(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: StupidArray
`public class StupidArray {  public static void main(String[] args) {    int i = 0;    while (true) {      i = args.length + 1;      args[i] = new String();    }  }}`

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
StupidArray.main([Ljava/lang/String;)V: Graph of 56 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 27 rules for P and 23 rules for R.

Combined rules. Obtained 1 rules for P and 0 rules for R.

Filtered ground terms:

89_0_main_ArrayLength(x1, x2, x3) → 89_0_main_ArrayLength(x2, x3)
Cond_89_0_main_ArrayLength(x1, x2, x3, x4) → Cond_89_0_main_ArrayLength(x1, x3, x4)

Filtered duplicate args:

89_0_main_ArrayLength(x1, x2) → 89_0_main_ArrayLength(x2)
Cond_89_0_main_ArrayLength(x1, x2, x3) → Cond_89_0_main_ArrayLength(x1, x3)

Combined rules. Obtained 1 rules for P and 0 rules for R.

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

(4) 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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0]))) → COND_89_0_MAIN_ARRAYLENGTH(x0[0] >= 0 && x0[0] > x0[0] + 1 && 1 <= x0[0] + 1, java.lang.Object(ARRAY(x0[0], x1[0])))
(1): COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1]))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))

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

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

The set Q is empty.

(5) 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 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0, x1))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0, 0), >(x0, +(x0, 1))), <=(1, +(x0, 1))), java.lang.Object(ARRAY(x0, x1))) the following chains were created:
• We consider the chain COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1]))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1]))), 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0]))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0]))), COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1]))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1]))) which results in the following constraint:

(1)    (java.lang.Object(ARRAY(x0[1], x1[1]))=java.lang.Object(ARRAY(x0[0], x1[0]))∧&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1)))=TRUEjava.lang.Object(ARRAY(x0[0], x1[0]))=java.lang.Object(ARRAY(x0[1]1, x1[1]1)) ⇒ 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0])))≥NonInfC∧89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0])))≥COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))∧(UIncreasing(COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))), ≥))

We simplified constraint (1) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(2)    (<=(1, +(x0[0], 1))=TRUE>=(x0[0], 0)=TRUE>(x0[0], +(x0[0], 1))=TRUE89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[1])))≥NonInfC∧89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[1])))≥COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[1])))∧(UIncreasing(COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(3)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[1] + [(-1)bni_17]x0[0] ≥ 0∧[-1 + (-1)bso_18] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(4)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[1] + [(-1)bni_17]x0[0] ≥ 0∧[-1 + (-1)bso_18] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(5)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[1] + [(-1)bni_17]x0[0] ≥ 0∧[-1 + (-1)bso_18] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(6)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))), ≥)∧[(-1)bni_17] = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x0[0] ≥ 0∧0 = 0∧[-1 + (-1)bso_18] ≥ 0)

We solved constraint (6) using rule (IDP_SMT_SPLIT).

For Pair COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0, x1))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0, x1))) the following chains were created:
• We consider the chain 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0]))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0]))), COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1]))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1]))), 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0]))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0]))) which results in the following constraint:

(7)    (&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1)))=TRUEjava.lang.Object(ARRAY(x0[0], x1[0]))=java.lang.Object(ARRAY(x0[1], x1[1]))∧java.lang.Object(ARRAY(x0[1], x1[1]))=java.lang.Object(ARRAY(x0[0]1, x1[0]1)) ⇒ COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])))≥NonInfC∧COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])))≥89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))∧(UIncreasing(89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))), ≥))

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

(8)    (<=(1, +(x0[0], 1))=TRUE>=(x0[0], 0)=TRUE>(x0[0], +(x0[0], 1))=TRUECOND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[0], x1[0])))≥NonInfC∧COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[0], x1[0])))≥89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0])))∧(UIncreasing(89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(9)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x1[0] + [(-1)bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(10)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x1[0] + [(-1)bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(11)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x1[0] + [(-1)bni_19]x0[0] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(12)    (x0[0] ≥ 0∧x0[0] ≥ 0∧[-2] ≥ 0 ⇒ (UIncreasing(89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))), ≥)∧[(-1)bni_19] = 0∧[(-1)bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x0[0] ≥ 0∧0 = 0∧[(-1)bso_20] ≥ 0)

We solved constraint (12) using rule (IDP_SMT_SPLIT).

To summarize, we get the following constraints P for the following pairs.
• 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0, x1))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0, 0), >(x0, +(x0, 1))), <=(1, +(x0, 1))), java.lang.Object(ARRAY(x0, x1)))

• COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0, x1))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0, x1)))

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(89_0_MAIN_ARRAYLENGTH(x1)) = [-1] + [-1]x1
POL(java.lang.Object(x1)) = [-1] + [-1]x1
POL(ARRAY(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(COND_89_0_MAIN_ARRAYLENGTH(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(<=(x1, x2)) = [-1]

The following pairs are in P>:

89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0]))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))
COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1]))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))

The following pairs are in Pbound:

89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[0], x1[0]))) → COND_89_0_MAIN_ARRAYLENGTH(&&(&&(>=(x0[0], 0), >(x0[0], +(x0[0], 1))), <=(1, +(x0[0], 1))), java.lang.Object(ARRAY(x0[0], x1[0])))
COND_89_0_MAIN_ARRAYLENGTH(TRUE, java.lang.Object(ARRAY(x0[1], x1[1]))) → 89_0_MAIN_ARRAYLENGTH(java.lang.Object(ARRAY(x0[1], x1[1])))

The following pairs are in P:
none

There are no usable rules.

(6) 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

R is empty.

The integer pair graph is empty.

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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