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 IDP
↳13 IDependencyGraphProof (⇔)
↳14 IDP
↳15 IDPNonInfProof (⇐)
↳16 AND
    ↳17 IDP
        ↳18 IDependencyGraphProof (⇔)
        ↳19 TRUE
    ↳20 IDP
        ↳21 IDependencyGraphProof (⇔)
        ↳22 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: Mod

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 164 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:
Load720(i90, i84, i90) → Cond_Load720(i90 > 0 && i84 >= i90, i90, i84, i90)
Cond_Load720(TRUE, i90, i84, i90) → Load793(i90, i90, i90, i84, i90)
Load793(i90, i90, i90, i94, i97) → Cond_Load793(i97 > 0, i90, i90, i90, i94, i97)
Cond_Load793(TRUE, i90, i90, i90, i94, i97) → Load793(i90, i90, i90, i94 + -1, i97 + -1)
Load793(i90, i90, i90, i94, 0) → Load720(i90, i94, i90)
The set Q consists of the following terms:
Load720(x0, x1, x0)
Cond_Load720(TRUE, x0, x1, x0)
Load793(x0, x0, x0, x1, x2)
Cond_Load793(TRUE, x0, x0, x0, x1, x2)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Load720(x1, x2, x3) → Load720(x2, x3)
Load793(x1, x2, x3, x4, x5) → Load793(x3, x4, x5)
Cond_Load793(x1, x2, x3, x4, x5, x6) → Cond_Load793(x1, x4, x5, x6)
Cond_Load720(x1, x2, x3, x4) → Cond_Load720(x1, x3, x4)

(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:
Load720(i84, i90) → Cond_Load720(i90 > 0 && i84 >= i90, i84, i90)
Cond_Load720(TRUE, i84, i90) → Load793(i90, i84, i90)
Load793(i90, i94, i97) → Cond_Load793(i97 > 0, i90, i94, i97)
Cond_Load793(TRUE, i90, i94, i97) → Load793(i90, i94 + -1, i97 + -1)
Load793(i90, i94, 0) → Load720(i94, i90)
The set Q consists of the following terms:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load720(i84, i90) → Cond_Load720(i90 > 0 && i84 >= i90, i84, i90)
Cond_Load720(TRUE, i84, i90) → Load793(i90, i84, i90)
Load793(i90, i94, i97) → Cond_Load793(i97 > 0, i90, i94, i97)
Cond_Load793(TRUE, i90, i94, i97) → Load793(i90, i94 + -1, i97 + -1)
Load793(i90, i94, 0) → Load720(i94, i90)

The integer pair graph contains the following rules and edges:
(0): LOAD720(i84[0], i90[0]) → COND_LOAD720(i90[0] > 0 && i84[0] >= i90[0], i84[0], i90[0])
(1): COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1])
(2): LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(i97[2] > 0, i90[2], i94[2], i97[2])
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)
(4): LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4])

(0) -> (1), if ((i90[0] →* i90[1])∧(i84[0] →* i84[1])∧(i90[0] > 0 && i84[0] >= i90[0] →* TRUE))


(1) -> (2), if ((i90[1] →* i97[2])∧(i90[1] →* i90[2])∧(i84[1] →* i94[2]))


(1) -> (4), if ((i84[1] →* i94[4])∧(i90[1] →* 0)∧(i90[1] →* i90[4]))


(2) -> (3), if ((i90[2] →* i90[3])∧(i94[2] →* i94[3])∧(i97[2] > 0 →* TRUE)∧(i97[2] →* i97[3]))


(3) -> (2), if ((i90[3] →* i90[2])∧(i97[3] + -1 →* i97[2])∧(i94[3] + -1 →* i94[2]))


(3) -> (4), if ((i97[3] + -1 →* 0)∧(i90[3] →* i90[4])∧(i94[3] + -1 →* i94[4]))


(4) -> (0), if ((i94[4] →* i84[0])∧(i90[4] →* i90[0]))



The set Q consists of the following terms:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD720(i84[0], i90[0]) → COND_LOAD720(i90[0] > 0 && i84[0] >= i90[0], i84[0], i90[0])
(1): COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1])
(2): LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(i97[2] > 0, i90[2], i94[2], i97[2])
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)
(4): LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4])

(0) -> (1), if ((i90[0] →* i90[1])∧(i84[0] →* i84[1])∧(i90[0] > 0 && i84[0] >= i90[0] →* TRUE))


(1) -> (2), if ((i90[1] →* i97[2])∧(i90[1] →* i90[2])∧(i84[1] →* i94[2]))


(1) -> (4), if ((i84[1] →* i94[4])∧(i90[1] →* 0)∧(i90[1] →* i90[4]))


(2) -> (3), if ((i90[2] →* i90[3])∧(i94[2] →* i94[3])∧(i97[2] > 0 →* TRUE)∧(i97[2] →* i97[3]))


(3) -> (2), if ((i90[3] →* i90[2])∧(i97[3] + -1 →* i97[2])∧(i94[3] + -1 →* i94[2]))


(3) -> (4), if ((i97[3] + -1 →* 0)∧(i90[3] →* i90[4])∧(i94[3] + -1 →* i94[4]))


(4) -> (0), if ((i94[4] →* i84[0])∧(i90[4] →* i90[0]))



The set Q consists of the following terms:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(11) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD720(i84, i90) → COND_LOAD720(&&(>(i90, 0), >=(i84, i90)), i84, i90) the following chains were created:
  • We consider the chain LOAD720(i84[0], i90[0]) → COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0]), COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1]) which results in the following constraint:

    (1)    (i90[0]=i90[1]∧i84[0]=i84[1]∧&&(>(i90[0], 0), >=(i84[0], i90[0]))=TRUE ⇒ LOAD720(i84[0], i90[0])≥NonInfC∧LOAD720(i84[0], i90[0])≥COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])∧(UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥))



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

    (2)    (>(i90[0], 0)=TRUE∧>=(i84[0], i90[0])=TRUE ⇒ LOAD720(i84[0], i90[0])≥NonInfC∧LOAD720(i84[0], i90[0])≥COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])∧(UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥))



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

    (3)    (i90[0] + [-1] ≥ 0∧i84[0] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[(-1)bso_23] + i90[0] ≥ 0)



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

    (4)    (i90[0] + [-1] ≥ 0∧i84[0] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[(-1)bso_23] + i90[0] ≥ 0)



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

    (5)    (i90[0] + [-1] ≥ 0∧i84[0] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[(-1)bso_23] + i90[0] ≥ 0)



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

    (6)    (i90[0] ≥ 0∧i84[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[1 + (-1)bso_23] + i90[0] ≥ 0)



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

    (7)    (i90[0] ≥ 0∧i84[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i90[0] + [bni_22]i84[0] ≥ 0∧[1 + (-1)bso_23] + i90[0] ≥ 0)







For Pair COND_LOAD720(TRUE, i84, i90) → LOAD793(i90, i84, i90) the following chains were created:
  • We consider the chain COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1]), LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) which results in the following constraint:

    (8)    (i90[1]=i97[2]∧i90[1]=i90[2]∧i84[1]=i94[2] ⇒ COND_LOAD720(TRUE, i84[1], i90[1])≥NonInfC∧COND_LOAD720(TRUE, i84[1], i90[1])≥LOAD793(i90[1], i84[1], i90[1])∧(UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥))



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

    (9)    (COND_LOAD720(TRUE, i84[1], i90[1])≥NonInfC∧COND_LOAD720(TRUE, i84[1], i90[1])≥LOAD793(i90[1], i84[1], i90[1])∧(UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥))



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

    (10)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (11)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (12)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (13)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)



  • We consider the chain COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1]), LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4]) which results in the following constraint:

    (14)    (i84[1]=i94[4]∧i90[1]=0∧i90[1]=i90[4] ⇒ COND_LOAD720(TRUE, i84[1], i90[1])≥NonInfC∧COND_LOAD720(TRUE, i84[1], i90[1])≥LOAD793(i90[1], i84[1], i90[1])∧(UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥))



    We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

    (15)    (COND_LOAD720(TRUE, i84[1], 0)≥NonInfC∧COND_LOAD720(TRUE, i84[1], 0)≥LOAD793(0, i84[1], 0)∧(UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥))



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

    (16)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (17)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (18)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (19)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧[(-1)bso_25] ≥ 0)







For Pair LOAD793(i90, i94, i97) → COND_LOAD793(>(i97, 0), i90, i94, i97) the following chains were created:
  • We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)) which results in the following constraint:

    (20)    (i90[2]=i90[3]∧i94[2]=i94[3]∧>(i97[2], 0)=TRUE∧i97[2]=i97[3] ⇒ LOAD793(i90[2], i94[2], i97[2])≥NonInfC∧LOAD793(i90[2], i94[2], i97[2])≥COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])∧(UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥))



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

    (21)    (>(i97[2], 0)=TRUE ⇒ LOAD793(i90[2], i94[2], i97[2])≥NonInfC∧LOAD793(i90[2], i94[2], i97[2])≥COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])∧(UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥))



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

    (22)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] + [bni_26]i94[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (23)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] + [bni_26]i94[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (24)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] + [bni_26]i94[2] ≥ 0∧[(-1)bso_27] ≥ 0)



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

    (25)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)



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

    (26)    (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[bni_26] = 0∧0 = 0∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)







For Pair COND_LOAD793(TRUE, i90, i94, i97) → LOAD793(i90, +(i94, -1), +(i97, -1)) the following chains were created:
  • We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)), LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) which results in the following constraint:

    (27)    (i90[2]=i90[3]∧i94[2]=i94[3]∧>(i97[2], 0)=TRUE∧i97[2]=i97[3]∧i90[3]=i90[2]1∧+(i97[3], -1)=i97[2]1∧+(i94[3], -1)=i94[2]1 ⇒ COND_LOAD793(TRUE, i90[3], i94[3], i97[3])≥NonInfC∧COND_LOAD793(TRUE, i90[3], i94[3], i97[3])≥LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))∧(UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥))



    We simplified constraint (27) using rules (III), (IV) which results in the following new constraint:

    (28)    (>(i97[2], 0)=TRUE ⇒ COND_LOAD793(TRUE, i90[2], i94[2], i97[2])≥NonInfC∧COND_LOAD793(TRUE, i90[2], i94[2], i97[2])≥LOAD793(i90[2], +(i94[2], -1), +(i97[2], -1))∧(UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥))



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

    (29)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (30)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (31)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (32)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)



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

    (33)    (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)



  • We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)), LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4]) which results in the following constraint:

    (34)    (i90[2]=i90[3]∧i94[2]=i94[3]∧>(i97[2], 0)=TRUE∧i97[2]=i97[3]∧+(i97[3], -1)=0∧i90[3]=i90[4]∧+(i94[3], -1)=i94[4] ⇒ COND_LOAD793(TRUE, i90[3], i94[3], i97[3])≥NonInfC∧COND_LOAD793(TRUE, i90[3], i94[3], i97[3])≥LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))∧(UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥))



    We simplified constraint (34) using rules (III), (IV) which results in the following new constraint:

    (35)    (>(i97[2], 0)=TRUE∧+(i97[2], -1)=0 ⇒ COND_LOAD793(TRUE, i90[2], i94[2], i97[2])≥NonInfC∧COND_LOAD793(TRUE, i90[2], i94[2], i97[2])≥LOAD793(i90[2], +(i94[2], -1), +(i97[2], -1))∧(UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥))



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

    (36)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (37)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (38)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (39)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)



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

    (40)    (i97[2] ≥ 0∧i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)







For Pair LOAD793(i90, i94, 0) → LOAD720(i94, i90) the following chains were created:
  • We consider the chain LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4]), LOAD720(i84[0], i90[0]) → COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0]) which results in the following constraint:

    (41)    (i94[4]=i84[0]∧i90[4]=i90[0] ⇒ LOAD793(i90[4], i94[4], 0)≥NonInfC∧LOAD793(i90[4], i94[4], 0)≥LOAD720(i94[4], i90[4])∧(UIncreasing(LOAD720(i94[4], i90[4])), ≥))



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

    (42)    (LOAD793(i90[4], i94[4], 0)≥NonInfC∧LOAD793(i90[4], i94[4], 0)≥LOAD720(i94[4], i90[4])∧(UIncreasing(LOAD720(i94[4], i90[4])), ≥))



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

    (43)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧[(-1)bso_31] ≥ 0)



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

    (44)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧[(-1)bso_31] ≥ 0)



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

    (45)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧[(-1)bso_31] ≥ 0)



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

    (46)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD720(i84, i90) → COND_LOAD720(&&(>(i90, 0), >=(i84, i90)), i84, i90)
    • (i90[0] ≥ 0∧i84[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i90[0] + [bni_22]i84[0] ≥ 0∧[1 + (-1)bso_23] + i90[0] ≥ 0)

  • COND_LOAD720(TRUE, i84, i90) → LOAD793(i90, i84, i90)
    • ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)
    • ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧[(-1)bso_25] ≥ 0)

  • LOAD793(i90, i94, i97) → COND_LOAD793(>(i97, 0), i90, i94, i97)
    • (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[bni_26] = 0∧0 = 0∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

  • COND_LOAD793(TRUE, i90, i94, i97) → LOAD793(i90, +(i94, -1), +(i97, -1))
    • (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
    • (i97[2] ≥ 0∧i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

  • LOAD793(i90, i94, 0) → LOAD720(i94, i90)
    • ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 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(LOAD720(x1, x2)) = [-1] + x1   
POL(COND_LOAD720(x1, x2, x3)) = [-1] + [-1]x3 + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(>=(x1, x2)) = [-1]   
POL(LOAD793(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(COND_LOAD793(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD720(i84[0], i90[0]) → COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])

The following pairs are in Pbound:

LOAD720(i84[0], i90[0]) → COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])

The following pairs are in P≥:

COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1])
LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])
COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))
LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4])

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

(12) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1])
(2): LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(i97[2] > 0, i90[2], i94[2], i97[2])
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)
(4): LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4])

(1) -> (2), if ((i90[1] →* i97[2])∧(i90[1] →* i90[2])∧(i84[1] →* i94[2]))


(3) -> (2), if ((i90[3] →* i90[2])∧(i97[3] + -1 →* i97[2])∧(i94[3] + -1 →* i94[2]))


(2) -> (3), if ((i90[2] →* i90[3])∧(i94[2] →* i94[3])∧(i97[2] > 0 →* TRUE)∧(i97[2] →* i97[3]))


(1) -> (4), if ((i84[1] →* i94[4])∧(i90[1] →* 0)∧(i90[1] →* i90[4]))


(3) -> (4), if ((i97[3] + -1 →* 0)∧(i90[3] →* i90[4])∧(i94[3] + -1 →* i94[4]))



The set Q consists of the following terms:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)
(2): LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(i97[2] > 0, i90[2], i94[2], i97[2])

(3) -> (2), if ((i90[3] →* i90[2])∧(i97[3] + -1 →* i97[2])∧(i94[3] + -1 →* i94[2]))


(2) -> (3), if ((i90[2] →* i90[3])∧(i94[2] →* i94[3])∧(i97[2] > 0 →* TRUE)∧(i97[2] →* i97[3]))



The set Q consists of the following terms:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(15) 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_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)) the following chains were created:
  • We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)), LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) which results in the following constraint:

    (1)    (i90[2]=i90[3]∧i94[2]=i94[3]∧>(i97[2], 0)=TRUE∧i97[2]=i97[3]∧i90[3]=i90[2]1∧+(i97[3], -1)=i97[2]1∧+(i94[3], -1)=i94[2]1 ⇒ COND_LOAD793(TRUE, i90[3], i94[3], i97[3])≥NonInfC∧COND_LOAD793(TRUE, i90[3], i94[3], i97[3])≥LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))∧(UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥))



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

    (2)    (>(i97[2], 0)=TRUE ⇒ COND_LOAD793(TRUE, i90[2], i94[2], i97[2])≥NonInfC∧COND_LOAD793(TRUE, i90[2], i94[2], i97[2])≥LOAD793(i90[2], +(i94[2], -1), +(i97[2], -1))∧(UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥))



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

    (3)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (4)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (5)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (6)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)



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

    (7)    (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)







For Pair LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) the following chains were created:
  • We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)) which results in the following constraint:

    (8)    (i90[2]=i90[3]∧i94[2]=i94[3]∧>(i97[2], 0)=TRUE∧i97[2]=i97[3] ⇒ LOAD793(i90[2], i94[2], i97[2])≥NonInfC∧LOAD793(i90[2], i94[2], i97[2])≥COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])∧(UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥))



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

    (9)    (>(i97[2], 0)=TRUE ⇒ LOAD793(i90[2], i94[2], i97[2])≥NonInfC∧LOAD793(i90[2], i94[2], i97[2])≥COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])∧(UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥))



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

    (10)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (11)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (12)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (13)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)



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

    (14)    (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))
    • (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

  • LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])
    • (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 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(COND_LOAD793(x1, x2, x3, x4)) = [-1] + x4   
POL(LOAD793(x1, x2, x3)) = [-1] + x3   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))

The following pairs are in Pbound:

COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))
LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])

The following pairs are in P≥:

LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])

There are no usable rules.

(16) Complex Obligation (AND)

(17) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(i97[2] > 0, i90[2], i94[2], i97[2])


The set Q consists of the following terms:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) 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:
Load720(x0, x1)
Cond_Load720(TRUE, x0, x1)
Load793(x0, x1, x2)
Cond_Load793(TRUE, x0, x1, x2)

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) TRUE