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 IDPNonInfProof (⇐)
↳10 IDP
↳11 IDependencyGraphProof (⇔)
↳12 TRUE

(0) Obligation:

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

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA9

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 230 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:
Load871(i12, i82, i75) → Cond_Load871(i75 >= 0 && i12 >= i75 && i82 > 0 && i75 + i82 > 0, i12, i82, i75)
Cond_Load871(TRUE, i12, i82, i75) → Load871(i12, i82, i75 + i82)
The set Q consists of the following terms:
Load871(x0, x1, x2)
Cond_Load871(TRUE, 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:
Load871(i12, i82, i75) → Cond_Load871(i75 >= 0 && i12 >= i75 && i82 > 0 && i75 + i82 > 0, i12, i82, i75)
Cond_Load871(TRUE, i12, i82, i75) → Load871(i12, i82, i75 + i82)

The integer pair graph contains the following rules and edges:
(0): LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(i75[0] >= 0 && i12[0] >= i75[0] && i82[0] > 0 && i75[0] + i82[0] > 0, i12[0], i82[0], i75[0])
(1): COND_LOAD871(TRUE, i12[1], i82[1], i75[1]) → LOAD871(i12[1], i82[1], i75[1] + i82[1])

(0) -> (1), if ((i12[0] →* i12[1])∧(i75[0] >= 0 && i12[0] >= i75[0] && i82[0] > 0 && i75[0] + i82[0] > 0 →* TRUE)∧(i82[0] →* i82[1])∧(i75[0] →* i75[1]))


(1) -> (0), if ((i75[1] + i82[1] →* i75[0])∧(i82[1] →* i82[0])∧(i12[1] →* i12[0]))



The set Q consists of the following terms:
Load871(x0, x1, x2)
Cond_Load871(TRUE, 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): LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(i75[0] >= 0 && i12[0] >= i75[0] && i82[0] > 0 && i75[0] + i82[0] > 0, i12[0], i82[0], i75[0])
(1): COND_LOAD871(TRUE, i12[1], i82[1], i75[1]) → LOAD871(i12[1], i82[1], i75[1] + i82[1])

(0) -> (1), if ((i12[0] →* i12[1])∧(i75[0] >= 0 && i12[0] >= i75[0] && i82[0] > 0 && i75[0] + i82[0] > 0 →* TRUE)∧(i82[0] →* i82[1])∧(i75[0] →* i75[1]))


(1) -> (0), if ((i75[1] + i82[1] →* i75[0])∧(i82[1] →* i82[0])∧(i12[1] →* i12[0]))



The set Q consists of the following terms:
Load871(x0, x1, x2)
Cond_Load871(TRUE, x0, x1, x2)

(9) 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 LOAD871(i12, i82, i75) → COND_LOAD871(&&(&&(&&(>=(i75, 0), >=(i12, i75)), >(i82, 0)), >(+(i75, i82), 0)), i12, i82, i75) the following chains were created:
  • We consider the chain LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0]), COND_LOAD871(TRUE, i12[1], i82[1], i75[1]) → LOAD871(i12[1], i82[1], +(i75[1], i82[1])) which results in the following constraint:

    (1)    (i12[0]=i12[1]∧&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0))=TRUE∧i82[0]=i82[1]∧i75[0]=i75[1] ⇒ LOAD871(i12[0], i82[0], i75[0])≥NonInfC∧LOAD871(i12[0], i82[0], i75[0])≥COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])∧(UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥))



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

    (2)    (>(+(i75[0], i82[0]), 0)=TRUE∧>(i82[0], 0)=TRUE∧>=(i75[0], 0)=TRUE∧>=(i12[0], i75[0])=TRUE ⇒ LOAD871(i12[0], i82[0], i75[0])≥NonInfC∧LOAD871(i12[0], i82[0], i75[0])≥COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])∧(UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥))



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

    (3)    (i75[0] + [-1] + i82[0] ≥ 0∧i82[0] + [-1] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i75[0] + [bni_15]i12[0] ≥ 0∧[(-1)bso_16] + i82[0] ≥ 0)



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

    (4)    (i75[0] + [-1] + i82[0] ≥ 0∧i82[0] + [-1] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i75[0] + [bni_15]i12[0] ≥ 0∧[(-1)bso_16] + i82[0] ≥ 0)



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

    (5)    (i75[0] + [-1] + i82[0] ≥ 0∧i82[0] + [-1] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i75[0] + [bni_15]i12[0] ≥ 0∧[(-1)bso_16] + i82[0] ≥ 0)



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

    (6)    (i75[0] + i82[0] ≥ 0∧i82[0] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i75[0] + [bni_15]i12[0] ≥ 0∧[1 + (-1)bso_16] + i82[0] ≥ 0)



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

    (7)    (i75[0] + i82[0] ≥ 0∧i82[0] ≥ 0∧i75[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i12[0] ≥ 0∧[1 + (-1)bso_16] + i82[0] ≥ 0)







For Pair COND_LOAD871(TRUE, i12, i82, i75) → LOAD871(i12, i82, +(i75, i82)) the following chains were created:
  • We consider the chain LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0]), COND_LOAD871(TRUE, i12[1], i82[1], i75[1]) → LOAD871(i12[1], i82[1], +(i75[1], i82[1])), LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0]) which results in the following constraint:

    (8)    (i12[0]=i12[1]∧&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0))=TRUE∧i82[0]=i82[1]∧i75[0]=i75[1]∧+(i75[1], i82[1])=i75[0]1∧i82[1]=i82[0]1∧i12[1]=i12[0]1 ⇒ COND_LOAD871(TRUE, i12[1], i82[1], i75[1])≥NonInfC∧COND_LOAD871(TRUE, i12[1], i82[1], i75[1])≥LOAD871(i12[1], i82[1], +(i75[1], i82[1]))∧(UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥))



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

    (9)    (>(+(i75[0], i82[0]), 0)=TRUE∧>(i82[0], 0)=TRUE∧>=(i75[0], 0)=TRUE∧>=(i12[0], i75[0])=TRUE ⇒ COND_LOAD871(TRUE, i12[0], i82[0], i75[0])≥NonInfC∧COND_LOAD871(TRUE, i12[0], i82[0], i75[0])≥LOAD871(i12[0], i82[0], +(i75[0], i82[0]))∧(UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥))



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

    (10)    (i75[0] + [-1] + i82[0] ≥ 0∧i82[0] + [-1] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i75[0] + [(-1)bni_17]i82[0] + [bni_17]i12[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (11)    (i75[0] + [-1] + i82[0] ≥ 0∧i82[0] + [-1] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i75[0] + [(-1)bni_17]i82[0] + [bni_17]i12[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (12)    (i75[0] + [-1] + i82[0] ≥ 0∧i82[0] + [-1] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i75[0] + [(-1)bni_17]i82[0] + [bni_17]i12[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (13)    (i75[0] + i82[0] ≥ 0∧i82[0] ≥ 0∧i75[0] ≥ 0∧i12[0] + [-1]i75[0] ≥ 0 ⇒ (UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i75[0] + [(-1)bni_17]i82[0] + [bni_17]i12[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (14)    (i75[0] + i82[0] ≥ 0∧i82[0] ≥ 0∧i75[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i82[0] + [bni_17]i12[0] ≥ 0∧[(-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD871(i12, i82, i75) → COND_LOAD871(&&(&&(&&(>=(i75, 0), >=(i12, i75)), >(i82, 0)), >(+(i75, i82), 0)), i12, i82, i75)
    • (i75[0] + i82[0] ≥ 0∧i82[0] ≥ 0∧i75[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i12[0] ≥ 0∧[1 + (-1)bso_16] + i82[0] ≥ 0)

  • COND_LOAD871(TRUE, i12, i82, i75) → LOAD871(i12, i82, +(i75, i82))
    • (i75[0] + i82[0] ≥ 0∧i82[0] ≥ 0∧i75[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(LOAD871(i12[1], i82[1], +(i75[1], i82[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i82[0] + [bni_17]i12[0] ≥ 0∧[(-1)bso_18] ≥ 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(LOAD871(x1, x2, x3)) = [-1] + [-1]x3 + x1   
POL(COND_LOAD871(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   

The following pairs are in P>:

LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])

The following pairs are in Pbound:

LOAD871(i12[0], i82[0], i75[0]) → COND_LOAD871(&&(&&(&&(>=(i75[0], 0), >=(i12[0], i75[0])), >(i82[0], 0)), >(+(i75[0], i82[0]), 0)), i12[0], i82[0], i75[0])

The following pairs are in P≥:

COND_LOAD871(TRUE, i12[1], i82[1], i75[1]) → LOAD871(i12[1], i82[1], +(i75[1], i82[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

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD871(TRUE, i12[1], i82[1], i75[1]) → LOAD871(i12[1], i82[1], i75[1] + i82[1])


The set Q consists of the following terms:
Load871(x0, x1, x2)
Cond_Load871(TRUE, x0, x1, x2)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE