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: GCD2

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 204 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:
Load995(i86, i90) → Cond_Load995(i86 >= 0 && i90 > 0 && !(i86 = i90), i86, i90)
Cond_Load995(TRUE, i86, i90) → Load1160(i90, i90, i90, i86, i90)
Load1160(i90, i90, i90, i104, i90) → Cond_Load1160(i90 > 0 && i104 > i90, i90, i90, i90, i104, i90)
Cond_Load1160(TRUE, i90, i90, i90, i104, i90) → Load1160(i90, i90, i90, i104 - i90, i90)
Load995(i90, i90) → Cond_Load9951(i90 > 0, i90, i90)
Cond_Load9951(TRUE, i90, i90) → Store1193(i90, i90, 0)
Store1193(i90, i90, i104) → Load995(i90, i104)
Load1160(i90, i90, i90, i104, i90) → Cond_Load11601(i90 > 0 && i104 <= i90, i90, i90, i90, i104, i90)
Cond_Load11601(TRUE, i90, i90, i90, i104, i90) → Load995(i90, i104)
The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x0, x0, x1, x0)
Cond_Load1160(TRUE, x0, x0, x0, x1, x0)
Cond_Load9951(TRUE, x0, x0)
Store1193(x0, x0, x1)
Cond_Load11601(TRUE, x0, x0, x0, x1, x0)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Cond_Load11601(x1, x2, x3, x4, x5, x6) → Cond_Load11601(x1, x5, x6)
Store1193(x1, x2, x3) → Store1193(x2, x3)
Cond_Load9951(x1, x2, x3) → Cond_Load9951(x1, x3)
Load1160(x1, x2, x3, x4, x5) → Load1160(x4, x5)
Cond_Load1160(x1, x2, x3, x4, x5, x6) → Cond_Load1160(x1, x5, x6)

(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:
Load995(i86, i90) → Cond_Load995(i86 >= 0 && i90 > 0 && !(i86 = i90), i86, i90)
Cond_Load995(TRUE, i86, i90) → Load1160(i86, i90)
Load1160(i104, i90) → Cond_Load1160(i90 > 0 && i104 > i90, i104, i90)
Cond_Load1160(TRUE, i104, i90) → Load1160(i104 - i90, i90)
Load995(i90, i90) → Cond_Load9951(i90 > 0, i90)
Cond_Load9951(TRUE, i90) → Store1193(i90, 0)
Store1193(i90, i104) → Load995(i90, i104)
Load1160(i104, i90) → Cond_Load11601(i90 > 0 && i104 <= i90, i104, i90)
Cond_Load11601(TRUE, i104, i90) → Load995(i90, i104)
The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(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:
Load995(i86, i90) → Cond_Load995(i86 >= 0 && i90 > 0 && !(i86 = i90), i86, i90)
Cond_Load995(TRUE, i86, i90) → Load1160(i86, i90)
Load1160(i104, i90) → Cond_Load1160(i90 > 0 && i104 > i90, i104, i90)
Cond_Load1160(TRUE, i104, i90) → Load1160(i104 - i90, i90)
Load995(i90, i90) → Cond_Load9951(i90 > 0, i90)
Cond_Load9951(TRUE, i90) → Store1193(i90, 0)
Store1193(i90, i104) → Load995(i90, i104)
Load1160(i104, i90) → Cond_Load11601(i90 > 0 && i104 <= i90, i104, i90)
Cond_Load11601(TRUE, i104, i90) → Load995(i90, i104)

The integer pair graph contains the following rules and edges:
(0): LOAD995(i86[0], i90[0]) → COND_LOAD995(i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]), i86[0], i90[0])
(1): COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
(2): LOAD1160(i104[2], i90[2]) → COND_LOAD1160(i90[2] > 0 && i104[2] > i90[2], i104[2], i90[2])
(3): COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(i104[3] - i90[3], i90[3])
(4): LOAD995(i90[4], i90[4]) → COND_LOAD9951(i90[4] > 0, i90[4])
(5): COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0)
(6): STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6])
(7): LOAD1160(i104[7], i90[7]) → COND_LOAD11601(i90[7] > 0 && i104[7] <= i90[7], i104[7], i90[7])
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])

(0) -> (1), if ((i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]) →* TRUE)∧(i86[0] →* i86[1])∧(i90[0] →* i90[1]))


(1) -> (2), if ((i90[1] →* i90[2])∧(i86[1] →* i104[2]))


(1) -> (7), if ((i86[1] →* i104[7])∧(i90[1] →* i90[7]))


(2) -> (3), if ((i90[2] →* i90[3])∧(i90[2] > 0 && i104[2] > i90[2] →* TRUE)∧(i104[2] →* i104[3]))


(3) -> (2), if ((i90[3] →* i90[2])∧(i104[3] - i90[3] →* i104[2]))


(3) -> (7), if ((i90[3] →* i90[7])∧(i104[3] - i90[3] →* i104[7]))


(4) -> (5), if ((i90[4] →* i90[5])∧(i90[4] > 0 →* TRUE))


(5) -> (6), if ((i90[5] →* i90[6])∧(0 →* i104[6]))


(6) -> (0), if ((i104[6] →* i90[0])∧(i90[6] →* i86[0]))


(6) -> (4), if ((i90[6] →* i90[4])∧(i104[6] →* i90[4]))


(7) -> (8), if ((i104[7] →* i104[8])∧(i90[7] > 0 && i104[7] <= i90[7] →* TRUE)∧(i90[7] →* i90[8]))


(8) -> (0), if ((i104[8] →* i90[0])∧(i90[8] →* i86[0]))


(8) -> (4), if ((i104[8] →* i90[4])∧(i90[8] →* i90[4]))



The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(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): LOAD995(i86[0], i90[0]) → COND_LOAD995(i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]), i86[0], i90[0])
(1): COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
(2): LOAD1160(i104[2], i90[2]) → COND_LOAD1160(i90[2] > 0 && i104[2] > i90[2], i104[2], i90[2])
(3): COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(i104[3] - i90[3], i90[3])
(4): LOAD995(i90[4], i90[4]) → COND_LOAD9951(i90[4] > 0, i90[4])
(5): COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0)
(6): STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6])
(7): LOAD1160(i104[7], i90[7]) → COND_LOAD11601(i90[7] > 0 && i104[7] <= i90[7], i104[7], i90[7])
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])

(0) -> (1), if ((i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]) →* TRUE)∧(i86[0] →* i86[1])∧(i90[0] →* i90[1]))


(1) -> (2), if ((i90[1] →* i90[2])∧(i86[1] →* i104[2]))


(1) -> (7), if ((i86[1] →* i104[7])∧(i90[1] →* i90[7]))


(2) -> (3), if ((i90[2] →* i90[3])∧(i90[2] > 0 && i104[2] > i90[2] →* TRUE)∧(i104[2] →* i104[3]))


(3) -> (2), if ((i90[3] →* i90[2])∧(i104[3] - i90[3] →* i104[2]))


(3) -> (7), if ((i90[3] →* i90[7])∧(i104[3] - i90[3] →* i104[7]))


(4) -> (5), if ((i90[4] →* i90[5])∧(i90[4] > 0 →* TRUE))


(5) -> (6), if ((i90[5] →* i90[6])∧(0 →* i104[6]))


(6) -> (0), if ((i104[6] →* i90[0])∧(i90[6] →* i86[0]))


(6) -> (4), if ((i90[6] →* i90[4])∧(i104[6] →* i90[4]))


(7) -> (8), if ((i104[7] →* i104[8])∧(i90[7] > 0 && i104[7] <= i90[7] →* TRUE)∧(i90[7] →* i90[8]))


(8) -> (0), if ((i104[8] →* i90[0])∧(i90[8] →* i86[0]))


(8) -> (4), if ((i104[8] →* i90[4])∧(i90[8] →* i90[4]))



The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(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 LOAD995(i86, i90) → COND_LOAD995(&&(&&(>=(i86, 0), >(i90, 0)), !(=(i86, i90))), i86, i90) the following chains were created:
  • We consider the chain LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]), COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]) which results in the following constraint:

    (1)    (&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0])))=TRUE∧i86[0]=i86[1]∧i90[0]=i90[1] ⇒ LOAD995(i86[0], i90[0])≥NonInfC∧LOAD995(i86[0], i90[0])≥COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])∧(UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥))



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

    (2)    (>=(i86[0], 0)=TRUE∧>(i90[0], 0)=TRUE∧<(i86[0], i90[0])=TRUE ⇒ LOAD995(i86[0], i90[0])≥NonInfC∧LOAD995(i86[0], i90[0])≥COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])∧(UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥))


    (3)    (>=(i86[0], 0)=TRUE∧>(i90[0], 0)=TRUE∧>(i86[0], i90[0])=TRUE ⇒ LOAD995(i86[0], i90[0])≥NonInfC∧LOAD995(i86[0], i90[0])≥COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])∧(UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥))



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

    (4)    (i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (5)    (i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (6)    (i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (7)    (i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (8)    (i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (9)    (i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (10)    (i86[0] ≥ 0∧i90[0] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (11)    ([1] + i90[0] + i86[0] ≥ 0∧i90[0] + [-1] ≥ 0∧i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (12)    (i86[0] ≥ 0∧i86[0] + i90[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]i86[0] + [bni_33]i90[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (13)    ([2] + i90[0] + i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_33 + (2)bni_33] + [(2)bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)







For Pair COND_LOAD995(TRUE, i86, i90) → LOAD1160(i86, i90) the following chains were created:
  • We consider the chain COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2]) which results in the following constraint:

    (14)    (i90[1]=i90[2]∧i86[1]=i104[2] ⇒ COND_LOAD995(TRUE, i86[1], i90[1])≥NonInfC∧COND_LOAD995(TRUE, i86[1], i90[1])≥LOAD1160(i86[1], i90[1])∧(UIncreasing(LOAD1160(i86[1], i90[1])), ≥))



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

    (15)    (COND_LOAD995(TRUE, i86[1], i90[1])≥NonInfC∧COND_LOAD995(TRUE, i86[1], i90[1])≥LOAD1160(i86[1], i90[1])∧(UIncreasing(LOAD1160(i86[1], i90[1])), ≥))



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

    (16)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)bso_36] ≥ 0)



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

    (17)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)bso_36] ≥ 0)



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

    (18)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)bso_36] ≥ 0)



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

    (19)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)



  • We consider the chain COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]) which results in the following constraint:

    (20)    (i86[1]=i104[7]∧i90[1]=i90[7] ⇒ COND_LOAD995(TRUE, i86[1], i90[1])≥NonInfC∧COND_LOAD995(TRUE, i86[1], i90[1])≥LOAD1160(i86[1], i90[1])∧(UIncreasing(LOAD1160(i86[1], i90[1])), ≥))



    We simplified constraint (20) using rule (IV) which results in the following new constraint:

    (21)    (COND_LOAD995(TRUE, i86[1], i90[1])≥NonInfC∧COND_LOAD995(TRUE, i86[1], i90[1])≥LOAD1160(i86[1], i90[1])∧(UIncreasing(LOAD1160(i86[1], i90[1])), ≥))



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

    (22)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)bso_36] ≥ 0)



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

    (23)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)bso_36] ≥ 0)



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

    (24)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)bso_36] ≥ 0)



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

    (25)    ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)







For Pair LOAD1160(i104, i90) → COND_LOAD1160(&&(>(i90, 0), >(i104, i90)), i104, i90) the following chains were created:
  • We consider the chain LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2]), COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(-(i104[3], i90[3]), i90[3]) which results in the following constraint:

    (26)    (i90[2]=i90[3]∧&&(>(i90[2], 0), >(i104[2], i90[2]))=TRUE∧i104[2]=i104[3] ⇒ LOAD1160(i104[2], i90[2])≥NonInfC∧LOAD1160(i104[2], i90[2])≥COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])∧(UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥))



    We simplified constraint (26) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (27)    (>(i90[2], 0)=TRUE∧>(i104[2], i90[2])=TRUE ⇒ LOAD1160(i104[2], i90[2])≥NonInfC∧LOAD1160(i104[2], i90[2])≥COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])∧(UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥))



    We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (28)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]i90[2] + [bni_37]i104[2] ≥ 0∧[(-1)bso_38] ≥ 0)



    We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (29)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]i90[2] + [bni_37]i104[2] ≥ 0∧[(-1)bso_38] ≥ 0)



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

    (30)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]i90[2] + [bni_37]i104[2] ≥ 0∧[(-1)bso_38] ≥ 0)



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

    (31)    (i90[2] ≥ 0∧i104[2] + [-2] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥)∧[(-1)Bound*bni_37] + [bni_37]i90[2] + [bni_37]i104[2] ≥ 0∧[(-1)bso_38] ≥ 0)



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

    (32)    (i90[2] ≥ 0∧i104[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥)∧[(-1)Bound*bni_37 + (2)bni_37] + [(2)bni_37]i90[2] + [bni_37]i104[2] ≥ 0∧[(-1)bso_38] ≥ 0)







For Pair COND_LOAD1160(TRUE, i104, i90) → LOAD1160(-(i104, i90), i90) the following chains were created:
  • We consider the chain LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2]), COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(-(i104[3], i90[3]), i90[3]), LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2]) which results in the following constraint:

    (33)    (i90[2]=i90[3]∧&&(>(i90[2], 0), >(i104[2], i90[2]))=TRUE∧i104[2]=i104[3]∧i90[3]=i90[2]1∧-(i104[3], i90[3])=i104[2]1 ⇒ COND_LOAD1160(TRUE, i104[3], i90[3])≥NonInfC∧COND_LOAD1160(TRUE, i104[3], i90[3])≥LOAD1160(-(i104[3], i90[3]), i90[3])∧(UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥))



    We simplified constraint (33) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (34)    (>(i90[2], 0)=TRUE∧>(i104[2], i90[2])=TRUE ⇒ COND_LOAD1160(TRUE, i104[2], i90[2])≥NonInfC∧COND_LOAD1160(TRUE, i104[2], i90[2])≥LOAD1160(-(i104[2], i90[2]), i90[2])∧(UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥))



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

    (35)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[(-1)bso_40] + i90[2] ≥ 0)



    We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (36)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[(-1)bso_40] + i90[2] ≥ 0)



    We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (37)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[(-1)bso_40] + i90[2] ≥ 0)



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

    (38)    (i90[2] ≥ 0∧i104[2] + [-2] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[1 + (-1)bso_40] + i90[2] ≥ 0)



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

    (39)    (i90[2] ≥ 0∧i104[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[1 + (-1)bso_40] + i90[2] ≥ 0)



  • We consider the chain LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2]), COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(-(i104[3], i90[3]), i90[3]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]) which results in the following constraint:

    (40)    (i90[2]=i90[3]∧&&(>(i90[2], 0), >(i104[2], i90[2]))=TRUE∧i104[2]=i104[3]∧i90[3]=i90[7]∧-(i104[3], i90[3])=i104[7] ⇒ COND_LOAD1160(TRUE, i104[3], i90[3])≥NonInfC∧COND_LOAD1160(TRUE, i104[3], i90[3])≥LOAD1160(-(i104[3], i90[3]), i90[3])∧(UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥))



    We simplified constraint (40) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (41)    (>(i90[2], 0)=TRUE∧>(i104[2], i90[2])=TRUE ⇒ COND_LOAD1160(TRUE, i104[2], i90[2])≥NonInfC∧COND_LOAD1160(TRUE, i104[2], i90[2])≥LOAD1160(-(i104[2], i90[2]), i90[2])∧(UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥))



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

    (42)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[(-1)bso_40] + i90[2] ≥ 0)



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

    (43)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[(-1)bso_40] + i90[2] ≥ 0)



    We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (44)    (i90[2] + [-1] ≥ 0∧i104[2] + [-1] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[(-1)bso_40] + i90[2] ≥ 0)



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

    (45)    (i90[2] ≥ 0∧i104[2] + [-2] + [-1]i90[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[1 + (-1)bso_40] + i90[2] ≥ 0)



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

    (46)    (i90[2] ≥ 0∧i104[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[1 + (-1)bso_40] + i90[2] ≥ 0)







For Pair LOAD995(i90, i90) → COND_LOAD9951(>(i90, 0), i90) the following chains were created:
  • We consider the chain LOAD995(i90[4], i90[4]) → COND_LOAD9951(>(i90[4], 0), i90[4]), COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0) which results in the following constraint:

    (47)    (i90[4]=i90[5]∧>(i90[4], 0)=TRUE ⇒ LOAD995(i90[4], i90[4])≥NonInfC∧LOAD995(i90[4], i90[4])≥COND_LOAD9951(>(i90[4], 0), i90[4])∧(UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥))



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

    (48)    (>(i90[4], 0)=TRUE ⇒ LOAD995(i90[4], i90[4])≥NonInfC∧LOAD995(i90[4], i90[4])≥COND_LOAD9951(>(i90[4], 0), i90[4])∧(UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥))



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

    (49)    (i90[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [(2)bni_41]i90[4] ≥ 0∧[-1 + (-1)bso_42] + i90[4] ≥ 0)



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

    (50)    (i90[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [(2)bni_41]i90[4] ≥ 0∧[-1 + (-1)bso_42] + i90[4] ≥ 0)



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

    (51)    (i90[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [(2)bni_41]i90[4] ≥ 0∧[-1 + (-1)bso_42] + i90[4] ≥ 0)



    We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (52)    (i90[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥)∧[bni_41 + (-1)Bound*bni_41] + [(2)bni_41]i90[4] ≥ 0∧[(-1)bso_42] + i90[4] ≥ 0)







For Pair COND_LOAD9951(TRUE, i90) → STORE1193(i90, 0) the following chains were created:
  • We consider the chain LOAD995(i90[4], i90[4]) → COND_LOAD9951(>(i90[4], 0), i90[4]), COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0), STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6]) which results in the following constraint:

    (53)    (i90[4]=i90[5]∧>(i90[4], 0)=TRUE∧i90[5]=i90[6]∧0=i104[6] ⇒ COND_LOAD9951(TRUE, i90[5])≥NonInfC∧COND_LOAD9951(TRUE, i90[5])≥STORE1193(i90[5], 0)∧(UIncreasing(STORE1193(i90[5], 0)), ≥))



    We simplified constraint (53) using rules (III), (IV) which results in the following new constraint:

    (54)    (>(i90[4], 0)=TRUE ⇒ COND_LOAD9951(TRUE, i90[4])≥NonInfC∧COND_LOAD9951(TRUE, i90[4])≥STORE1193(i90[4], 0)∧(UIncreasing(STORE1193(i90[5], 0)), ≥))



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

    (55)    (i90[4] + [-1] ≥ 0 ⇒ (UIncreasing(STORE1193(i90[5], 0)), ≥)∧[(-1)Bound*bni_43] + [bni_43]i90[4] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (56)    (i90[4] + [-1] ≥ 0 ⇒ (UIncreasing(STORE1193(i90[5], 0)), ≥)∧[(-1)Bound*bni_43] + [bni_43]i90[4] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (57)    (i90[4] + [-1] ≥ 0 ⇒ (UIncreasing(STORE1193(i90[5], 0)), ≥)∧[(-1)Bound*bni_43] + [bni_43]i90[4] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (58)    (i90[4] ≥ 0 ⇒ (UIncreasing(STORE1193(i90[5], 0)), ≥)∧[(-1)Bound*bni_43 + bni_43] + [bni_43]i90[4] ≥ 0∧[(-1)bso_44] ≥ 0)







For Pair STORE1193(i90, i104) → LOAD995(i90, i104) the following chains were created:
  • We consider the chain STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]) which results in the following constraint:

    (59)    (i104[6]=i90[0]∧i90[6]=i86[0] ⇒ STORE1193(i90[6], i104[6])≥NonInfC∧STORE1193(i90[6], i104[6])≥LOAD995(i90[6], i104[6])∧(UIncreasing(LOAD995(i90[6], i104[6])), ≥))



    We simplified constraint (59) using rule (IV) which results in the following new constraint:

    (60)    (STORE1193(i90[6], i104[6])≥NonInfC∧STORE1193(i90[6], i104[6])≥LOAD995(i90[6], i104[6])∧(UIncreasing(LOAD995(i90[6], i104[6])), ≥))



    We simplified constraint (60) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (61)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧[1 + (-1)bso_46] ≥ 0)



    We simplified constraint (61) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (62)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧[1 + (-1)bso_46] ≥ 0)



    We simplified constraint (62) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (63)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧[1 + (-1)bso_46] ≥ 0)



    We simplified constraint (63) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (64)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_46] ≥ 0)



  • We consider the chain STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6]), LOAD995(i90[4], i90[4]) → COND_LOAD9951(>(i90[4], 0), i90[4]) which results in the following constraint:

    (65)    (i90[6]=i90[4]∧i104[6]=i90[4] ⇒ STORE1193(i90[6], i104[6])≥NonInfC∧STORE1193(i90[6], i104[6])≥LOAD995(i90[6], i104[6])∧(UIncreasing(LOAD995(i90[6], i104[6])), ≥))



    We simplified constraint (65) using rule (III) which results in the following new constraint:

    (66)    (STORE1193(i104[6], i104[6])≥NonInfC∧STORE1193(i104[6], i104[6])≥LOAD995(i104[6], i104[6])∧(UIncreasing(LOAD995(i90[6], i104[6])), ≥))



    We simplified constraint (66) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (67)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧[1 + (-1)bso_46] ≥ 0)



    We simplified constraint (67) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (68)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧[1 + (-1)bso_46] ≥ 0)



    We simplified constraint (68) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (69)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧[1 + (-1)bso_46] ≥ 0)



    We simplified constraint (69) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (70)    ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧0 = 0∧[1 + (-1)bso_46] ≥ 0)







For Pair LOAD1160(i104, i90) → COND_LOAD11601(&&(>(i90, 0), <=(i104, i90)), i104, i90) the following chains were created:
  • We consider the chain LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]), COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]) which results in the following constraint:

    (71)    (i104[7]=i104[8]∧&&(>(i90[7], 0), <=(i104[7], i90[7]))=TRUE∧i90[7]=i90[8] ⇒ LOAD1160(i104[7], i90[7])≥NonInfC∧LOAD1160(i104[7], i90[7])≥COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])∧(UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥))



    We simplified constraint (71) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (72)    (>(i90[7], 0)=TRUE∧<=(i104[7], i90[7])=TRUE ⇒ LOAD1160(i104[7], i90[7])≥NonInfC∧LOAD1160(i104[7], i90[7])≥COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])∧(UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥))



    We simplified constraint (72) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (73)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]i90[7] + [bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (73) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (74)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]i90[7] + [bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (74) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (75)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] + [bni_47]i90[7] + [bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (75) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (76)    (i90[7] ≥ 0∧[1] + i90[7] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_47] + [bni_47]i90[7] + [bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (76) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (77)    (i90[7] ≥ 0∧[1] + i90[7] + [-1]i104[7] ≥ 0∧i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_47] + [bni_47]i90[7] + [bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)


    (78)    (i90[7] ≥ 0∧[1] + i90[7] + i104[7] ≥ 0∧i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_47] + [bni_47]i90[7] + [(-1)bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)







For Pair COND_LOAD11601(TRUE, i104, i90) → LOAD995(i90, i104) the following chains were created:
  • We consider the chain COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]) which results in the following constraint:

    (79)    (i104[8]=i90[0]∧i90[8]=i86[0] ⇒ COND_LOAD11601(TRUE, i104[8], i90[8])≥NonInfC∧COND_LOAD11601(TRUE, i104[8], i90[8])≥LOAD995(i90[8], i104[8])∧(UIncreasing(LOAD995(i90[8], i104[8])), ≥))



    We simplified constraint (79) using rule (IV) which results in the following new constraint:

    (80)    (COND_LOAD11601(TRUE, i104[8], i90[8])≥NonInfC∧COND_LOAD11601(TRUE, i104[8], i90[8])≥LOAD995(i90[8], i104[8])∧(UIncreasing(LOAD995(i90[8], i104[8])), ≥))



    We simplified constraint (80) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (81)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)bso_50] ≥ 0)



    We simplified constraint (81) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (82)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)bso_50] ≥ 0)



    We simplified constraint (82) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (83)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)bso_50] ≥ 0)



    We simplified constraint (83) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (84)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_50] ≥ 0)



  • We consider the chain COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i90[4], i90[4]) → COND_LOAD9951(>(i90[4], 0), i90[4]) which results in the following constraint:

    (85)    (i104[8]=i90[4]∧i90[8]=i90[4] ⇒ COND_LOAD11601(TRUE, i104[8], i90[8])≥NonInfC∧COND_LOAD11601(TRUE, i104[8], i90[8])≥LOAD995(i90[8], i104[8])∧(UIncreasing(LOAD995(i90[8], i104[8])), ≥))



    We simplified constraint (85) using rule (III) which results in the following new constraint:

    (86)    (COND_LOAD11601(TRUE, i90[8], i90[8])≥NonInfC∧COND_LOAD11601(TRUE, i90[8], i90[8])≥LOAD995(i90[8], i90[8])∧(UIncreasing(LOAD995(i90[8], i104[8])), ≥))



    We simplified constraint (86) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (87)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)bso_50] ≥ 0)



    We simplified constraint (87) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (88)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)bso_50] ≥ 0)



    We simplified constraint (88) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (89)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)bso_50] ≥ 0)



    We simplified constraint (89) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (90)    ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧0 = 0∧[(-1)bso_50] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD995(i86, i90) → COND_LOAD995(&&(&&(>=(i86, 0), >(i90, 0)), !(=(i86, i90))), i86, i90)
    • (i86[0] ≥ 0∧i86[0] + i90[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]i86[0] + [bni_33]i90[0] ≥ 0∧[(-1)bso_34] ≥ 0)
    • ([2] + i90[0] + i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_33 + (2)bni_33] + [(2)bni_33]i90[0] + [bni_33]i86[0] ≥ 0∧[(-1)bso_34] ≥ 0)

  • COND_LOAD995(TRUE, i86, i90) → LOAD1160(i86, i90)
    • ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)
    • ((UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

  • LOAD1160(i104, i90) → COND_LOAD1160(&&(>(i90, 0), >(i104, i90)), i104, i90)
    • (i90[2] ≥ 0∧i104[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])), ≥)∧[(-1)Bound*bni_37 + (2)bni_37] + [(2)bni_37]i90[2] + [bni_37]i104[2] ≥ 0∧[(-1)bso_38] ≥ 0)

  • COND_LOAD1160(TRUE, i104, i90) → LOAD1160(-(i104, i90), i90)
    • (i90[2] ≥ 0∧i104[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[1 + (-1)bso_40] + i90[2] ≥ 0)
    • (i90[2] ≥ 0∧i104[2] ≥ 0 ⇒ (UIncreasing(LOAD1160(-(i104[3], i90[3]), i90[3])), ≥)∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]i90[2] + [bni_39]i104[2] ≥ 0∧[1 + (-1)bso_40] + i90[2] ≥ 0)

  • LOAD995(i90, i90) → COND_LOAD9951(>(i90, 0), i90)
    • (i90[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD9951(>(i90[4], 0), i90[4])), ≥)∧[bni_41 + (-1)Bound*bni_41] + [(2)bni_41]i90[4] ≥ 0∧[(-1)bso_42] + i90[4] ≥ 0)

  • COND_LOAD9951(TRUE, i90) → STORE1193(i90, 0)
    • (i90[4] ≥ 0 ⇒ (UIncreasing(STORE1193(i90[5], 0)), ≥)∧[(-1)Bound*bni_43 + bni_43] + [bni_43]i90[4] ≥ 0∧[(-1)bso_44] ≥ 0)

  • STORE1193(i90, i104) → LOAD995(i90, i104)
    • ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_46] ≥ 0)
    • ((UIncreasing(LOAD995(i90[6], i104[6])), ≥)∧0 = 0∧[1 + (-1)bso_46] ≥ 0)

  • LOAD1160(i104, i90) → COND_LOAD11601(&&(>(i90, 0), <=(i104, i90)), i104, i90)
    • (i90[7] ≥ 0∧[1] + i90[7] + [-1]i104[7] ≥ 0∧i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_47] + [bni_47]i90[7] + [bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)
    • (i90[7] ≥ 0∧[1] + i90[7] + i104[7] ≥ 0∧i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_47] + [bni_47]i90[7] + [(-1)bni_47]i104[7] ≥ 0∧[(-1)bso_48] ≥ 0)

  • COND_LOAD11601(TRUE, i104, i90) → LOAD995(i90, i104)
    • ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_50] ≥ 0)
    • ((UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧0 = 0∧[(-1)bso_50] ≥ 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(LOAD995(x1, x2)) = [-1] + x2 + x1   
POL(COND_LOAD995(x1, x2, x3)) = [-1] + x3 + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   
POL(LOAD1160(x1, x2)) = [-1] + x2 + x1   
POL(COND_LOAD1160(x1, x2, x3)) = [-1] + x3 + x2 + [-1]x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(COND_LOAD9951(x1, x2)) = x2   
POL(STORE1193(x1, x2)) = x2 + x1   
POL(COND_LOAD11601(x1, x2, x3)) = [-1] + x3 + x2 + [-1]x1   
POL(<=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(-(i104[3], i90[3]), i90[3])
STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6])

The following pairs are in Pbound:

LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])
LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])
COND_LOAD1160(TRUE, i104[3], i90[3]) → LOAD1160(-(i104[3], i90[3]), i90[3])
LOAD995(i90[4], i90[4]) → COND_LOAD9951(>(i90[4], 0), i90[4])
COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0)

The following pairs are in P≥:

LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])
COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
LOAD1160(i104[2], i90[2]) → COND_LOAD1160(&&(>(i90[2], 0), >(i104[2], i90[2])), i104[2], i90[2])
LOAD995(i90[4], i90[4]) → COND_LOAD9951(>(i90[4], 0), i90[4])
COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0)
LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])
COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])

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

&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
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): LOAD995(i86[0], i90[0]) → COND_LOAD995(i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]), i86[0], i90[0])
(1): COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
(2): LOAD1160(i104[2], i90[2]) → COND_LOAD1160(i90[2] > 0 && i104[2] > i90[2], i104[2], i90[2])
(4): LOAD995(i90[4], i90[4]) → COND_LOAD9951(i90[4] > 0, i90[4])
(5): COND_LOAD9951(TRUE, i90[5]) → STORE1193(i90[5], 0)
(7): LOAD1160(i104[7], i90[7]) → COND_LOAD11601(i90[7] > 0 && i104[7] <= i90[7], i104[7], i90[7])
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])

(8) -> (0), if ((i104[8] →* i90[0])∧(i90[8] →* i86[0]))


(0) -> (1), if ((i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]) →* TRUE)∧(i86[0] →* i86[1])∧(i90[0] →* i90[1]))


(1) -> (2), if ((i90[1] →* i90[2])∧(i86[1] →* i104[2]))


(8) -> (4), if ((i104[8] →* i90[4])∧(i90[8] →* i90[4]))


(4) -> (5), if ((i90[4] →* i90[5])∧(i90[4] > 0 →* TRUE))


(1) -> (7), if ((i86[1] →* i104[7])∧(i90[1] →* i90[7]))


(7) -> (8), if ((i104[7] →* i104[8])∧(i90[7] > 0 && i104[7] <= i90[7] →* TRUE)∧(i90[7] →* i90[8]))



The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])
(7): LOAD1160(i104[7], i90[7]) → COND_LOAD11601(i90[7] > 0 && i104[7] <= i90[7], i104[7], i90[7])
(1): COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
(0): LOAD995(i86[0], i90[0]) → COND_LOAD995(i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]), i86[0], i90[0])

(8) -> (0), if ((i104[8] →* i90[0])∧(i90[8] →* i86[0]))


(0) -> (1), if ((i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]) →* TRUE)∧(i86[0] →* i86[1])∧(i90[0] →* i90[1]))


(1) -> (7), if ((i86[1] →* i104[7])∧(i90[1] →* i90[7]))


(7) -> (8), if ((i104[7] →* i104[8])∧(i90[7] > 0 && i104[7] <= i90[7] →* TRUE)∧(i90[7] →* i90[8]))



The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(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_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]) the following chains were created:
  • We consider the chain LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]), COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]), COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]), COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]) which results in the following constraint:

    (1)    (&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0])))=TRUE∧i86[0]=i86[1]∧i90[0]=i90[1]∧i86[1]=i104[7]∧i90[1]=i90[7]∧i104[7]=i104[8]∧&&(>(i90[7], 0), <=(i104[7], i90[7]))=TRUE∧i90[7]=i90[8]∧i104[8]=i90[0]1∧i90[8]=i86[0]1∧&&(&&(>=(i86[0]1, 0), >(i90[0]1, 0)), !(=(i86[0]1, i90[0]1)))=TRUE∧i86[0]1=i86[1]1∧i90[0]1=i90[1]1 ⇒ COND_LOAD11601(TRUE, i104[8], i90[8])≥NonInfC∧COND_LOAD11601(TRUE, i104[8], i90[8])≥LOAD995(i90[8], i104[8])∧(UIncreasing(LOAD995(i90[8], i104[8])), ≥))



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

    (2)    (>(i90[0], 0)=TRUE∧<=(i86[0], i90[0])=TRUE∧>=(i86[0], 0)=TRUE∧>=(i90[0], 0)=TRUE∧>(i86[0], 0)=TRUE∧<(i86[0], i90[0])=TRUE∧<(i90[0], i86[0])=TRUE ⇒ COND_LOAD11601(TRUE, i86[0], i90[0])≥NonInfC∧COND_LOAD11601(TRUE, i86[0], i90[0])≥LOAD995(i90[0], i86[0])∧(UIncreasing(LOAD995(i90[8], i104[8])), ≥))



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

    (3)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (4)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (6)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (8)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (10)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (11)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (12)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (13)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (14)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



    We solved constraint (11) using rule (IDP_SMT_SPLIT).We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (15)    (i90[0] ≥ 0∧[1] + i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧[1] + i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23 + (-1)bni_23] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



    We solved constraint (13) using rule (IDP_SMT_SPLIT).We solved constraint (14) using rule (IDP_SMT_SPLIT).We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (16)    (i86[0] + i90[0] ≥ 0∧[1] + i90[0] ≥ 0∧i86[0] ≥ 0∧[1] + i86[0] + i90[0] ≥ 0∧i86[0] + [-1] ≥ 0∧i90[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23 + (-1)bni_23] + [(-1)bni_23]i86[0] + [(-1)bni_23]i90[0] ≥ 0∧[-1 + (-1)bso_24] + i86[0] ≥ 0)



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

    (17)    ([1] + i86[0] + i90[0] ≥ 0∧[1] + i90[0] ≥ 0∧[1] + i86[0] ≥ 0∧[2] + i86[0] + i90[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23 + (-2)bni_23] + [(-1)bni_23]i86[0] + [(-1)bni_23]i90[0] ≥ 0∧[(-1)bso_24] + i86[0] ≥ 0)







For Pair LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]) the following chains were created:
  • We consider the chain COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]), COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]), COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]) which results in the following constraint:

    (18)    (i104[8]=i90[0]∧i90[8]=i86[0]∧&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0])))=TRUE∧i86[0]=i86[1]∧i90[0]=i90[1]∧i86[1]=i104[7]∧i90[1]=i90[7]∧i104[7]=i104[8]1∧&&(>(i90[7], 0), <=(i104[7], i90[7]))=TRUE∧i90[7]=i90[8]1∧i104[8]1=i90[0]1∧i90[8]1=i86[0]1 ⇒ LOAD1160(i104[7], i90[7])≥NonInfC∧LOAD1160(i104[7], i90[7])≥COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])∧(UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥))



    We simplified constraint (18) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (19)    (>(i90[0], 0)=TRUE∧<=(i86[0], i90[0])=TRUE∧>=(i86[0], 0)=TRUE∧<(i86[0], i90[0])=TRUE ⇒ LOAD1160(i86[0], i90[0])≥NonInfC∧LOAD1160(i86[0], i90[0])≥COND_LOAD11601(&&(>(i90[0], 0), <=(i86[0], i90[0])), i86[0], i90[0])∧(UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥))



    We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (20)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i90[0] + [(-1)bni_25]i86[0] ≥ 0∧[-1 + (-1)bso_26] + [2]i90[0] + [-1]i86[0] ≥ 0)



    We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (21)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i90[0] + [(-1)bni_25]i86[0] ≥ 0∧[-1 + (-1)bso_26] + [2]i90[0] + [-1]i86[0] ≥ 0)



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

    (23)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i90[0] + [(-1)bni_25]i86[0] ≥ 0∧[-1 + (-1)bso_26] + [2]i90[0] + [-1]i86[0] ≥ 0)



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

    (24)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] + [-1] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i90[0] + [(-1)bni_25]i86[0] ≥ 0∧[-1 + (-1)bso_26] + [2]i90[0] + [-1]i86[0] ≥ 0)



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

    (25)    (i90[0] + [-1] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i86[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i90[0] + [(-1)bni_25]i86[0] ≥ 0∧[-1 + (-1)bso_26] + [2]i90[0] + [-1]i86[0] ≥ 0)



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

    (26)    (i90[0] ≥ 0∧[1] + i90[0] + [-1]i86[0] ≥ 0∧i86[0] ≥ 0∧i90[0] + [-1]i86[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i90[0] + [(-1)bni_25]i86[0] ≥ 0∧[1 + (-1)bso_26] + [2]i90[0] + [-1]i86[0] ≥ 0)



    We solved constraint (25) using rule (IDP_SMT_SPLIT).We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (i86[0] + i90[0] ≥ 0∧[1] + i90[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i90[0] ≥ 0∧[1 + (-1)bso_26] + i86[0] + [2]i90[0] ≥ 0)







For Pair COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]) the following chains were created:
  • We consider the chain LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]), COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]), COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]), COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]) which results in the following constraint:

    (28)    (i104[7]=i104[8]∧&&(>(i90[7], 0), <=(i104[7], i90[7]))=TRUE∧i90[7]=i90[8]∧i104[8]=i90[0]∧i90[8]=i86[0]∧&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0])))=TRUE∧i86[0]=i86[1]∧i90[0]=i90[1]∧i86[1]=i104[7]1∧i90[1]=i90[7]1∧i104[7]1=i104[8]1∧&&(>(i90[7]1, 0), <=(i104[7]1, i90[7]1))=TRUE∧i90[7]1=i90[8]1 ⇒ COND_LOAD995(TRUE, i86[1], i90[1])≥NonInfC∧COND_LOAD995(TRUE, i86[1], i90[1])≥LOAD1160(i86[1], i90[1])∧(UIncreasing(LOAD1160(i86[1], i90[1])), ≥))



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

    (29)    (>(i90[7], 0)=TRUE∧<=(i104[7], i90[7])=TRUE∧>(i104[7], 0)=TRUE∧<=(i90[7], i104[7])=TRUE∧>=(i90[7], 0)=TRUE∧<(i90[7], i104[7])=TRUE ⇒ COND_LOAD995(TRUE, i90[7], i104[7])≥NonInfC∧COND_LOAD995(TRUE, i90[7], i104[7])≥LOAD1160(i90[7], i104[7])∧(UIncreasing(LOAD1160(i86[1], i90[1])), ≥))



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

    (30)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1]i90[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] + [-1]i90[7] ≥ 0 ⇒ (UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]i104[7] + [(-1)bni_27]i90[7] ≥ 0∧[1 + (-1)bso_28] + [-2]i104[7] ≥ 0)



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

    (31)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1]i90[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] + [-1]i90[7] ≥ 0 ⇒ (UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]i104[7] + [(-1)bni_27]i90[7] ≥ 0∧[1 + (-1)bso_28] + [-2]i104[7] ≥ 0)



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

    (33)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1]i90[7] ≥ 0∧i90[7] ≥ 0∧i90[7] + [-1] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]i104[7] + [(-1)bni_27]i90[7] ≥ 0∧[1 + (-1)bso_28] + [-2]i104[7] ≥ 0)



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

    (34)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1]i90[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] + [-1]i90[7] ≥ 0 ⇒ (UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]i104[7] + [(-1)bni_27]i90[7] ≥ 0∧[1 + (-1)bso_28] + [-2]i104[7] ≥ 0)



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

    (35)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1]i90[7] ≥ 0∧i90[7] ≥ 0∧i90[7] + [-1] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(LOAD1160(i86[1], i90[1])), ≥)∧[(-1)Bound*bni_27] + [(-1)bni_27]i104[7] + [(-1)bni_27]i90[7] ≥ 0∧[1 + (-1)bso_28] + [-2]i104[7] ≥ 0)



    We solved constraint (34) using rule (IDP_SMT_SPLIT).We solved constraint (35) using rule (IDP_SMT_SPLIT).




For Pair LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]) the following chains were created:
  • We consider the chain COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]), COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8]), LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0]), COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1]), LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7]) which results in the following constraint:

    (36)    (i86[1]=i104[7]∧i90[1]=i90[7]∧i104[7]=i104[8]∧&&(>(i90[7], 0), <=(i104[7], i90[7]))=TRUE∧i90[7]=i90[8]∧i104[8]=i90[0]∧i90[8]=i86[0]∧&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0])))=TRUE∧i86[0]=i86[1]1∧i90[0]=i90[1]1∧i86[1]1=i104[7]1∧i90[1]1=i90[7]1 ⇒ LOAD995(i86[0], i90[0])≥NonInfC∧LOAD995(i86[0], i90[0])≥COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])∧(UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥))



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

    (37)    (>(i90[7], 0)=TRUE∧<=(i104[7], i90[7])=TRUE∧>=(i90[7], 0)=TRUE∧>(i104[7], 0)=TRUE∧<(i90[7], i104[7])=TRUE ⇒ LOAD995(i90[7], i104[7])≥NonInfC∧LOAD995(i90[7], i104[7])≥COND_LOAD995(&&(&&(>=(i90[7], 0), >(i104[7], 0)), !(=(i90[7], i104[7]))), i90[7], i104[7])∧(UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥))



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

    (38)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1] + [-1]i90[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (39)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1] + [-1]i90[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (41)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i90[7] + [-1] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (42)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i104[7] + [-1] + [-1]i90[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (43)    (i90[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0∧i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i90[7] + [-1] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[bni_29 + (-1)Bound*bni_29] + [(-1)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



    We solved constraint (42) using rule (IDP_SMT_SPLIT).We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (44)    (i90[7] ≥ 0∧[1] + i90[7] + [-1]i104[7] ≥ 0∧[1] + i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i90[7] + [-1]i104[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_29] + [(-1)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (45)    (i104[7] + i90[7] ≥ 0∧[1] + i90[7] ≥ 0∧[1] + i104[7] + i90[7] ≥ 0∧i104[7] + [-1] ≥ 0∧i90[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_29] + [(-2)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (46)    ([1] + i104[7] + i90[7] ≥ 0∧[1] + i90[7] ≥ 0∧[2] + i104[7] + i90[7] ≥ 0∧i104[7] ≥ 0∧i90[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_29 + (-2)bni_29] + [(-2)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])
    • ([1] + i86[0] + i90[0] ≥ 0∧[1] + i90[0] ≥ 0∧[1] + i86[0] ≥ 0∧[2] + i86[0] + i90[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(LOAD995(i90[8], i104[8])), ≥)∧[(-1)Bound*bni_23 + (-2)bni_23] + [(-1)bni_23]i86[0] + [(-1)bni_23]i90[0] ≥ 0∧[(-1)bso_24] + i86[0] ≥ 0)

  • LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])
    • (i86[0] + i90[0] ≥ 0∧[1] + i90[0] ≥ 0∧i86[0] ≥ 0∧i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i90[0] ≥ 0∧[1 + (-1)bso_26] + i86[0] + [2]i90[0] ≥ 0)

  • COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])

  • LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])
    • ([1] + i104[7] + i90[7] ≥ 0∧[1] + i90[7] ≥ 0∧[2] + i104[7] + i90[7] ≥ 0∧i104[7] ≥ 0∧i90[7] ≥ 0 ⇒ (UIncreasing(COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])), ≥)∧[(-1)Bound*bni_29 + (-2)bni_29] + [(-2)bni_29]i104[7] + [(-1)bni_29]i90[7] ≥ 0∧[1 + (-1)bso_30] ≥ 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) = [1]   
POL(COND_LOAD11601(x1, x2, x3)) = [1] + [-1]x3 + [-1]x1   
POL(LOAD995(x1, x2)) = [1] + [-1]x2 + [-1]x1   
POL(LOAD1160(x1, x2)) = [-1] + x2 + [-1]x1   
POL(&&(x1, x2)) = [1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<=(x1, x2)) = [-1]   
POL(COND_LOAD995(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + x1   
POL(>=(x1, x2)) = [-1]   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   

The following pairs are in P>:

LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])
COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
LOAD995(i86[0], i90[0]) → COND_LOAD995(&&(&&(>=(i86[0], 0), >(i90[0], 0)), !(=(i86[0], i90[0]))), i86[0], i90[0])

The following pairs are in Pbound:

LOAD1160(i104[7], i90[7]) → COND_LOAD11601(&&(>(i90[7], 0), <=(i104[7], i90[7])), i104[7], i90[7])
COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])

The following pairs are in P≥:

COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])

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

(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:
none


R is empty.

The integer pair graph contains the following rules and edges:
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])


The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])
(0): LOAD995(i86[0], i90[0]) → COND_LOAD995(i86[0] >= 0 && i90[0] > 0 && !(i86[0] = i90[0]), i86[0], i90[0])

(8) -> (0), if ((i104[8] →* i90[0])∧(i90[8] →* i86[0]))



The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(22) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD995(TRUE, i86[1], i90[1]) → LOAD1160(i86[1], i90[1])
(6): STORE1193(i90[6], i104[6]) → LOAD995(i90[6], i104[6])
(7): LOAD1160(i104[7], i90[7]) → COND_LOAD11601(i90[7] > 0 && i104[7] <= i90[7], i104[7], i90[7])
(8): COND_LOAD11601(TRUE, i104[8], i90[8]) → LOAD995(i90[8], i104[8])

(1) -> (7), if ((i86[1] →* i104[7])∧(i90[1] →* i90[7]))


(7) -> (8), if ((i104[7] →* i104[8])∧(i90[7] > 0 && i104[7] <= i90[7] →* TRUE)∧(i90[7] →* i90[8]))



The set Q consists of the following terms:
Load995(x0, x1)
Cond_Load995(TRUE, x0, x1)
Load1160(x0, x1)
Cond_Load1160(TRUE, x0, x1)
Cond_Load9951(TRUE, x0)
Store1193(x0, x1)
Cond_Load11601(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(26) TRUE