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 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 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: LogIterative

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 183 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:
Load739(i54, i50, i54, i52) → Cond_Load739(i54 > 1 && i50 >= i54 && i52 + 1 > 0, i54, i50, i54, i52)
Cond_Load739(TRUE, i54, i50, i54, i52) → Load739(i54, i50 / i54, i54, i52 + 1)
The set Q consists of the following terms:
Load739(x0, x1, x0, x2)
Cond_Load739(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:

Load739(x1, x2, x3, x4) → Load739(x2, x3, x4)
Cond_Load739(x1, x2, x3, x4, x5) → Cond_Load739(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:
Load739(i50, i54, i52) → Cond_Load739(i54 > 1 && i50 >= i54 && i52 + 1 > 0, i50, i54, i52)
Cond_Load739(TRUE, i50, i54, i52) → Load739(i50 / i54, i54, i52 + 1)
The set Q consists of the following terms:
Load739(x0, x1, x2)
Cond_Load739(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:
Load739(i50, i54, i52) → Cond_Load739(i54 > 1 && i50 >= i54 && i52 + 1 > 0, i50, i54, i52)
Cond_Load739(TRUE, i50, i54, i52) → Load739(i50 / i54, i54, i52 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(i54[0] > 1 && i50[0] >= i54[0] && i52[0] + 1 > 0, i50[0], i54[0], i52[0])
(1): COND_LOAD739(TRUE, i50[1], i54[1], i52[1]) → LOAD739(i50[1] / i54[1], i54[1], i52[1] + 1)

(0) -> (1), if ((i50[0] →* i50[1])∧(i54[0] > 1 && i50[0] >= i54[0] && i52[0] + 1 > 0 →* TRUE)∧(i54[0] →* i54[1])∧(i52[0] →* i52[1]))


(1) -> (0), if ((i50[1] / i54[1] →* i50[0])∧(i54[1] →* i54[0])∧(i52[1] + 1 →* i52[0]))



The set Q consists of the following terms:
Load739(x0, x1, x2)
Cond_Load739(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): LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(i54[0] > 1 && i50[0] >= i54[0] && i52[0] + 1 > 0, i50[0], i54[0], i52[0])
(1): COND_LOAD739(TRUE, i50[1], i54[1], i52[1]) → LOAD739(i50[1] / i54[1], i54[1], i52[1] + 1)

(0) -> (1), if ((i50[0] →* i50[1])∧(i54[0] > 1 && i50[0] >= i54[0] && i52[0] + 1 > 0 →* TRUE)∧(i54[0] →* i54[1])∧(i52[0] →* i52[1]))


(1) -> (0), if ((i50[1] / i54[1] →* i50[0])∧(i54[1] →* i54[0])∧(i52[1] + 1 →* i52[0]))



The set Q consists of the following terms:
Load739(x0, x1, x2)
Cond_Load739(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 LOAD739(i50, i54, i52) → COND_LOAD739(&&(&&(>(i54, 1), >=(i50, i54)), >(+(i52, 1), 0)), i50, i54, i52) the following chains were created:
  • We consider the chain LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0]), COND_LOAD739(TRUE, i50[1], i54[1], i52[1]) → LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1)) which results in the following constraint:

    (1)    (i50[0]=i50[1]∧&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0))=TRUE∧i54[0]=i54[1]∧i52[0]=i52[1] ⇒ LOAD739(i50[0], i54[0], i52[0])≥NonInfC∧LOAD739(i50[0], i54[0], i52[0])≥COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])∧(UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥))



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

    (2)    (>(+(i52[0], 1), 0)=TRUE∧>(i54[0], 1)=TRUE∧>=(i50[0], i54[0])=TRUE ⇒ LOAD739(i50[0], i54[0], i52[0])≥NonInfC∧LOAD739(i50[0], i54[0], i52[0])≥COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])∧(UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥))



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

    (3)    (i52[0] ≥ 0∧i54[0] + [-2] ≥ 0∧i50[0] + [-1]i54[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i50[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (4)    (i52[0] ≥ 0∧i54[0] + [-2] ≥ 0∧i50[0] + [-1]i54[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i50[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (5)    (i52[0] ≥ 0∧i54[0] + [-2] ≥ 0∧i50[0] + [-1]i54[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i50[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (6)    (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] + [-2] + [-1]i54[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i50[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (7)    (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i54[0] + [bni_15]i50[0] ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_LOAD739(TRUE, i50, i54, i52) → LOAD739(/(i50, i54), i54, +(i52, 1)) the following chains were created:
  • We consider the chain LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0]), COND_LOAD739(TRUE, i50[1], i54[1], i52[1]) → LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1)), LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0]) which results in the following constraint:

    (8)    (i50[0]=i50[1]∧&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0))=TRUE∧i54[0]=i54[1]∧i52[0]=i52[1]∧/(i50[1], i54[1])=i50[0]1∧i54[1]=i54[0]1∧+(i52[1], 1)=i52[0]1 ⇒ COND_LOAD739(TRUE, i50[1], i54[1], i52[1])≥NonInfC∧COND_LOAD739(TRUE, i50[1], i54[1], i52[1])≥LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))∧(UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥))



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

    (9)    (>(+(i52[0], 1), 0)=TRUE∧>(i54[0], 1)=TRUE∧>=(i50[0], i54[0])=TRUE ⇒ COND_LOAD739(TRUE, i50[0], i54[0], i52[0])≥NonInfC∧COND_LOAD739(TRUE, i50[0], i54[0], i52[0])≥LOAD739(/(i50[0], i54[0]), i54[0], +(i52[0], 1))∧(UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥))



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

    (10)    (i52[0] ≥ 0∧i54[0] + [-2] ≥ 0∧i50[0] + [-1]i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i50[0] ≥ 0∧[(-1)bso_21] + i50[0] + [-1]max{i50[0], [-1]i50[0]} + min{max{i54[0], [-1]i54[0]} + [-1], max{i50[0], [-1]i50[0]}} ≥ 0)



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

    (11)    (i52[0] ≥ 0∧i54[0] + [-2] ≥ 0∧i50[0] + [-1]i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i50[0] ≥ 0∧[(-1)bso_21] + i50[0] + [-1]max{i50[0], [-1]i50[0]} + min{max{i54[0], [-1]i54[0]} + [-1], max{i50[0], [-1]i50[0]}} ≥ 0)



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

    (12)    (i52[0] ≥ 0∧i54[0] + [-2] ≥ 0∧i50[0] + [-1]i54[0] ≥ 0∧[2]i50[0] ≥ 0∧[2]i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i50[0] ≥ 0∧[-1 + (-1)bso_21] + i54[0] ≥ 0)



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

    (13)    (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] + [-2] + [-1]i54[0] ≥ 0∧[2]i50[0] ≥ 0∧[4] + [2]i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i50[0] ≥ 0∧[1 + (-1)bso_21] + i54[0] ≥ 0)



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

    (14)    (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] ≥ 0∧[4] + [2]i54[0] + [2]i50[0] ≥ 0∧[4] + [2]i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i54[0] + [bni_17]i50[0] ≥ 0∧[1 + (-1)bso_21] + i54[0] ≥ 0)



    We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:

    (15)    (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] ≥ 0∧[2] + i54[0] + i50[0] ≥ 0∧[2] + i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i54[0] + [bni_17]i50[0] ≥ 0∧[1 + (-1)bso_21] + i54[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD739(i50, i54, i52) → COND_LOAD739(&&(&&(>(i54, 1), >=(i50, i54)), >(+(i52, 1), 0)), i50, i54, i52)
    • (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i54[0] + [bni_15]i50[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_LOAD739(TRUE, i50, i54, i52) → LOAD739(/(i50, i54), i54, +(i52, 1))
    • (i52[0] ≥ 0∧i54[0] ≥ 0∧i50[0] ≥ 0∧[2] + i54[0] + i50[0] ≥ 0∧[2] + i54[0] ≥ 0 ⇒ (UIncreasing(LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i54[0] + [bni_17]i50[0] ≥ 0∧[1 + (-1)bso_21] + i54[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) = [2]   
POL(LOAD739(x1, x2, x3)) = [-1] + x1   
POL(COND_LOAD739(x1, x2, x3, x4)) = [-1] + x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(>=(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(0) = 0   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, i54[0])1 @ {LOAD739_3/0}) = max{x1, [-1]x1} + [-1]min{max{x2, [-1]x2} + [-1], max{x1, [-1]x1}}   

The following pairs are in P>:

COND_LOAD739(TRUE, i50[1], i54[1], i52[1]) → LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))

The following pairs are in Pbound:

LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])
COND_LOAD739(TRUE, i50[1], i54[1], i52[1]) → LOAD739(/(i50[1], i54[1]), i54[1], +(i52[1], 1))

The following pairs are in P≥:

LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(&&(&&(>(i54[0], 1), >=(i50[0], i54[0])), >(+(i52[0], 1), 0)), i50[0], i54[0], i52[0])

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

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

(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): LOAD739(i50[0], i54[0], i52[0]) → COND_LOAD739(i54[0] > 1 && i50[0] >= i54[0] && i52[0] + 1 > 0, i50[0], i54[0], i52[0])


The set Q consists of the following terms:
Load739(x0, x1, x2)
Cond_Load739(TRUE, 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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load739(x0, x1, x2)
Cond_Load739(TRUE, x0, x1, x2)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE