Termination of the given JBC could be proven:

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

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 215 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:
Load845(i108, i111, i110) → Cond_Load845(i111 > 0 && 1 = i111 % 2, i108, i111, i110)
Cond_Load845(TRUE, i108, i111, i110) → Load888(i108, i111, i110 * i108)
Load888(i108, i111, i110) → Load845(i108 * i108, i111 / 2, i110)
Load845(i108, i111, i110) → Cond_Load8451(i111 % 2 >= 0 && !(i111 % 2 = 1) && i111 > 0, i108, i111, i110)
Cond_Load8451(TRUE, i108, i111, i110) → Load845(i108 * i108, i111 / 2, i110)
The set Q consists of the following terms:
Load845(x0, x1, x2)
Cond_Load845(TRUE, x0, x1, x2)
Load888(x0, x1, x2)
Cond_Load8451(TRUE, x0, x1, x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load845(i108, i111, i110) → Cond_Load845(i111 > 0 && 1 = i111 % 2, i108, i111, i110)
Cond_Load845(TRUE, i108, i111, i110) → Load888(i108, i111, i110 * i108)
Load888(i108, i111, i110) → Load845(i108 * i108, i111 / 2, i110)
Load845(i108, i111, i110) → Cond_Load8451(i111 % 2 >= 0 && !(i111 % 2 = 1) && i111 > 0, i108, i111, i110)
Cond_Load8451(TRUE, i108, i111, i110) → Load845(i108 * i108, i111 / 2, i110)

The integer pair graph contains the following rules and edges:
(0): LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(i111[0] > 0 && 1 = i111[0] % 2, i108[0], i111[0], i110[0])
(1): COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], i110[1] * i108[1])
(2): LOAD888(i108[2], i111[2], i110[2]) → LOAD845(i108[2] * i108[2], i111[2] / 2, i110[2])
(3): LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0, i108[3], i111[3], i110[3])
(4): COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(i108[4] * i108[4], i111[4] / 2, i110[4])

(0) -> (1), if ((i111[0] > 0 && 1 = i111[0] % 2 →* TRUE)∧(i111[0] →* i111[1])∧(i110[0] →* i110[1])∧(i108[0] →* i108[1]))


(1) -> (2), if ((i108[1] →* i108[2])∧(i110[1] * i108[1] →* i110[2])∧(i111[1] →* i111[2]))


(2) -> (0), if ((i110[2] →* i110[0])∧(i108[2] * i108[2] →* i108[0])∧(i111[2] / 2 →* i111[0]))


(2) -> (3), if ((i110[2] →* i110[3])∧(i111[2] / 2 →* i111[3])∧(i108[2] * i108[2] →* i108[3]))


(3) -> (4), if ((i110[3] →* i110[4])∧(i108[3] →* i108[4])∧(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0 →* TRUE)∧(i111[3] →* i111[4]))


(4) -> (0), if ((i108[4] * i108[4] →* i108[0])∧(i110[4] →* i110[0])∧(i111[4] / 2 →* i111[0]))


(4) -> (3), if ((i111[4] / 2 →* i111[3])∧(i110[4] →* i110[3])∧(i108[4] * i108[4] →* i108[3]))



The set Q consists of the following terms:
Load845(x0, x1, x2)
Cond_Load845(TRUE, x0, x1, x2)
Load888(x0, x1, x2)
Cond_Load8451(TRUE, x0, x1, x2)

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(i111[0] > 0 && 1 = i111[0] % 2, i108[0], i111[0], i110[0])
(1): COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], i110[1] * i108[1])
(2): LOAD888(i108[2], i111[2], i110[2]) → LOAD845(i108[2] * i108[2], i111[2] / 2, i110[2])
(3): LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0, i108[3], i111[3], i110[3])
(4): COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(i108[4] * i108[4], i111[4] / 2, i110[4])

(0) -> (1), if ((i111[0] > 0 && 1 = i111[0] % 2 →* TRUE)∧(i111[0] →* i111[1])∧(i110[0] →* i110[1])∧(i108[0] →* i108[1]))


(1) -> (2), if ((i108[1] →* i108[2])∧(i110[1] * i108[1] →* i110[2])∧(i111[1] →* i111[2]))


(2) -> (0), if ((i110[2] →* i110[0])∧(i108[2] * i108[2] →* i108[0])∧(i111[2] / 2 →* i111[0]))


(2) -> (3), if ((i110[2] →* i110[3])∧(i111[2] / 2 →* i111[3])∧(i108[2] * i108[2] →* i108[3]))


(3) -> (4), if ((i110[3] →* i110[4])∧(i108[3] →* i108[4])∧(i111[3] % 2 >= 0 && !(i111[3] % 2 = 1) && i111[3] > 0 →* TRUE)∧(i111[3] →* i111[4]))


(4) -> (0), if ((i108[4] * i108[4] →* i108[0])∧(i110[4] →* i110[0])∧(i111[4] / 2 →* i111[0]))


(4) -> (3), if ((i111[4] / 2 →* i111[3])∧(i110[4] →* i110[3])∧(i108[4] * i108[4] →* i108[3]))



The set Q consists of the following terms:
Load845(x0, x1, x2)
Cond_Load845(TRUE, x0, x1, x2)
Load888(x0, x1, x2)
Cond_Load8451(TRUE, x0, x1, x2)

(9) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair LOAD845(i108, i111, i110) → COND_LOAD845(&&(>(i111, 0), =(1, %(i111, 2))), i108, i111, i110) the following chains were created:
  • We consider the chain LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0]), COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1])) which results in the following constraint:

    (1)    (&&(>(i111[0], 0), =(1, %(i111[0], 2)))=TRUE∧i111[0]=i111[1]∧i110[0]=i110[1]∧i108[0]=i108[1] ⇒ LOAD845(i108[0], i111[0], i110[0])≥NonInfC∧LOAD845(i108[0], i111[0], i110[0])≥COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])∧(UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥))



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

    (2)    (>(i111[0], 0)=TRUE∧>=(1, %(i111[0], 2))=TRUE∧<=(1, %(i111[0], 2))=TRUE ⇒ LOAD845(i108[0], i111[0], i110[0])≥NonInfC∧LOAD845(i108[0], i111[0], i110[0])≥COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])∧(UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥))



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

    (3)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (4)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (5)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (6)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)



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

    (7)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)



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

    (8)    (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)







For Pair COND_LOAD845(TRUE, i108, i111, i110) → LOAD888(i108, i111, *(i110, i108)) the following chains were created:
  • We consider the chain LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0]), COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1])) which results in the following constraint:

    (9)    (&&(>(i111[0], 0), =(1, %(i111[0], 2)))=TRUE∧i111[0]=i111[1]∧i110[0]=i110[1]∧i108[0]=i108[1] ⇒ COND_LOAD845(TRUE, i108[1], i111[1], i110[1])≥NonInfC∧COND_LOAD845(TRUE, i108[1], i111[1], i110[1])≥LOAD888(i108[1], i111[1], *(i110[1], i108[1]))∧(UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥))



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

    (10)    (>(i111[0], 0)=TRUE∧>=(1, %(i111[0], 2))=TRUE∧<=(1, %(i111[0], 2))=TRUE ⇒ COND_LOAD845(TRUE, i108[0], i111[0], i110[0])≥NonInfC∧COND_LOAD845(TRUE, i108[0], i111[0], i110[0])≥LOAD888(i108[0], i111[0], *(i110[0], i108[0]))∧(UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥))



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

    (11)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (12)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (13)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (14)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)



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

    (15)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)



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

    (16)    (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)







For Pair LOAD888(i108, i111, i110) → LOAD845(*(i108, i108), /(i111, 2), i110) the following chains were created:
  • We consider the chain LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0]), COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1])), LOAD888(i108[2], i111[2], i110[2]) → LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2]) which results in the following constraint:

    (17)    (&&(>(i111[0], 0), =(1, %(i111[0], 2)))=TRUE∧i111[0]=i111[1]∧i110[0]=i110[1]∧i108[0]=i108[1]∧i108[1]=i108[2]∧*(i110[1], i108[1])=i110[2]∧i111[1]=i111[2] ⇒ LOAD888(i108[2], i111[2], i110[2])≥NonInfC∧LOAD888(i108[2], i111[2], i110[2])≥LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])∧(UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥))



    We simplified constraint (17) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

    (18)    (>(i111[0], 0)=TRUE∧>=(1, %(i111[0], 2))=TRUE∧<=(1, %(i111[0], 2))=TRUE ⇒ LOAD888(i108[1], i111[0], *(i110[1], i108[1]))≥NonInfC∧LOAD888(i108[1], i111[0], *(i110[1], i108[1]))≥LOAD845(*(i108[1], i108[1]), /(i111[0], 2), *(i110[1], i108[1]))∧(UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥))



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

    (19)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧[(-1)bso_38] + i111[0] + [-1]max{i111[0], [-1]i111[0]} ≥ 0)



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

    (20)    (i111[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧[(-1)bso_38] + i111[0] + [-1]max{i111[0], [-1]i111[0]} ≥ 0)



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

    (21)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧[(-1)bso_38] ≥ 0)



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

    (22)    (i111[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2]i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)



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

    (23)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)



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

    (24)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i111[0] ≥ 0∧i108[1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)


    (25)    (i111[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0∧[2] + [2]i111[0] ≥ 0∧i108[1] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)



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

    (26)    (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)



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

    (27)    (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)







For Pair LOAD845(i108, i111, i110) → COND_LOAD8451(&&(&&(>=(%(i111, 2), 0), !(=(%(i111, 2), 1))), >(i111, 0)), i108, i111, i110) the following chains were created:
  • We consider the chain LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3]), COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4]) which results in the following constraint:

    (28)    (i110[3]=i110[4]∧i108[3]=i108[4]∧&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0))=TRUE∧i111[3]=i111[4] ⇒ LOAD845(i108[3], i111[3], i110[3])≥NonInfC∧LOAD845(i108[3], i111[3], i110[3])≥COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])∧(UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥))



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

    (29)    (>(i111[3], 0)=TRUE∧>=(%(i111[3], 2), 0)=TRUE∧<(%(i111[3], 2), 1)=TRUE ⇒ LOAD845(i108[3], i111[3], i110[3])≥NonInfC∧LOAD845(i108[3], i111[3], i110[3])≥COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])∧(UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥))


    (30)    (>(i111[3], 0)=TRUE∧>=(%(i111[3], 2), 0)=TRUE∧>(%(i111[3], 2), 1)=TRUE ⇒ LOAD845(i108[3], i111[3], i110[3])≥NonInfC∧LOAD845(i108[3], i111[3], i110[3])≥COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])∧(UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥))



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

    (31)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



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

    (32)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



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

    (33)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



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

    (34)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



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

    (35)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



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

    (36)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧[1 + (-1)bso_40] ≥ 0)



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

    (37)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)



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

    (38)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)



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

    (39)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)



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

    (40)    (i111[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)



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

    (41)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)



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

    (42)    (i111[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)







For Pair COND_LOAD8451(TRUE, i108, i111, i110) → LOAD845(*(i108, i108), /(i111, 2), i110) the following chains were created:
  • We consider the chain LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3]), COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4]) which results in the following constraint:

    (43)    (i110[3]=i110[4]∧i108[3]=i108[4]∧&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0))=TRUE∧i111[3]=i111[4] ⇒ COND_LOAD8451(TRUE, i108[4], i111[4], i110[4])≥NonInfC∧COND_LOAD8451(TRUE, i108[4], i111[4], i110[4])≥LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])∧(UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥))



    We simplified constraint (43) using rules (III), (IDP_BOOLEAN) which results in the following new constraints:

    (44)    (>(i111[3], 0)=TRUE∧>=(%(i111[3], 2), 0)=TRUE∧<(%(i111[3], 2), 1)=TRUE ⇒ COND_LOAD8451(TRUE, i108[3], i111[3], i110[3])≥NonInfC∧COND_LOAD8451(TRUE, i108[3], i111[3], i110[3])≥LOAD845(*(i108[3], i108[3]), /(i111[3], 2), i110[3])∧(UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥))


    (45)    (>(i111[3], 0)=TRUE∧>=(%(i111[3], 2), 0)=TRUE∧>(%(i111[3], 2), 1)=TRUE ⇒ COND_LOAD8451(TRUE, i108[3], i111[3], i110[3])≥NonInfC∧COND_LOAD8451(TRUE, i108[3], i111[3], i110[3])≥LOAD845(*(i108[3], i108[3]), /(i111[3], 2), i110[3])∧(UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥))



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

    (46)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)



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

    (47)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)



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

    (48)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧[-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)



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

    (49)    (i111[3] + [-1] ≥ 0∧max{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-2] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] + i111[3] + [-1]max{i111[3], [-1]i111[3]} ≥ 0)



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

    (50)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (51)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (52)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (53)    (i111[3] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



    We simplified constraint (52) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (54)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (55)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)


    (56)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (57)    (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (58)    (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (59)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (60)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)


    (61)    (i111[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧0 ≥ 0∧[2] + [2]i111[3] ≥ 0∧i108[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (62)    (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)



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

    (63)    (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD845(i108, i111, i110) → COND_LOAD845(&&(>(i111, 0), =(1, %(i111, 2))), i108, i111, i110)
    • (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_30 + bni_30] + [bni_30]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

  • COND_LOAD845(TRUE, i108, i111, i110) → LOAD888(i108, i111, *(i110, i108))
    • (i111[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(LOAD888(i108[1], i111[1], *(i110[1], i108[1]))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_32 + bni_32] + [bni_32]i111[0] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

  • LOAD888(i108, i111, i110) → LOAD845(*(i108, i108), /(i111, 2), i110)
    • (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
    • (i111[0] ≥ 0∧[1] ≥ 0∧i108[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[0] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i111[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

  • LOAD845(i108, i111, i110) → COND_LOAD8451(&&(&&(>=(%(i111, 2), 0), !(=(%(i111, 2), 1))), >(i111, 0)), i108, i111, i110)
    • (i111[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)
    • (i111[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_39 + bni_39] + [bni_39]i111[3] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_40] ≥ 0)

  • COND_LOAD8451(TRUE, i108, i111, i110) → LOAD845(*(i108, i108), /(i111, 2), i110)
    • (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
    • (i111[3] ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
    • (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 0)
    • (i111[3] ≥ 0∧0 ≥ 0∧i108[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i111[3] ≥ 0 ⇒ (UIncreasing(LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_41] + [bni_41]i111[3] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_42] ≥ 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(LOAD845(x1, x2, x3)) = x2   
POL(COND_LOAD845(x1, x2, x3, x4)) = x3   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(=(x1, x2)) = [-1]   
POL(1) = [1]   
POL(2) = [2]   
POL(LOAD888(x1, x2, x3)) = [-1] + x2   
POL(*(x1, x2)) = x1·x2   
POL(COND_LOAD8451(x1, x2, x3, x4)) = [-1] + x3   
POL(>=(x1, x2)) = [-1]   
POL(!(x1)) = [-1]   

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

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}   
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}   
POL(/(x1, 2)1 @ {LOAD845_3/1}) = max{x1, [-1]x1} + [-1]   

The following pairs are in P>:

COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1]))
LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])

The following pairs are in Pbound:

LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])
COND_LOAD845(TRUE, i108[1], i111[1], i110[1]) → LOAD888(i108[1], i111[1], *(i110[1], i108[1]))
LOAD888(i108[2], i111[2], i110[2]) → LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])
LOAD845(i108[3], i111[3], i110[3]) → COND_LOAD8451(&&(&&(>=(%(i111[3], 2), 0), !(=(%(i111[3], 2), 1))), >(i111[3], 0)), i108[3], i111[3], i110[3])
COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])

The following pairs are in P≥:

LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(&&(>(i111[0], 0), =(1, %(i111[0], 2))), i108[0], i111[0], i110[0])
LOAD888(i108[2], i111[2], i110[2]) → LOAD845(*(i108[2], i108[2]), /(i111[2], 2), i110[2])
COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(*(i108[4], i108[4]), /(i111[4], 2), i110[4])

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

/1 →

(10) Complex Obligation (AND)

(11) 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): LOAD845(i108[0], i111[0], i110[0]) → COND_LOAD845(i111[0] > 0 && 1 = i111[0] % 2, i108[0], i111[0], i110[0])
(2): LOAD888(i108[2], i111[2], i110[2]) → LOAD845(i108[2] * i108[2], i111[2] / 2, i110[2])
(4): COND_LOAD8451(TRUE, i108[4], i111[4], i110[4]) → LOAD845(i108[4] * i108[4], i111[4] / 2, i110[4])

(2) -> (0), if ((i110[2] →* i110[0])∧(i108[2] * i108[2] →* i108[0])∧(i111[2] / 2 →* i111[0]))


(4) -> (0), if ((i108[4] * i108[4] →* i108[0])∧(i110[4] →* i110[0])∧(i111[4] / 2 →* i111[0]))



The set Q consists of the following terms:
Load845(x0, x1, x2)
Cond_Load845(TRUE, x0, x1, x2)
Load888(x0, x1, x2)
Cond_Load8451(TRUE, x0, x1, x2)

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) 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:
Load845(x0, x1, x2)
Cond_Load845(TRUE, x0, x1, x2)
Load888(x0, x1, x2)
Cond_Load8451(TRUE, x0, x1, x2)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE