Termination of the given JBC could be proven:

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

(0) Obligation:

JBC Problem based on JBC Program:
No human-readable program information known.

Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: AG313

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 163 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:
Load665(i44, i113, i44, i109) → Cond_Load665(i44 > 0 && i113 > 0 && i109 + 1 > 0, i44, i113, i44, i109)
Cond_Load665(TRUE, i44, i113, i44, i109) → Load665(i44, i113 - 1 - i44 - 1, i44, i109 + 1)
The set Q consists of the following terms:
Load665(x0, x1, x0, x2)
Cond_Load665(TRUE, x0, x1, x0, x2)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Load665(x1, x2, x3, x4) → Load665(x2, x3, x4)
Cond_Load665(x1, x2, x3, x4, x5) → Cond_Load665(x1, x3, x4, x5)

(6) 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:
Load665(i113, i44, i109) → Cond_Load665(i44 > 0 && i113 > 0 && i109 + 1 > 0, i113, i44, i109)
Cond_Load665(TRUE, i113, i44, i109) → Load665(i113 - 1 - i44 - 1, i44, i109 + 1)
The set Q consists of the following terms:
Load665(x0, x1, x2)
Cond_Load665(TRUE, x0, x1, x2)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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


The ITRS R consists of the following rules:
Load665(i113, i44, i109) → Cond_Load665(i44 > 0 && i113 > 0 && i109 + 1 > 0, i113, i44, i109)
Cond_Load665(TRUE, i113, i44, i109) → Load665(i113 - 1 - i44 - 1, i44, i109 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0, i113[0], i44[0], i109[0])
(1): COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(i113[1] - 1 - i44[1] - 1, i44[1], i109[1] + 1)

(0) -> (1), if ((i109[0] →* i109[1])∧(i113[0] →* i113[1])∧(i44[0] →* i44[1])∧(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0 →* TRUE))


(1) -> (0), if ((i109[1] + 1 →* i109[0])∧(i44[1] →* i44[0])∧(i113[1] - 1 - i44[1] - 1 →* i113[0]))



The set Q consists of the following terms:
Load665(x0, x1, x2)
Cond_Load665(TRUE, x0, x1, x2)

(9) 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.

(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): LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0, i113[0], i44[0], i109[0])
(1): COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(i113[1] - 1 - i44[1] - 1, i44[1], i109[1] + 1)

(0) -> (1), if ((i109[0] →* i109[1])∧(i113[0] →* i113[1])∧(i44[0] →* i44[1])∧(i44[0] > 0 && i113[0] > 0 && i109[0] + 1 > 0 →* TRUE))


(1) -> (0), if ((i109[1] + 1 →* i109[0])∧(i44[1] →* i44[0])∧(i113[1] - 1 - i44[1] - 1 →* i113[0]))



The set Q consists of the following terms:
Load665(x0, x1, x2)
Cond_Load665(TRUE, 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 LOAD665(i113, i44, i109) → COND_LOAD665(&&(&&(>(i44, 0), >(i113, 0)), >(+(i109, 1), 0)), i113, i44, i109) the following chains were created:
  • We consider the chain LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0]), COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1)) which results in the following constraint:

    (1)    (i109[0]=i109[1]∧i113[0]=i113[1]∧i44[0]=i44[1]∧&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0))=TRUE ⇒ LOAD665(i113[0], i44[0], i109[0])≥NonInfC∧LOAD665(i113[0], i44[0], i109[0])≥COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])∧(UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥))



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

    (2)    (>(+(i109[0], 1), 0)=TRUE∧>(i44[0], 0)=TRUE∧>(i113[0], 0)=TRUE ⇒ LOAD665(i113[0], i44[0], i109[0])≥NonInfC∧LOAD665(i113[0], i44[0], i109[0])≥COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])∧(UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥))



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

    (3)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[(-1)bso_15] + i44[0] ≥ 0)



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

    (4)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[(-1)bso_15] + i44[0] ≥ 0)



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

    (5)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[(-1)bso_15] + i44[0] ≥ 0)



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

    (6)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i113[0] ≥ 0∧[1 + (-1)bso_15] + i44[0] ≥ 0)



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

    (7)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i113[0] ≥ 0∧[1 + (-1)bso_15] + i44[0] ≥ 0)







For Pair COND_LOAD665(TRUE, i113, i44, i109) → LOAD665(-(-(i113, 1), -(i44, 1)), i44, +(i109, 1)) the following chains were created:
  • We consider the chain LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0]), COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1)), LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0]) which results in the following constraint:

    (8)    (i109[0]=i109[1]∧i113[0]=i113[1]∧i44[0]=i44[1]∧&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0))=TRUE∧+(i109[1], 1)=i109[0]1∧i44[1]=i44[0]1∧-(-(i113[1], 1), -(i44[1], 1))=i113[0]1 ⇒ COND_LOAD665(TRUE, i113[1], i44[1], i109[1])≥NonInfC∧COND_LOAD665(TRUE, i113[1], i44[1], i109[1])≥LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))∧(UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥))



    We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(+(i109[0], 1), 0)=TRUE∧>(i44[0], 0)=TRUE∧>(i113[0], 0)=TRUE ⇒ COND_LOAD665(TRUE, i113[0], i44[0], i109[0])≥NonInfC∧COND_LOAD665(TRUE, i113[0], i44[0], i109[0])≥LOAD665(-(-(i113[0], 1), -(i44[0], 1)), i44[0], +(i109[0], 1))∧(UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥))



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

    (10)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (11)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (12)    (i109[0] ≥ 0∧i44[0] + [-1] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (13)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16 + (-1)bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-1)bso_17] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD665(i113, i44, i109) → COND_LOAD665(&&(&&(>(i44, 0), >(i113, 0)), >(+(i109, 1), 0)), i113, i44, i109)
    • (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i113[0] ≥ 0∧[1 + (-1)bso_15] + i44[0] ≥ 0)

  • COND_LOAD665(TRUE, i113, i44, i109) → LOAD665(-(-(i113, 1), -(i44, 1)), i44, +(i109, 1))
    • (i109[0] ≥ 0∧i44[0] ≥ 0∧i113[0] ≥ 0 ⇒ (UIncreasing(LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i44[0] + [bni_16]i113[0] ≥ 0∧[(-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) = [1]   
POL(LOAD665(x1, x2, x3)) = x1   
POL(COND_LOAD665(x1, x2, x3, x4)) = [-1]x3 + x2   
POL(&&(x1, x2)) = [1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(-(x1, x2)) = x1 + [-1]x2   

The following pairs are in P>:

LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])

The following pairs are in Pbound:

LOAD665(i113[0], i44[0], i109[0]) → COND_LOAD665(&&(&&(>(i44[0], 0), >(i113[0], 0)), >(+(i109[0], 1), 0)), i113[0], i44[0], i109[0])

The following pairs are in P≥:

COND_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(-(-(i113[1], 1), -(i44[1], 1)), i44[1], +(i109[1], 1))

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

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

(12) 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_LOAD665(TRUE, i113[1], i44[1], i109[1]) → LOAD665(i113[1] - 1 - i44[1] - 1, i44[1], i109[1] + 1)


The set Q consists of the following terms:
Load665(x0, x1, x2)
Cond_Load665(TRUE, x0, x1, x2)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE