Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 ITRStoIDPProof (⇔)
↳6 IDP
↳7 UsableRulesProof (⇔)
↳8 IDP
↳9 ItpfGraphProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Loop1 
/**
 * A very simple loop over an array.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that all calls terminate
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Loop1 {
    public static void main(String[] args) {
	for (int i = 0; i < args.length; i++) {}
    }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 58 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load182(java.lang.Object(ARRAY(i11, a37data)), i7) → Cond_Load182(i7 >= 0 && i7 < i11 && i7 + 1 > 0, java.lang.Object(ARRAY(i11, a37data)), i7)
Cond_Load182(TRUE, java.lang.Object(ARRAY(i11, a37data)), i7) → Load182(java.lang.Object(ARRAY(i11, a37data)), i7 + 1)
The set Q consists of the following terms:
Load182(java.lang.Object(ARRAY(x0, x1)), x2)
Cond_Load182(TRUE, java.lang.Object(ARRAY(x0, x1)), x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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:

Boolean, Integer


The ITRS R consists of the following rules:
Load182(java.lang.Object(ARRAY(i11, a37data)), i7) → Cond_Load182(i7 >= 0 && i7 < i11 && i7 + 1 > 0, java.lang.Object(ARRAY(i11, a37data)), i7)
Cond_Load182(TRUE, java.lang.Object(ARRAY(i11, a37data)), i7) → Load182(java.lang.Object(ARRAY(i11, a37data)), i7 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])
(1): COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1] + 1)

(0) -> (1), if ((i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0 →* TRUE)∧(java.lang.Object(ARRAY(i11[0], a37data[0])) →* java.lang.Object(ARRAY(i11[1], a37data[1])))∧(i7[0] →* i7[1]))


(1) -> (0), if ((java.lang.Object(ARRAY(i11[1], a37data[1])) →* java.lang.Object(ARRAY(i11[0], a37data[0])))∧(i7[1] + 1 →* i7[0]))



The set Q consists of the following terms:
Load182(java.lang.Object(ARRAY(x0, x1)), x2)
Cond_Load182(TRUE, java.lang.Object(ARRAY(x0, x1)), x2)

(7) 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.

(8) 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): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])
(1): COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1] + 1)

(0) -> (1), if ((i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0 →* TRUE)∧(java.lang.Object(ARRAY(i11[0], a37data[0])) →* java.lang.Object(ARRAY(i11[1], a37data[1])))∧(i7[0] →* i7[1]))


(1) -> (0), if ((java.lang.Object(ARRAY(i11[1], a37data[1])) →* java.lang.Object(ARRAY(i11[0], a37data[0])))∧(i7[1] + 1 →* i7[0]))



The set Q consists of the following terms:
Load182(java.lang.Object(ARRAY(x0, x1)), x2)
Cond_Load182(TRUE, java.lang.Object(ARRAY(x0, x1)), x2)

(9) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])
(1): COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1] + 1)

(0) -> (1), if ((i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0 →* TRUE)∧((i11[0] →* i11[1])∧(a37data[0] →* a37data[1]))∧(i7[0] →* i7[1]))


(1) -> (0), if (((i11[1] →* i11[0])∧(a37data[1] →* a37data[0]))∧(i7[1] + 1 →* i7[0]))



The set Q consists of the following terms:
Load182(java.lang.Object(ARRAY(x0, x1)), x2)
Cond_Load182(TRUE, java.lang.Object(ARRAY(x0, x1)), x2)

(11) 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 LOAD182(java.lang.Object(ARRAY(i11, a37data)), i7) → COND_LOAD182(&&(&&(>=(i7, 0), <(i7, i11)), >(+(i7, 1), 0)), java.lang.Object(ARRAY(i11, a37data)), i7) the following chains were created:
  • We consider the chain LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]), COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1)) which results in the following constraint:

    (1)    (&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0))=TRUE∧i11[0]=i11[1]∧a37data[0]=a37data[1]∧i7[0]=i7[1] ⇒ LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥NonInfC∧LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])∧(UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥))



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

    (2)    (>(+(i7[0], 1), 0)=TRUE∧>=(i7[0], 0)=TRUE∧<(i7[0], i11[0])=TRUE ⇒ LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥NonInfC∧LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])∧(UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥))



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

    (3)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (4)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (5)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (6)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i11[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)







For Pair COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11, a37data)), i7) → LOAD182(java.lang.Object(ARRAY(i11, a37data)), +(i7, 1)) the following chains were created:
  • We consider the chain COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1)) which results in the following constraint:

    (8)    (COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1])≥NonInfC∧COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1])≥LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))∧(UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥))



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

    (9)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧[1 + (-1)bso_17] ≥ 0)



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

    (10)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧[1 + (-1)bso_17] ≥ 0)



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

    (11)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧[1 + (-1)bso_17] ≥ 0)



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

    (12)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD182(java.lang.Object(ARRAY(i11, a37data)), i7) → COND_LOAD182(&&(&&(>=(i7, 0), <(i7, i11)), >(+(i7, 1), 0)), java.lang.Object(ARRAY(i11, a37data)), i7)
    • (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i11[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

  • COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11, a37data)), i7) → LOAD182(java.lang.Object(ARRAY(i11, a37data)), +(i7, 1))
    • ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 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(LOAD182(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(java.lang.Object(x1)) = x1   
POL(ARRAY(x1, x2)) = [-1]x1   
POL(COND_LOAD182(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   

The following pairs are in P>:

COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))

The following pairs are in Pbound:

LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

The following pairs are in P≥:

LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

There are no usable rules.

(12) Complex Obligation (AND)

(13) 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): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])


The set Q consists of the following terms:
Load182(java.lang.Object(ARRAY(x0, x1)), x2)
Cond_Load182(TRUE, java.lang.Object(ARRAY(x0, x1)), x2)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(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


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1] + 1)


The set Q consists of the following terms:
Load182(java.lang.Object(ARRAY(x0, x1)), x2)
Cond_Load182(TRUE, java.lang.Object(ARRAY(x0, x1)), x2)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE