Termination proof of /tmp/tmpSS

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPtoQDPProof (⇒)

        ↳7 QDP

        ↳8 UsableRulesProof (⇔)

        ↳9 QDP

        ↳10 QReductionProof (⇔)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 IDP

        ↳15 IDPNonInfProof (⇒)

        ↳16 IDP

        ↳17 IDependencyGraphProof (⇔)

        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: AProVERec01

public class AProVERec01 {
	public static void main(String[] args){
		List a = new List(args[0].length(), null);
		rec(args[1].length(), a);
	}

	public static void rec(int y, List res){
		int x = 3 * y;
		if(x < 100000){
			rec(x+1, res);
		}
		x = x*2;
		res.add(x);
	}
}

class List {
	int val = 0;
	List next = null;
	
	List (int v,List n){
		val = v;
		next = n;
	}

	public void add(int newVal){
		if (next == null) {
			next = new List(newVal, null);
		} else {
			next.add(newVal);
		}
	}
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
AProVERec01.main([Ljava/lang/String;)V: Graph of 143 nodes with 0 SCCs.

AProVERec01.rec(ILList;)V: Graph of 45 nodes with 0 SCCs.

List.add(I)V: Graph of 42 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 12 rules for P and 30 rules for R.

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

Filtered ground terms:

1213_0_add_FieldAccess(x1, x2, x3, x4) → 1213_0_add_FieldAccess(x2, x3, x4)
List(x1, x2) → List(x2)
1339_0_add_Return(x1) → 1339_0_add_Return
1276_0_add_Return(x1) → 1276_0_add_Return

Filtered duplicate args:

1213_0_add_FieldAccess(x1, x2, x3) → 1213_0_add_FieldAccess(x2, x3)

Filtered unneeded arguments:

1236_1_add_InvokeMethod(x1, x2, x3) → 1236_1_add_InvokeMethod(x1, x2)

Finished conversion. Obtained 1 rules for P and 2 rules for R. System has no predefined symbols.

Log for SCC 1:

Generated 15 rules for P and 77 rules for R.

Combined rules. Obtained 1 rules for P and 7 rules for R.

Filtered ground terms:

951_0_rec_Load(x1, x2, x3, x4) → 951_0_rec_Load(x2, x3)
Cond_951_0_rec_Load(x1, x2, x3, x4, x5) → Cond_951_0_rec_Load(x1, x3, x4)
1339_0_add_Return(x1) → 1339_0_add_Return
1276_0_add_Return(x1) → 1276_0_add_Return
List(x1) → List
1216_0_add_NONNULL(x1, x2, x3, x4) → 1216_0_add_NONNULL(x3, x4)
1131_0_rec_Return(x1) → 1131_0_rec_Return
1204_1_rec_InvokeMethod(x1, x2, x3) → 1204_1_rec_InvokeMethod(x1, x3)

Filtered duplicate args:

989_1_rec_InvokeMethod(x1, x2, x3, x4, x5) → 989_1_rec_InvokeMethod(x1, x3, x4, x5)

Filtered unneeded arguments:

989_1_rec_InvokeMethod(x1, x2, x3, x4) → 989_1_rec_InvokeMethod(x1, x2, x4)
1204_1_rec_InvokeMethod(x1, x2) → 1204_1_rec_InvokeMethod(x1)
1236_1_add_InvokeMethod(x1, x2, x3) → 1236_1_add_InvokeMethod(x1, x2)

Combined rules. Obtained 1 rules for P and 7 rules for R.

Finished conversion. Obtained 1 rules for P and 7 rules for R. System has predefined symbols.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The following domains are used:
none

The ITRS R consists of the following rules:
1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return

The integer pair graph contains the following rules and edges:
(0): 1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(List(java.lang.Object(x0[0])))) → 1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

(0) -> (0), if ((x1[0] →^* x1[0]')∧(java.lang.Object(x0[0]) →^* java.lang.Object(List(java.lang.Object(x0[0]')))))

The set Q consists of the following terms:
1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0))
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0))

(6) IDPtoQDPProof (SOUND transformation)

Represented integers and predefined function symbols by Terms

(7) Obligation:

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

1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(List(java.lang.Object(x0[0])))) → 1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

The TRS R consists of the following rules:

1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return

The set Q consists of the following terms:

1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0))
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0))

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

(8) 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.

(9) Obligation:

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

1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(List(java.lang.Object(x0[0])))) → 1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

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

1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0))
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0))

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

(10) 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].

1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0))
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0))

(11) Obligation:

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

1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(List(java.lang.Object(x0[0])))) → 1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(x0[0]))

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

(12) 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:

1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(List(java.lang.Object(x0[0])))) → 1213_0_ADD_FIELDACCESS(x1[0], java.lang.Object(x0[0]))
The graph contains the following edges 1 >= 1, 2 > 2

(13) YES

(14) Obligation:

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

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
989_1_rec_InvokeMethod(1131_0_rec_Return, x1, java.lang.Object(x0)) → 1204_1_rec_InvokeMethod(1216_0_add_NONNULL(x1 * 2, x3))
1204_1_rec_InvokeMethod(1276_0_add_Return) → 1131_0_rec_Return
1204_1_rec_InvokeMethod(1339_0_add_Return) → 1131_0_rec_Return
1216_0_add_NONNULL(x0, NULL) → 1276_0_add_Return
1216_0_add_NONNULL(x0, java.lang.Object(x1)) → 1236_1_add_InvokeMethod(1216_0_add_NONNULL(x0, x2), java.lang.Object(List))
1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return

The integer pair graph contains the following rules and edges:
(0): 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(100000 > 3 * x0[0] && 0 <= 3 * x0[0], x0[0], java.lang.Object(x1[0]))
(1): COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1])) → 951_0_REC_LOAD(3 * x0[1] + 1, java.lang.Object(x1[1]))

(0) -> (1), if ((100000 > 3 * x0[0] && 0 <= 3 * x0[0] →^* TRUE)∧(x0[0] →^* x0[1])∧(java.lang.Object(x1[0]) →^* java.lang.Object(x1[1])))

(1) -> (0), if ((3 * x0[1] + 1 →^* x0[0])∧(java.lang.Object(x1[1]) →^* java.lang.Object(x1[0])))

The set Q consists of the following terms:
989_1_rec_InvokeMethod(1131_0_rec_Return, x0, java.lang.Object(x1))
1204_1_rec_InvokeMethod(1276_0_add_Return)
1204_1_rec_InvokeMethod(1339_0_add_Return)
1216_0_add_NONNULL(x0, NULL)
1216_0_add_NONNULL(x0, java.lang.Object(x1))
1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0))
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0))

(15) 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 951_0_REC_LOAD(x0, java.lang.Object(x1)) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0)), <=(0, *(3, x0))), x0, java.lang.Object(x1)) the following chains were created:

We consider the chain 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0])), COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1])) → 951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1])) which results in the following constraint:
(1)    (&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0])))=TRUE∧x0[0]=x0[1]∧java.lang.Object(x1[0])=java.lang.Object(x1[1]) ⇒ 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0]))≥NonInfC∧951_0_REC_LOAD(x0[0], java.lang.Object(x1[0]))≥COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))∧(U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(100000, *(3, x0[0]))=TRUE∧<=(0, *(3, x0[0]))=TRUE ⇒ 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0]))≥NonInfC∧951_0_REC_LOAD(x0[0], java.lang.Object(x1[0]))≥COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))∧(U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    ([99999] + [-3]x0[0] ≥ 0∧[3]x0[0] ≥ 0 ⇒ (U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥)∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    ([99999] + [-3]x0[0] ≥ 0∧[3]x0[0] ≥ 0 ⇒ (U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥)∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    ([99999] + [-3]x0[0] ≥ 0∧[3]x0[0] ≥ 0 ⇒ (U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥)∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(6)    ([33333] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

For Pair COND_951_0_REC_LOAD(TRUE, x0, java.lang.Object(x1)) → 951_0_REC_LOAD(+(*(3, x0), 1), java.lang.Object(x1)) the following chains were created:

We consider the chain 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0])), COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1])) → 951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1])), 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0])) which results in the following constraint:
(7)    (&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0])))=TRUE∧x0[0]=x0[1]∧java.lang.Object(x1[0])=java.lang.Object(x1[1])∧+(*(3, x0[1]), 1)=x0[0]1∧java.lang.Object(x1[1])=java.lang.Object(x1[0]1) ⇒ COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1]))≥NonInfC∧COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1]))≥951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))∧(U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥))

We simplified constraint (7) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(8)    (>(100000, *(3, x0[0]))=TRUE∧<=(0, *(3, x0[0]))=TRUE ⇒ COND_951_0_REC_LOAD(TRUE, x0[0], java.lang.Object(x1[0]))≥NonInfC∧COND_951_0_REC_LOAD(TRUE, x0[0], java.lang.Object(x1[0]))≥951_0_REC_LOAD(+(*(3, x0[0]), 1), java.lang.Object(x1[0]))∧(U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ([99999] + [-3]x0[0] ≥ 0∧[3]x0[0] ≥ 0 ⇒ (U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥)∧[(-1)Bound*bni_34] + [(-1)bni_34]x0[0] ≥ 0∧[1 + (-1)bso_35] + [2]x0[0] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ([99999] + [-3]x0[0] ≥ 0∧[3]x0[0] ≥ 0 ⇒ (U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥)∧[(-1)Bound*bni_34] + [(-1)bni_34]x0[0] ≥ 0∧[1 + (-1)bso_35] + [2]x0[0] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ([99999] + [-3]x0[0] ≥ 0∧[3]x0[0] ≥ 0 ⇒ (U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥)∧[(-1)Bound*bni_34] + [(-1)bni_34]x0[0] ≥ 0∧[1 + (-1)bso_35] + [2]x0[0] ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
(12)    ([33333] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [(-1)bni_34]x0[0] ≥ 0∧0 = 0∧[1 + (-1)bso_35] + [2]x0[0] ≥ 0)

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

951_0_REC_LOAD(x0, java.lang.Object(x1)) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0)), <=(0, *(3, x0))), x0, java.lang.Object(x1))
- ([33333] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [(-1)bni_32]x0[0] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)
COND_951_0_REC_LOAD(TRUE, x0, java.lang.Object(x1)) → 951_0_REC_LOAD(+(*(3, x0), 1), java.lang.Object(x1))
- ([33333] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [(-1)bni_34]x0[0] ≥ 0∧0 = 0∧[1 + (-1)bso_35] + [2]x0[0] ≥ 0)

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(989_1_rec_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁
POL(1131_0_rec_Return) = [-1]
POL(java.lang.Object(x₁)) = [-1]
POL(1204_1_rec_InvokeMethod(x₁)) = [-1] + [-1]x₁
POL(1216_0_add_NONNULL(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(*(x₁, x₂)) = x₁·x₂
POL(2) = [2]
POL(1276_0_add_Return) = [-1]
POL(1339_0_add_Return) = [-1]
POL(NULL) = [-1]
POL(1236_1_add_InvokeMethod(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(List) = [-1]
POL(951_0_REC_LOAD(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(COND_951_0_REC_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(100000) = [100000]
POL(3) = [3]
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1])) → 951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))

The following pairs are in P_bound:

951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))
COND_951_0_REC_LOAD(TRUE, x0[1], java.lang.Object(x1[1])) → 951_0_REC_LOAD(+(*(3, x0[1]), 1), java.lang.Object(x1[1]))

The following pairs are in P_≥:

951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(&&(>(100000, *(3, x0[0])), <=(0, *(3, x0[0]))), x0[0], java.lang.Object(x1[0]))

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

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

(16) Obligation:

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

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
989_1_rec_InvokeMethod(1131_0_rec_Return, x1, java.lang.Object(x0)) → 1204_1_rec_InvokeMethod(1216_0_add_NONNULL(x1 * 2, x3))
1204_1_rec_InvokeMethod(1276_0_add_Return) → 1131_0_rec_Return
1204_1_rec_InvokeMethod(1339_0_add_Return) → 1131_0_rec_Return
1216_0_add_NONNULL(x0, NULL) → 1276_0_add_Return
1216_0_add_NONNULL(x0, java.lang.Object(x1)) → 1236_1_add_InvokeMethod(1216_0_add_NONNULL(x0, x2), java.lang.Object(List))
1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0)) → 1339_0_add_Return

The integer pair graph contains the following rules and edges:
(0): 951_0_REC_LOAD(x0[0], java.lang.Object(x1[0])) → COND_951_0_REC_LOAD(100000 > 3 * x0[0] && 0 <= 3 * x0[0], x0[0], java.lang.Object(x1[0]))

The set Q consists of the following terms:
989_1_rec_InvokeMethod(1131_0_rec_Return, x0, java.lang.Object(x1))
1204_1_rec_InvokeMethod(1276_0_add_Return)
1204_1_rec_InvokeMethod(1339_0_add_Return)
1216_0_add_NONNULL(x0, NULL)
1216_0_add_NONNULL(x0, java.lang.Object(x1))
1236_1_add_InvokeMethod(1276_0_add_Return, java.lang.Object(x0))
1236_1_add_InvokeMethod(1339_0_add_Return, java.lang.Object(x0))

(17) 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) IDPtoQDPProof (SOUND transformation)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) IDPNonInfProof (SOUND transformation)

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE