], java.lang.Object(x0[0]))
(1): 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[1]))
(2): 3021_1_APPLYTREECHANGER_INVOKEMETHOD(3721_0_applyTreeChanger_Return, java.lang.Object(x0[2]), java.lang.Object(x1[2])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[2]))
(3): 3021_1_APPLYTREECHANGER_INVOKEMETHOD(3924_0_applyTreeChanger_Return, java.lang.Object(x0[3]), java.lang.Object(x1[3])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[3]))
(4): 3021_1_APPLYTREECHANGER_INVOKEMETHOD(4463_0_applyTreeChanger_Return, java.lang.Object(x0[4]), java.lang.Object(x1[4])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[4]))
(5): 3021_1_APPLYTREECHANGER_INVOKEMETHOD(3028_0_applyTreeChanger_Return, java.lang.Object(x0[5]), java.lang.Object(x1[5])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[5]))
(6): 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[6]))

(0) -> (2), if ((2509_0_applyTreeChanger_FieldAccess(java.lang.Object(x0[0])) →* 3721_0_applyTreeChanger_Return)∧(x1[0] →* java.lang.Object(x0[2]))∧(java.lang.Object(x0[0]) →* java.lang.Object(x1[2])))


(0) -> (3), if ((2509_0_applyTreeChanger_FieldAccess(java.lang.Object(x0[0])) →* 3924_0_applyTreeChanger_Return)∧(x1[0] →* java.lang.Object(x0[3]))∧(java.lang.Object(x0[0]) →* java.lang.Object(x1[3])))


(0) -> (4), if ((2509_0_applyTreeChanger_FieldAccess(java.lang.Object(x0[0])) →* 4463_0_applyTreeChanger_Return)∧(x1[0] →* java.lang.Object(x0[4]))∧(java.lang.Object(x0[0]) →* java.lang.Object(x1[4])))


(0) -> (5), if ((2509_0_applyTreeChanger_FieldAccess(java.lang.Object(x0[0])) →* 3028_0_applyTreeChanger_Return)∧(x1[0] →* java.lang.Object(x0[5]))∧(java.lang.Object(x0[0]) →* java.lang.Object(x1[5])))


(1) -> (0), if ((java.lang.Object(x0[1]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))))


(1) -> (1), if ((java.lang.Object(x0[1]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]'), x1[1]'))))


(1) -> (6), if ((java.lang.Object(x0[1]) →* java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))))


(2) -> (0), if ((java.lang.Object(x0[2]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))))


(2) -> (1), if ((java.lang.Object(x0[2]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))))


(2) -> (6), if ((java.lang.Object(x0[2]) →* java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))))


(3) -> (0), if ((java.lang.Object(x0[3]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))))


(3) -> (1), if ((java.lang.Object(x0[3]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))))


(3) -> (6), if ((java.lang.Object(x0[3]) →* java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))))


(4) -> (0), if ((java.lang.Object(x0[4]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))))


(4) -> (1), if ((java.lang.Object(x0[4]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))))


(4) -> (6), if ((java.lang.Object(x0[4]) →* java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))))


(5) -> (0), if ((java.lang.Object(x0[5]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))))


(5) -> (1), if ((java.lang.Object(x0[5]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))))


(5) -> (6), if ((java.lang.Object(x0[5]) →* java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))))


(6) -> (0), if ((java.lang.Object(x0[6]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))))


(6) -> (1), if ((java.lang.Object(x0[6]) →* java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))))


(6) -> (6), if ((java.lang.Object(x0[6]) →* java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6]')))))



The set Q consists of the following terms:
4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))

(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:

2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[0]), x1[0]))) → 3021_1_APPLYTREECHANGER_INVOKEMETHOD(2509_0_applyTreeChanger_FieldAccess(java.lang.Object(x0[0])), x1[0], java.lang.Object(x0[0]))
2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[1]))
3021_1_APPLYTREECHANGER_INVOKEMETHOD(3721_0_applyTreeChanger_Return, java.lang.Object(x0[2]), java.lang.Object(x1[2])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[2]))
3021_1_APPLYTREECHANGER_INVOKEMETHOD(3924_0_applyTreeChanger_Return, java.lang.Object(x0[3]), java.lang.Object(x1[3])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[3]))
3021_1_APPLYTREECHANGER_INVOKEMETHOD(4463_0_applyTreeChanger_Return, java.lang.Object(x0[4]), java.lang.Object(x1[4])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[4]))
3021_1_APPLYTREECHANGER_INVOKEMETHOD(3028_0_applyTreeChanger_Return, java.lang.Object(x0[5]), java.lang.Object(x1[5])) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[5]))
2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[6]))

The TRS R consists of the following rules:

4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return

The set Q consists of the following terms:

4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))

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

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(14) Obligation:

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

2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[1]))
2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[6]))

The TRS R consists of the following rules:

4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0)) → 4463_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0)) → 3721_0_applyTreeChanger_Return

The set Q consists of the following terms:

4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))

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

(15) 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.

(16) Obligation:

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

2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[1]))
2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[6]))

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

4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))

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

(17) 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].

4106_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
4106_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3028_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3721_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(3924_0_applyTreeChanger_Return, java.lang.Object(x0))
3105_1_applyTreeChanger_InvokeMethod(4463_0_applyTreeChanger_Return, java.lang.Object(x0))

(18) Obligation:

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

2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[1]))
2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[6]))

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

(19) 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:

  • 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(java.lang.Object(x0[1]), x1[1]))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[1]))
    The graph contains the following edges 1 > 1

  • 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(BinTreeChanger.Tree(NULL, java.lang.Object(x0[6])))) → 2509_0_APPLYTREECHANGER_FIELDACCESS(java.lang.Object(x0[6]))
    The graph contains the following edges 1 > 1

(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:
3247_0_insert_Load(x0) → 2638_0_insert_GT(x0)
2638_0_insert_GT(x0) → 2874_0_insert_Return
2638_0_insert_GT(x0) → 2754_1_insert_InvokeMethod(2638_0_insert_GT(x0))
2754_1_insert_InvokeMethod(2815_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2816_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2853_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2874_0_insert_Return) → 2874_0_insert_Return
2638_0_insert_GT(x0) → 2853_0_insert_Return
2638_0_insert_GT(x0) → 2749_1_insert_InvokeMethod(2638_0_insert_GT(x0))
2749_1_insert_InvokeMethod(2815_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2816_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2853_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2874_0_insert_Return) → 2853_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_2935_1_CREATETREE_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])
(1): COND_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 2935_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:
3247_0_insert_Load(x0)
2638_0_insert_GT(x0)
2754_1_insert_InvokeMethod(2815_0_insert_Return)
2754_1_insert_InvokeMethod(2816_0_insert_Return)
2754_1_insert_InvokeMethod(2853_0_insert_Return)
2754_1_insert_InvokeMethod(2874_0_insert_Return)
2749_1_insert_InvokeMethod(2815_0_insert_Return)
2749_1_insert_InvokeMethod(2816_0_insert_Return)
2749_1_insert_InvokeMethod(2853_0_insert_Return)
2749_1_insert_InvokeMethod(2874_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 2935_1_CREATETREE_INVOKEMETHOD(x0, x1) → COND_2935_1_CREATETREE_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1) the following chains were created:
  • We consider the chain 2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_2935_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0]), COND_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 2935_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] ⇒ 2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥COND_2935_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(UIncreasing(COND_2935_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 ⇒ 2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥NonInfC∧2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0])≥COND_2935_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])∧(UIncreasing(COND_2935_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 ⇒ (UIncreasing(COND_2935_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 ⇒ (UIncreasing(COND_2935_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 ⇒ (UIncreasing(COND_2935_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 ⇒ (UIncreasing(COND_2935_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_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0, x1) → 2935_1_CREATETREE_INVOKEMETHOD(x0, +(x1, 1)) the following chains were created:
  • We consider the chain COND_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 2935_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1)) which results in the following constraint:

    (7)    (COND_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1])≥NonInfC∧COND_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1])≥2935_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))∧(UIncreasing(2935_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥))



    We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (8)    ((UIncreasing(2935_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)    ((UIncreasing(2935_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)    ((UIncreasing(2935_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)    ((UIncreasing(2935_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.
  • 2935_1_CREATETREE_INVOKEMETHOD(x0, x1) → COND_2935_1_CREATETREE_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), x0, x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_2935_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_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0, x1) → 2935_1_CREATETREE_INVOKEMETHOD(x0, +(x1, 1))
    • ((UIncreasing(2935_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(3247_0_insert_Load(x1)) = [-1]   
POL(2638_0_insert_GT(x1)) = [-1]   
POL(2874_0_insert_Return) = [-1]   
POL(2754_1_insert_InvokeMethod(x1)) = [-1]   
POL(2815_0_insert_Return) = [-1]   
POL(2816_0_insert_Return) = [-1]   
POL(2853_0_insert_Return) = [-1]   
POL(2749_1_insert_InvokeMethod(x1)) = [-1]   
POL(2935_1_CREATETREE_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [2]x1   
POL(COND_2935_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_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 2935_1_CREATETREE_INVOKEMETHOD(x0[1], +(x1[1], 1))

The following pairs are in Pbound:

2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_2935_1_CREATETREE_INVOKEMETHOD(&&(>=(x1[0], 0), >(x0[0], +(x1[0], 1))), x0[0], x1[0])

The following pairs are in P≥:

2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_2935_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:
3247_0_insert_Load(x0) → 2638_0_insert_GT(x0)
2638_0_insert_GT(x0) → 2874_0_insert_Return
2638_0_insert_GT(x0) → 2754_1_insert_InvokeMethod(2638_0_insert_GT(x0))
2754_1_insert_InvokeMethod(2815_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2816_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2853_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2874_0_insert_Return) → 2874_0_insert_Return
2638_0_insert_GT(x0) → 2853_0_insert_Return
2638_0_insert_GT(x0) → 2749_1_insert_InvokeMethod(2638_0_insert_GT(x0))
2749_1_insert_InvokeMethod(2815_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2816_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2853_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2874_0_insert_Return) → 2853_0_insert_Return

The integer pair graph contains the following rules and edges:
(0): 2935_1_CREATETREE_INVOKEMETHOD(x0[0], x1[0]) → COND_2935_1_CREATETREE_INVOKEMETHOD(x1[0] >= 0 && x0[0] > x1[0] + 1, x0[0], x1[0])


The set Q consists of the following terms:
3247_0_insert_Load(x0)
2638_0_insert_GT(x0)
2754_1_insert_InvokeMethod(2815_0_insert_Return)
2754_1_insert_InvokeMethod(2816_0_insert_Return)
2754_1_insert_InvokeMethod(2853_0_insert_Return)
2754_1_insert_InvokeMethod(2874_0_insert_Return)
2749_1_insert_InvokeMethod(2815_0_insert_Return)
2749_1_insert_InvokeMethod(2816_0_insert_Return)
2749_1_insert_InvokeMethod(2853_0_insert_Return)
2749_1_insert_InvokeMethod(2874_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:
3247_0_insert_Load(x0) → 2638_0_insert_GT(x0)
2638_0_insert_GT(x0) → 2874_0_insert_Return
2638_0_insert_GT(x0) → 2754_1_insert_InvokeMethod(2638_0_insert_GT(x0))
2754_1_insert_InvokeMethod(2815_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2816_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2853_0_insert_Return) → 2874_0_insert_Return
2754_1_insert_InvokeMethod(2874_0_insert_Return) → 2874_0_insert_Return
2638_0_insert_GT(x0) → 2853_0_insert_Return
2638_0_insert_GT(x0) → 2749_1_insert_InvokeMethod(2638_0_insert_GT(x0))
2749_1_insert_InvokeMethod(2815_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2816_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2853_0_insert_Return) → 2853_0_insert_Return
2749_1_insert_InvokeMethod(2874_0_insert_Return) → 2853_0_insert_Return

The integer pair graph contains the following rules and edges:
(1): COND_2935_1_CREATETREE_INVOKEMETHOD(TRUE, x0[1], x1[1]) → 2935_1_CREATETREE_INVOKEMETHOD(x0[1], x1[1] + 1)


The set Q consists of the following terms:
3247_0_insert_Load(x0)
2638_0_insert_GT(x0)
2754_1_insert_InvokeMethod(2815_0_insert_Return)
2754_1_insert_InvokeMethod(2816_0_insert_Return)
2754_1_insert_InvokeMethod(2853_0_insert_Return)
2754_1_insert_InvokeMethod(2874_0_insert_Return)
2749_1_insert_InvokeMethod(2815_0_insert_Return)
2749_1_insert_InvokeMethod(2816_0_insert_Return)
2749_1_insert_InvokeMethod(2853_0_insert_Return)
2749_1_insert_InvokeMethod(2874_0_insert_Return)

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE