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 IDP
        ↳16 IDPNonInfProof (⇐)
        ↳17 AND
            ↳18 IDP
                ↳19 IDependencyGraphProof (⇔)
                ↳20 TRUE
            ↳21 IDP
                ↳22 IDependencyGraphProof (⇔)
                ↳23 TRUE
    ↳24 IDP
        ↳25 IDependencyGraphProof (⇔)
        ↳26 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: GCD4

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 196 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:
Load1169(i100, i98) → Cond_Load1169(i100 > 0 && i98 > 0, i100, i98)
Cond_Load1169(TRUE, i100, i98) → Load1302(i100, i98, i98, i100, i98, i100, i98)
Load1302(i100, i98, i98, i100, i98, i107, i98) → Cond_Load1302(i98 > 0 && i107 >= i98, i100, i98, i98, i100, i98, i107, i98)
Cond_Load1302(TRUE, i100, i98, i98, i100, i98, i107, i98) → Load1302(i100, i98, i98, i100, i98, i107 - i98, i98)
Load1302(i100, i98, i98, i100, i98, i107, i98) → Cond_Load13021(i107 < i98, i100, i98, i98, i100, i98, i107, i98)
Cond_Load13021(TRUE, i100, i98, i98, i100, i98, i107, i98) → Load1169(i98, i107)
The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x1, x0, x1, x2, x1)
Cond_Load1302(TRUE, x0, x1, x1, x0, x1, x2, x1)
Cond_Load13021(TRUE, x0, x1, x1, x0, x1, x2, x1)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Cond_Load13021(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_Load13021(x1, x5, x7, x8)
Load1302(x1, x2, x3, x4, x5, x6, x7) → Load1302(x4, x6, x7)
Cond_Load1302(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_Load1302(x1, x5, x7, x8)

(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:
Load1169(i100, i98) → Cond_Load1169(i100 > 0 && i98 > 0, i100, i98)
Cond_Load1169(TRUE, i100, i98) → Load1302(i100, i100, i98)
Load1302(i100, i107, i98) → Cond_Load1302(i98 > 0 && i107 >= i98, i100, i107, i98)
Cond_Load1302(TRUE, i100, i107, i98) → Load1302(i100, i107 - i98, i98)
Load1302(i100, i107, i98) → Cond_Load13021(i107 < i98, i100, i107, i98)
Cond_Load13021(TRUE, i100, i107, i98) → Load1169(i98, i107)
The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(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:
Load1169(i100, i98) → Cond_Load1169(i100 > 0 && i98 > 0, i100, i98)
Cond_Load1169(TRUE, i100, i98) → Load1302(i100, i100, i98)
Load1302(i100, i107, i98) → Cond_Load1302(i98 > 0 && i107 >= i98, i100, i107, i98)
Cond_Load1302(TRUE, i100, i107, i98) → Load1302(i100, i107 - i98, i98)
Load1302(i100, i107, i98) → Cond_Load13021(i107 < i98, i100, i107, i98)
Cond_Load13021(TRUE, i100, i107, i98) → Load1169(i98, i107)

The integer pair graph contains the following rules and edges:
(0): LOAD1169(i100[0], i98[0]) → COND_LOAD1169(i100[0] > 0 && i98[0] > 0, i100[0], i98[0])
(1): COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])
(3): COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], i107[3] - i98[3], i98[3])
(4): LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(i107[4] < i98[4], i100[4], i107[4], i98[4])
(5): COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5])

(0) -> (1), if ((i100[0] →* i100[1])∧(i98[0] →* i98[1])∧(i100[0] > 0 && i98[0] > 0 →* TRUE))


(1) -> (2), if ((i100[1] →* i107[2])∧(i100[1] →* i100[2])∧(i98[1] →* i98[2]))


(1) -> (4), if ((i98[1] →* i98[4])∧(i100[1] →* i107[4])∧(i100[1] →* i100[4]))


(2) -> (3), if ((i98[2] →* i98[3])∧(i100[2] →* i100[3])∧(i107[2] →* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2] →* TRUE))


(3) -> (2), if ((i107[3] - i98[3] →* i107[2])∧(i100[3] →* i100[2])∧(i98[3] →* i98[2]))


(3) -> (4), if ((i98[3] →* i98[4])∧(i100[3] →* i100[4])∧(i107[3] - i98[3] →* i107[4]))


(4) -> (5), if ((i100[4] →* i100[5])∧(i98[4] →* i98[5])∧(i107[4] →* i107[5])∧(i107[4] < i98[4] →* TRUE))


(5) -> (0), if ((i107[5] →* i98[0])∧(i98[5] →* i100[0]))



The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(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): LOAD1169(i100[0], i98[0]) → COND_LOAD1169(i100[0] > 0 && i98[0] > 0, i100[0], i98[0])
(1): COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])
(3): COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], i107[3] - i98[3], i98[3])
(4): LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(i107[4] < i98[4], i100[4], i107[4], i98[4])
(5): COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5])

(0) -> (1), if ((i100[0] →* i100[1])∧(i98[0] →* i98[1])∧(i100[0] > 0 && i98[0] > 0 →* TRUE))


(1) -> (2), if ((i100[1] →* i107[2])∧(i100[1] →* i100[2])∧(i98[1] →* i98[2]))


(1) -> (4), if ((i98[1] →* i98[4])∧(i100[1] →* i107[4])∧(i100[1] →* i100[4]))


(2) -> (3), if ((i98[2] →* i98[3])∧(i100[2] →* i100[3])∧(i107[2] →* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2] →* TRUE))


(3) -> (2), if ((i107[3] - i98[3] →* i107[2])∧(i100[3] →* i100[2])∧(i98[3] →* i98[2]))


(3) -> (4), if ((i98[3] →* i98[4])∧(i100[3] →* i100[4])∧(i107[3] - i98[3] →* i107[4]))


(4) -> (5), if ((i100[4] →* i100[5])∧(i98[4] →* i98[5])∧(i107[4] →* i107[5])∧(i107[4] < i98[4] →* TRUE))


(5) -> (0), if ((i107[5] →* i98[0])∧(i98[5] →* i100[0]))



The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(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 LOAD1169(i100, i98) → COND_LOAD1169(&&(>(i100, 0), >(i98, 0)), i100, i98) the following chains were created:
  • We consider the chain LOAD1169(i100[0], i98[0]) → COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0]), COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1]) which results in the following constraint:

    (1)    (i100[0]=i100[1]∧i98[0]=i98[1]∧&&(>(i100[0], 0), >(i98[0], 0))=TRUE ⇒ LOAD1169(i100[0], i98[0])≥NonInfC∧LOAD1169(i100[0], i98[0])≥COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])∧(UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥))



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

    (2)    (>(i100[0], 0)=TRUE∧>(i98[0], 0)=TRUE ⇒ LOAD1169(i100[0], i98[0])≥NonInfC∧LOAD1169(i100[0], i98[0])≥COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])∧(UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥))



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

    (3)    (i100[0] + [-1] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (4)    (i100[0] + [-1] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (5)    (i100[0] + [-1] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (6)    (i100[0] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (7)    (i100[0] ≥ 0∧i98[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)







For Pair COND_LOAD1169(TRUE, i100, i98) → LOAD1302(i100, i100, i98) the following chains were created:
  • We consider the chain COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1]), LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) which results in the following constraint:

    (8)    (i100[1]=i107[2]∧i100[1]=i100[2]∧i98[1]=i98[2] ⇒ COND_LOAD1169(TRUE, i100[1], i98[1])≥NonInfC∧COND_LOAD1169(TRUE, i100[1], i98[1])≥LOAD1302(i100[1], i100[1], i98[1])∧(UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥))



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

    (9)    (COND_LOAD1169(TRUE, i100[1], i98[1])≥NonInfC∧COND_LOAD1169(TRUE, i100[1], i98[1])≥LOAD1302(i100[1], i100[1], i98[1])∧(UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥))



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

    (10)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)



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

    (11)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)



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

    (12)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)



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

    (13)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)



  • We consider the chain COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1]), LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4]) which results in the following constraint:

    (14)    (i98[1]=i98[4]∧i100[1]=i107[4]∧i100[1]=i100[4] ⇒ COND_LOAD1169(TRUE, i100[1], i98[1])≥NonInfC∧COND_LOAD1169(TRUE, i100[1], i98[1])≥LOAD1302(i100[1], i100[1], i98[1])∧(UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥))



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

    (15)    (COND_LOAD1169(TRUE, i100[1], i98[1])≥NonInfC∧COND_LOAD1169(TRUE, i100[1], i98[1])≥LOAD1302(i100[1], i100[1], i98[1])∧(UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥))



    We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (16)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)



    We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (17)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)



    We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (18)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)



    We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (19)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)







For Pair LOAD1302(i100, i107, i98) → COND_LOAD1302(&&(>(i98, 0), >=(i107, i98)), i100, i107, i98) the following chains were created:
  • We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]) which results in the following constraint:

    (20)    (i98[2]=i98[3]∧i100[2]=i100[3]∧i107[2]=i107[3]∧&&(>(i98[2], 0), >=(i107[2], i98[2]))=TRUE ⇒ LOAD1302(i100[2], i107[2], i98[2])≥NonInfC∧LOAD1302(i100[2], i107[2], i98[2])≥COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])∧(UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥))



    We simplified constraint (20) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (21)    (>(i98[2], 0)=TRUE∧>=(i107[2], i98[2])=TRUE ⇒ LOAD1302(i100[2], i107[2], i98[2])≥NonInfC∧LOAD1302(i100[2], i107[2], i98[2])≥COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])∧(UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥))



    We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (22)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (23)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (23) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (24)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (24) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (25)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (26)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)



    We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)







For Pair COND_LOAD1302(TRUE, i100, i107, i98) → LOAD1302(i100, -(i107, i98), i98) the following chains were created:
  • We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]), LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) which results in the following constraint:

    (28)    (i98[2]=i98[3]∧i100[2]=i100[3]∧i107[2]=i107[3]∧&&(>(i98[2], 0), >=(i107[2], i98[2]))=TRUE∧-(i107[3], i98[3])=i107[2]1∧i100[3]=i100[2]1∧i98[3]=i98[2]1 ⇒ COND_LOAD1302(TRUE, i100[3], i107[3], i98[3])≥NonInfC∧COND_LOAD1302(TRUE, i100[3], i107[3], i98[3])≥LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])∧(UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥))



    We simplified constraint (28) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (29)    (>(i98[2], 0)=TRUE∧>=(i107[2], i98[2])=TRUE ⇒ COND_LOAD1302(TRUE, i100[2], i107[2], i98[2])≥NonInfC∧COND_LOAD1302(TRUE, i100[2], i107[2], i98[2])≥LOAD1302(i100[2], -(i107[2], i98[2]), i98[2])∧(UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥))



    We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (30)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (31)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (32)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (32) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (33)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (34)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (35)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)



  • We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]), LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4]) which results in the following constraint:

    (36)    (i98[2]=i98[3]∧i100[2]=i100[3]∧i107[2]=i107[3]∧&&(>(i98[2], 0), >=(i107[2], i98[2]))=TRUE∧i98[3]=i98[4]∧i100[3]=i100[4]∧-(i107[3], i98[3])=i107[4] ⇒ COND_LOAD1302(TRUE, i100[3], i107[3], i98[3])≥NonInfC∧COND_LOAD1302(TRUE, i100[3], i107[3], i98[3])≥LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])∧(UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥))



    We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (37)    (>(i98[2], 0)=TRUE∧>=(i107[2], i98[2])=TRUE ⇒ COND_LOAD1302(TRUE, i100[2], i107[2], i98[2])≥NonInfC∧COND_LOAD1302(TRUE, i100[2], i107[2], i98[2])≥LOAD1302(i100[2], -(i107[2], i98[2]), i98[2])∧(UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥))



    We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (38)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (39)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (40)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (40) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (41)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (42)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)



    We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (43)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)







For Pair LOAD1302(i100, i107, i98) → COND_LOAD13021(<(i107, i98), i100, i107, i98) the following chains were created:
  • We consider the chain LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4]), COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5]) which results in the following constraint:

    (44)    (i100[4]=i100[5]∧i98[4]=i98[5]∧i107[4]=i107[5]∧<(i107[4], i98[4])=TRUE ⇒ LOAD1302(i100[4], i107[4], i98[4])≥NonInfC∧LOAD1302(i100[4], i107[4], i98[4])≥COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])∧(UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥))



    We simplified constraint (44) using rule (IV) which results in the following new constraint:

    (45)    (<(i107[4], i98[4])=TRUE ⇒ LOAD1302(i100[4], i107[4], i98[4])≥NonInfC∧LOAD1302(i100[4], i107[4], i98[4])≥COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])∧(UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥))



    We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (46)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)



    We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (47)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)



    We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (48)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)



    We simplified constraint (48) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (49)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧0 = 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)



    We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (50)    (i98[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)



    We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (51)    (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [(-1)bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)


    (52)    (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)







For Pair COND_LOAD13021(TRUE, i100, i107, i98) → LOAD1169(i98, i107) the following chains were created:
  • We consider the chain COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5]), LOAD1169(i100[0], i98[0]) → COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0]) which results in the following constraint:

    (53)    (i107[5]=i98[0]∧i98[5]=i100[0] ⇒ COND_LOAD13021(TRUE, i100[5], i107[5], i98[5])≥NonInfC∧COND_LOAD13021(TRUE, i100[5], i107[5], i98[5])≥LOAD1169(i98[5], i107[5])∧(UIncreasing(LOAD1169(i98[5], i107[5])), ≥))



    We simplified constraint (53) using rule (IV) which results in the following new constraint:

    (54)    (COND_LOAD13021(TRUE, i100[5], i107[5], i98[5])≥NonInfC∧COND_LOAD13021(TRUE, i100[5], i107[5], i98[5])≥LOAD1169(i98[5], i107[5])∧(UIncreasing(LOAD1169(i98[5], i107[5])), ≥))



    We simplified constraint (54) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (55)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧[1 + (-1)bso_39] ≥ 0)



    We simplified constraint (55) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (56)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧[1 + (-1)bso_39] ≥ 0)



    We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (57)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧[1 + (-1)bso_39] ≥ 0)



    We simplified constraint (57) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (58)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_39] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1169(i100, i98) → COND_LOAD1169(&&(>(i100, 0), >(i98, 0)), i100, i98)
    • (i100[0] ≥ 0∧i98[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

  • COND_LOAD1169(TRUE, i100, i98) → LOAD1302(i100, i100, i98)
    • ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)
    • ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

  • LOAD1302(i100, i107, i98) → COND_LOAD1302(&&(>(i98, 0), >=(i107, i98)), i100, i107, i98)
    • (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

  • COND_LOAD1302(TRUE, i100, i107, i98) → LOAD1302(i100, -(i107, i98), i98)
    • (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)
    • (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

  • LOAD1302(i100, i107, i98) → COND_LOAD13021(<(i107, i98), i100, i107, i98)
    • (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [(-1)bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)
    • (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)

  • COND_LOAD13021(TRUE, i100, i107, i98) → LOAD1169(i98, i107)
    • ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_39] ≥ 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(LOAD1169(x1, x2)) = [-1] + x2   
POL(COND_LOAD1169(x1, x2, x3)) = [-1] + x3   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(LOAD1302(x1, x2, x3)) = [-1] + x3   
POL(COND_LOAD1302(x1, x2, x3, x4)) = [-1] + x4   
POL(>=(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(COND_LOAD13021(x1, x2, x3, x4)) = x3   
POL(<(x1, x2)) = [1]   

The following pairs are in P>:

COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5])

The following pairs are in Pbound:

LOAD1169(i100[0], i98[0]) → COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])
LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])
COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])

The following pairs are in P≥:

LOAD1169(i100[0], i98[0]) → COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])
COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1])
LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])
COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])
LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])

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

FALSE1 → &&(FALSE, FALSE)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): LOAD1169(i100[0], i98[0]) → COND_LOAD1169(i100[0] > 0 && i98[0] > 0, i100[0], i98[0])
(1): COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])
(3): COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], i107[3] - i98[3], i98[3])
(4): LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(i107[4] < i98[4], i100[4], i107[4], i98[4])

(0) -> (1), if ((i100[0] →* i100[1])∧(i98[0] →* i98[1])∧(i100[0] > 0 && i98[0] > 0 →* TRUE))


(1) -> (2), if ((i100[1] →* i107[2])∧(i100[1] →* i100[2])∧(i98[1] →* i98[2]))


(3) -> (2), if ((i107[3] - i98[3] →* i107[2])∧(i100[3] →* i100[2])∧(i98[3] →* i98[2]))


(2) -> (3), if ((i98[2] →* i98[3])∧(i100[2] →* i100[3])∧(i107[2] →* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2] →* TRUE))


(1) -> (4), if ((i98[1] →* i98[4])∧(i100[1] →* i107[4])∧(i100[1] →* i100[4]))


(3) -> (4), if ((i98[3] →* i98[4])∧(i100[3] →* i100[4])∧(i107[3] - i98[3] →* i107[4]))



The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(TRUE, x0, x1, x2)

(14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(15) 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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], i107[3] - i98[3], i98[3])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])

(3) -> (2), if ((i107[3] - i98[3] →* i107[2])∧(i100[3] →* i100[2])∧(i98[3] →* i98[2]))


(2) -> (3), if ((i98[2] →* i98[3])∧(i100[2] →* i100[3])∧(i107[2] →* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2] →* TRUE))



The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(TRUE, x0, x1, x2)

(16) 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 COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]) the following chains were created:
  • We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]), LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) which results in the following constraint:

    (1)    (i98[2]=i98[3]∧i100[2]=i100[3]∧i107[2]=i107[3]∧&&(>(i98[2], 0), >=(i107[2], i98[2]))=TRUE∧-(i107[3], i98[3])=i107[2]1∧i100[3]=i100[2]1∧i98[3]=i98[2]1 ⇒ COND_LOAD1302(TRUE, i100[3], i107[3], i98[3])≥NonInfC∧COND_LOAD1302(TRUE, i100[3], i107[3], i98[3])≥LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])∧(UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥))



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

    (2)    (>(i98[2], 0)=TRUE∧>=(i107[2], i98[2])=TRUE ⇒ COND_LOAD1302(TRUE, i100[2], i107[2], i98[2])≥NonInfC∧COND_LOAD1302(TRUE, i100[2], i107[2], i98[2])≥LOAD1302(i100[2], -(i107[2], i98[2]), i98[2])∧(UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥))



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

    (3)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)



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

    (4)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)



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

    (5)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)



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

    (6)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)



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

    (7)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-3)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[1 + (-1)bso_16] + [2]i98[2] ≥ 0)



    We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (8)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[1 + (-1)bso_16] + [2]i98[2] ≥ 0)







For Pair LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) the following chains were created:
  • We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]) which results in the following constraint:

    (9)    (i98[2]=i98[3]∧i100[2]=i100[3]∧i107[2]=i107[3]∧&&(>(i98[2], 0), >=(i107[2], i98[2]))=TRUE ⇒ LOAD1302(i100[2], i107[2], i98[2])≥NonInfC∧LOAD1302(i100[2], i107[2], i98[2])≥COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])∧(UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥))



    We simplified constraint (9) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (10)    (>(i98[2], 0)=TRUE∧>=(i107[2], i98[2])=TRUE ⇒ LOAD1302(i100[2], i107[2], i98[2])≥NonInfC∧LOAD1302(i100[2], i107[2], i98[2])≥COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])∧(UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥))



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

    (11)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (12)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (13)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (14)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)



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

    (15)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (16)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])
    • (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[1 + (-1)bso_16] + [2]i98[2] ≥ 0)

  • LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])
    • (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧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) = [1]   
POL(FALSE) = [2]   
POL(COND_LOAD1302(x1, x2, x3, x4)) = [-1] + [-1]x4 + [2]x3 + [-1]x1   
POL(LOAD1302(x1, x2, x3)) = [-1] + [-1]x3 + [2]x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(>=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])

The following pairs are in Pbound:

COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])
LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])

The following pairs are in P≥:

LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])

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

(17) Complex Obligation (AND)

(18) 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:
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])


The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(TRUE, x0, x1, x2)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE

(21) 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:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(TRUE, x0, x1, x2)

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(23) TRUE

(24) 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_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1])
(4): LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(i107[4] < i98[4], i100[4], i107[4], i98[4])
(5): COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5])

(1) -> (4), if ((i98[1] →* i98[4])∧(i100[1] →* i107[4])∧(i100[1] →* i100[4]))


(4) -> (5), if ((i100[4] →* i100[5])∧(i98[4] →* i98[5])∧(i107[4] →* i107[5])∧(i107[4] < i98[4] →* TRUE))



The set Q consists of the following terms:
Load1169(x0, x1)
Cond_Load1169(TRUE, x0, x1)
Load1302(x0, x1, x2)
Cond_Load1302(TRUE, x0, x1, x2)
Cond_Load13021(TRUE, x0, x1, x2)

(25) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(26) TRUE