Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 DuplicateArgsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 IDP
        ↳19 IDPNonInfProof (⇐)
        ↳20 AND
            ↳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: DivMinus2

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 195 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:
Load783(i95, i88, i95, i90) → Cond_Load783(i95 > 0 && i88 >= i95, i95, i88, i95, i90)
Cond_Load783(TRUE, i95, i88, i95, i90) → Load866(i95, i95, i90, i95, i88, i95)
Load866(i95, i95, i90, i95, i100, i103) → Cond_Load866(i103 > 0, i95, i95, i90, i95, i100, i103)
Cond_Load866(TRUE, i95, i95, i90, i95, i100, i103) → Load866(i95, i95, i90, i95, i100 + -1, i103 + -1)
Load866(i95, i95, i90, i95, i100, 0) → Load783(i95, i100, i95, i90 + 1)
The set Q consists of the following terms:
Load783(x0, x1, x0, x2)
Cond_Load783(TRUE, x0, x1, x0, x2)
Load866(x0, x0, x1, x0, x2, x3)
Cond_Load866(TRUE, x0, x0, x1, x0, x2, x3)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Load783(x1, x2, x3, x4) → Load783(x2, x3, x4)
Load866(x1, x2, x3, x4, x5, x6) → Load866(x3, x4, x5, x6)
Cond_Load866(x1, x2, x3, x4, x5, x6, x7) → Cond_Load866(x1, x4, x5, x6, x7)
Cond_Load783(x1, x2, x3, x4, x5) → Cond_Load783(x1, x3, x4, x5)

(6) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load783(i88, i95, i90) → Cond_Load783(i95 > 0 && i88 >= i95, i88, i95, i90)
Cond_Load783(TRUE, i88, i95, i90) → Load866(i90, i95, i88, i95)
Load866(i90, i95, i100, i103) → Cond_Load866(i103 > 0, i90, i95, i100, i103)
Cond_Load866(TRUE, i90, i95, i100, i103) → Load866(i90, i95, i100 + -1, i103 + -1)
Load866(i90, i95, i100, 0) → Load783(i100, i95, i90 + 1)
The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(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:
Load783(i88, i95, i90) → Cond_Load783(i95 > 0 && i88 >= i95, i88, i95, i90)
Cond_Load783(TRUE, i88, i95, i90) → Load866(i90, i95, i88, i95)
Load866(i90, i95, i100, i103) → Cond_Load866(i103 > 0, i90, i95, i100, i103)
Cond_Load866(TRUE, i90, i95, i100, i103) → Load866(i90, i95, i100 + -1, i103 + -1)
Load866(i90, i95, i100, 0) → Load783(i100, i95, i90 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(i95[0] > 0 && i88[0] >= i95[0], i88[0], i95[0], i90[0])
(1): COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1])
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)
(4): LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], i90[4] + 1)

(0) -> (1), if ((i88[0] →* i88[1])∧(i95[0] > 0 && i88[0] >= i95[0] →* TRUE)∧(i90[0] →* i90[1])∧(i95[0] →* i95[1]))


(1) -> (2), if ((i95[1] →* i95[2])∧(i90[1] →* i90[2])∧(i95[1] →* i103[2])∧(i88[1] →* i100[2]))


(1) -> (4), if ((i95[1] →* 0)∧(i88[1] →* i100[4])∧(i90[1] →* i90[4])∧(i95[1] →* i95[4]))


(2) -> (3), if ((i100[2] →* i100[3])∧(i90[2] →* i90[3])∧(i103[2] →* i103[3])∧(i95[2] →* i95[3])∧(i103[2] > 0 →* TRUE))


(3) -> (2), if ((i103[3] + -1 →* i103[2])∧(i95[3] →* i95[2])∧(i90[3] →* i90[2])∧(i100[3] + -1 →* i100[2]))


(3) -> (4), if ((i100[3] + -1 →* i100[4])∧(i95[3] →* i95[4])∧(i103[3] + -1 →* 0)∧(i90[3] →* i90[4]))


(4) -> (0), if ((i100[4] →* i88[0])∧(i95[4] →* i95[0])∧(i90[4] + 1 →* i90[0]))



The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(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): LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(i95[0] > 0 && i88[0] >= i95[0], i88[0], i95[0], i90[0])
(1): COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1])
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)
(4): LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], i90[4] + 1)

(0) -> (1), if ((i88[0] →* i88[1])∧(i95[0] > 0 && i88[0] >= i95[0] →* TRUE)∧(i90[0] →* i90[1])∧(i95[0] →* i95[1]))


(1) -> (2), if ((i95[1] →* i95[2])∧(i90[1] →* i90[2])∧(i95[1] →* i103[2])∧(i88[1] →* i100[2]))


(1) -> (4), if ((i95[1] →* 0)∧(i88[1] →* i100[4])∧(i90[1] →* i90[4])∧(i95[1] →* i95[4]))


(2) -> (3), if ((i100[2] →* i100[3])∧(i90[2] →* i90[3])∧(i103[2] →* i103[3])∧(i95[2] →* i95[3])∧(i103[2] > 0 →* TRUE))


(3) -> (2), if ((i103[3] + -1 →* i103[2])∧(i95[3] →* i95[2])∧(i90[3] →* i90[2])∧(i100[3] + -1 →* i100[2]))


(3) -> (4), if ((i100[3] + -1 →* i100[4])∧(i95[3] →* i95[4])∧(i103[3] + -1 →* 0)∧(i90[3] →* i90[4]))


(4) -> (0), if ((i100[4] →* i88[0])∧(i95[4] →* i95[0])∧(i90[4] + 1 →* i90[0]))



The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(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 LOAD783(i88, i95, i90) → COND_LOAD783(&&(>(i95, 0), >=(i88, i95)), i88, i95, i90) the following chains were created:
  • We consider the chain LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0]), COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]) which results in the following constraint:

    (1)    (i88[0]=i88[1]∧&&(>(i95[0], 0), >=(i88[0], i95[0]))=TRUE∧i90[0]=i90[1]∧i95[0]=i95[1] ⇒ LOAD783(i88[0], i95[0], i90[0])≥NonInfC∧LOAD783(i88[0], i95[0], i90[0])≥COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])∧(UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥))



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

    (2)    (>(i95[0], 0)=TRUE∧>=(i88[0], i95[0])=TRUE ⇒ LOAD783(i88[0], i95[0], i90[0])≥NonInfC∧LOAD783(i88[0], i95[0], i90[0])≥COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])∧(UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥))



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

    (3)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧[(-1)bso_27] + i95[0] ≥ 0)



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

    (4)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧[(-1)bso_27] + i95[0] ≥ 0)



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

    (5)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧[(-1)bso_27] + i95[0] ≥ 0)



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

    (6)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[(-1)bso_27] + i95[0] ≥ 0)



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

    (7)    (i95[0] ≥ 0∧i88[0] + [-1] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[1 + (-1)bso_27] + i95[0] ≥ 0)



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

    (8)    (i95[0] ≥ 0∧i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i95[0] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[1 + (-1)bso_27] + i95[0] ≥ 0)







For Pair COND_LOAD783(TRUE, i88, i95, i90) → LOAD866(i90, i95, i88, i95) the following chains were created:
  • We consider the chain COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]), LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) which results in the following constraint:

    (9)    (i95[1]=i95[2]∧i90[1]=i90[2]∧i95[1]=i103[2]∧i88[1]=i100[2] ⇒ COND_LOAD783(TRUE, i88[1], i95[1], i90[1])≥NonInfC∧COND_LOAD783(TRUE, i88[1], i95[1], i90[1])≥LOAD866(i90[1], i95[1], i88[1], i95[1])∧(UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥))



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

    (10)    (COND_LOAD783(TRUE, i88[1], i95[1], i90[1])≥NonInfC∧COND_LOAD783(TRUE, i88[1], i95[1], i90[1])≥LOAD866(i90[1], i95[1], i88[1], i95[1])∧(UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥))



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

    (11)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (12)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (13)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (14)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)



  • We consider the chain COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)) which results in the following constraint:

    (15)    (i95[1]=0∧i88[1]=i100[4]∧i90[1]=i90[4]∧i95[1]=i95[4] ⇒ COND_LOAD783(TRUE, i88[1], i95[1], i90[1])≥NonInfC∧COND_LOAD783(TRUE, i88[1], i95[1], i90[1])≥LOAD866(i90[1], i95[1], i88[1], i95[1])∧(UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥))



    We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:

    (16)    (COND_LOAD783(TRUE, i88[1], 0, i90[1])≥NonInfC∧COND_LOAD783(TRUE, i88[1], 0, i90[1])≥LOAD866(i90[1], 0, i88[1], 0)∧(UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥))



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

    (17)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (18)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (19)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)



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

    (20)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)







For Pair LOAD866(i90, i95, i100, i103) → COND_LOAD866(>(i103, 0), i90, i95, i100, i103) the following chains were created:
  • We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)) which results in the following constraint:

    (21)    (i100[2]=i100[3]∧i90[2]=i90[3]∧i103[2]=i103[3]∧i95[2]=i95[3]∧>(i103[2], 0)=TRUE ⇒ LOAD866(i90[2], i95[2], i100[2], i103[2])≥NonInfC∧LOAD866(i90[2], i95[2], i100[2], i103[2])≥COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])∧(UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥))



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

    (22)    (>(i103[2], 0)=TRUE ⇒ LOAD866(i90[2], i95[2], i100[2], i103[2])≥NonInfC∧LOAD866(i90[2], i95[2], i100[2], i103[2])≥COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])∧(UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥))



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

    (23)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] + [(2)bni_30]i100[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (24)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] + [(2)bni_30]i100[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (25)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] + [(2)bni_30]i100[2] ≥ 0∧[(-1)bso_31] ≥ 0)



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

    (26)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)



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

    (27)    (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_30] = 0∧0 = 0∧0 = 0∧[(-2)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)







For Pair COND_LOAD866(TRUE, i90, i95, i100, i103) → LOAD866(i90, i95, +(i100, -1), +(i103, -1)) the following chains were created:
  • We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) which results in the following constraint:

    (28)    (i100[2]=i100[3]∧i90[2]=i90[3]∧i103[2]=i103[3]∧i95[2]=i95[3]∧>(i103[2], 0)=TRUE∧+(i103[3], -1)=i103[2]1∧i95[3]=i95[2]1∧i90[3]=i90[2]1∧+(i100[3], -1)=i100[2]1 ⇒ COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥NonInfC∧COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))



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

    (29)    (>(i103[2], 0)=TRUE ⇒ COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥NonInfC∧COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥LOAD866(i90[2], i95[2], +(i100[2], -1), +(i103[2], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))



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

    (30)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (31)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (32)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (33)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)



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

    (34)    (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)



  • We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)) which results in the following constraint:

    (35)    (i100[2]=i100[3]∧i90[2]=i90[3]∧i103[2]=i103[3]∧i95[2]=i95[3]∧>(i103[2], 0)=TRUE∧+(i100[3], -1)=i100[4]∧i95[3]=i95[4]∧+(i103[3], -1)=0∧i90[3]=i90[4] ⇒ COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥NonInfC∧COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))



    We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:

    (36)    (>(i103[2], 0)=TRUE∧+(i103[2], -1)=0 ⇒ COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥NonInfC∧COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥LOAD866(i90[2], i95[2], +(i100[2], -1), +(i103[2], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))



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

    (37)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (38)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (39)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)



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

    (40)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)



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

    (41)    (i103[2] ≥ 0∧i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)







For Pair LOAD866(i90, i95, i100, 0) → LOAD783(i100, i95, +(i90, 1)) the following chains were created:
  • We consider the chain COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)), LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0]) which results in the following constraint:

    (42)    (i95[1]=0∧i88[1]=i100[4]∧i90[1]=i90[4]∧i95[1]=i95[4]∧i100[4]=i88[0]∧i95[4]=i95[0]∧+(i90[4], 1)=i90[0] ⇒ LOAD866(i90[4], i95[4], i100[4], 0)≥NonInfC∧LOAD866(i90[4], i95[4], i100[4], 0)≥LOAD783(i100[4], i95[4], +(i90[4], 1))∧(UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥))



    We simplified constraint (42) using rules (III), (IV) which results in the following new constraint:

    (43)    (LOAD866(i90[1], 0, i88[1], 0)≥NonInfC∧LOAD866(i90[1], 0, i88[1], 0)≥LOAD783(i88[1], 0, +(i90[1], 1))∧(UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥))



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

    (44)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧[(-1)bso_35] ≥ 0)



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

    (45)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧[(-1)bso_35] ≥ 0)



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

    (46)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧[(-1)bso_35] ≥ 0)



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

    (47)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)



  • We consider the chain COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)), LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0]) which results in the following constraint:

    (48)    (+(i100[3], -1)=i100[4]∧i95[3]=i95[4]∧+(i103[3], -1)=0∧i90[3]=i90[4]∧i100[4]=i88[0]∧i95[4]=i95[0]∧+(i90[4], 1)=i90[0] ⇒ LOAD866(i90[4], i95[4], i100[4], 0)≥NonInfC∧LOAD866(i90[4], i95[4], i100[4], 0)≥LOAD783(i100[4], i95[4], +(i90[4], 1))∧(UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥))



    We simplified constraint (48) using rules (III), (IV) which results in the following new constraint:

    (49)    (+(i103[3], -1)=0 ⇒ LOAD866(i90[3], i95[3], +(i100[3], -1), 0)≥NonInfC∧LOAD866(i90[3], i95[3], +(i100[3], -1), 0)≥LOAD783(+(i100[3], -1), i95[3], +(i90[3], 1))∧(UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥))



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

    (50)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 ≥ 0∧[(-1)bso_35] ≥ 0)



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

    (51)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 ≥ 0∧[(-1)bso_35] ≥ 0)



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

    (52)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 ≥ 0∧[(-1)bso_35] ≥ 0)



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

    (53)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD783(i88, i95, i90) → COND_LOAD783(&&(>(i95, 0), >=(i88, i95)), i88, i95, i90)
    • (i95[0] ≥ 0∧i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i95[0] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[1 + (-1)bso_27] + i95[0] ≥ 0)

  • COND_LOAD783(TRUE, i88, i95, i90) → LOAD866(i90, i95, i88, i95)
    • ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
    • ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

  • LOAD866(i90, i95, i100, i103) → COND_LOAD866(>(i103, 0), i90, i95, i100, i103)
    • (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_30] = 0∧0 = 0∧0 = 0∧[(-2)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

  • COND_LOAD866(TRUE, i90, i95, i100, i103) → LOAD866(i90, i95, +(i100, -1), +(i103, -1))
    • (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)
    • (i103[2] ≥ 0∧i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

  • LOAD866(i90, i95, i100, 0) → LOAD783(i100, i95, +(i90, 1))
    • ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)
    • (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 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) = [3]   
POL(LOAD783(x1, x2, x3)) = [-1] + [2]x1   
POL(COND_LOAD783(x1, x2, x3, x4)) = [-1] + [-1]x3 + [2]x2   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(>=(x1, x2)) = [-1]   
POL(LOAD866(x1, x2, x3, x4)) = [-1] + [-1]x4 + [2]x3   
POL(COND_LOAD866(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [2]x4   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(1) = [1]   

The following pairs are in P>:

LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])
COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))

The following pairs are in Pbound:

LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])

The following pairs are in P≥:

COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1])
LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])
LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1))

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

FALSE1 → &&(TRUE, 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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1])
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(4): LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], i90[4] + 1)

(1) -> (2), if ((i95[1] →* i95[2])∧(i90[1] →* i90[2])∧(i95[1] →* i103[2])∧(i88[1] →* i100[2]))


(1) -> (4), if ((i95[1] →* 0)∧(i88[1] →* i100[4])∧(i90[1] →* i90[4])∧(i95[1] →* i95[4]))



The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1])
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)
(4): LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], i90[4] + 1)

(1) -> (2), if ((i95[1] →* i95[2])∧(i90[1] →* i90[2])∧(i95[1] →* i103[2])∧(i88[1] →* i100[2]))


(3) -> (2), if ((i103[3] + -1 →* i103[2])∧(i95[3] →* i95[2])∧(i90[3] →* i90[2])∧(i100[3] + -1 →* i100[2]))


(2) -> (3), if ((i100[2] →* i100[3])∧(i90[2] →* i90[3])∧(i103[2] →* i103[3])∧(i95[2] →* i95[3])∧(i103[2] > 0 →* TRUE))


(1) -> (4), if ((i95[1] →* 0)∧(i88[1] →* i100[4])∧(i90[1] →* i90[4])∧(i95[1] →* i95[4]))


(3) -> (4), if ((i100[3] + -1 →* i100[4])∧(i95[3] →* i95[4])∧(i103[3] + -1 →* 0)∧(i90[3] →* i90[4]))



The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])

(3) -> (2), if ((i103[3] + -1 →* i103[2])∧(i95[3] →* i95[2])∧(i90[3] →* i90[2])∧(i100[3] + -1 →* i100[2]))


(2) -> (3), if ((i100[2] →* i100[3])∧(i90[2] →* i90[3])∧(i103[2] →* i103[3])∧(i95[2] →* i95[3])∧(i103[2] > 0 →* TRUE))



The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(19) 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_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)) the following chains were created:
  • We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) which results in the following constraint:

    (1)    (i100[2]=i100[3]∧i90[2]=i90[3]∧i103[2]=i103[3]∧i95[2]=i95[3]∧>(i103[2], 0)=TRUE∧+(i103[3], -1)=i103[2]1∧i95[3]=i95[2]1∧i90[3]=i90[2]1∧+(i100[3], -1)=i100[2]1 ⇒ COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥NonInfC∧COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))



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

    (2)    (>(i103[2], 0)=TRUE ⇒ COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥NonInfC∧COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥LOAD866(i90[2], i95[2], +(i100[2], -1), +(i103[2], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))



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

    (3)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (4)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (5)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (6)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)



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

    (7)    (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)







For Pair LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) the following chains were created:
  • We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)) which results in the following constraint:

    (8)    (i100[2]=i100[3]∧i90[2]=i90[3]∧i103[2]=i103[3]∧i95[2]=i95[3]∧>(i103[2], 0)=TRUE ⇒ LOAD866(i90[2], i95[2], i100[2], i103[2])≥NonInfC∧LOAD866(i90[2], i95[2], i100[2], i103[2])≥COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])∧(UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥))



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

    (9)    (>(i103[2], 0)=TRUE ⇒ LOAD866(i90[2], i95[2], i100[2], i103[2])≥NonInfC∧LOAD866(i90[2], i95[2], i100[2], i103[2])≥COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])∧(UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥))



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

    (10)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (11)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (12)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (13)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)



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

    (14)    (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))
    • (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

  • LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])
    • (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [2]   
POL(FALSE) = 0   
POL(COND_LOAD866(x1, x2, x3, x4, x5)) = [2] + x5   
POL(LOAD866(x1, x2, x3, x4)) = [2] + x4   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))

The following pairs are in Pbound:

COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))
LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])

The following pairs are in P≥:

LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])

There are no usable rules.

(20) Complex Obligation (AND)

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])


The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load783(x0, x1, x2)
Cond_Load783(TRUE, x0, x1, x2)
Load866(x0, x1, x2, x3)
Cond_Load866(TRUE, x0, x1, x2, x3)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE