### (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:

List(x1, x2) → List(x2)

Filtered duplicate args:

Filtered unneeded arguments:

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:

List(x1) → List
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)

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.

### (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:

The integer pair graph contains the following rules and edges:

(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:

### (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:

The TRS R consists of the following rules:

The set Q consists of the following terms:

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:

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

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].

### (11) Obligation:

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

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:

The graph contains the following edges 1 >= 1, 2 > 2

### (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))

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]))

(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))

### (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])))=TRUEx0[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]))∧(UIncreasing(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]))=TRUE951_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]))∧(UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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:

We simplified constraint (7) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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)

• ([33333] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(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 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(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(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>:

The following pairs are in Pbound:

The following pairs are in P:

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

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

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))