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: PastaB14

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 175 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:
Load661(i104, i104) → Cond_Load661(i104 > 0, i104, i104)
Cond_Load661(TRUE, i104, i104) → Load759(i104, i104)
Load759(i120, i123) → Cond_Load759(i123 > 0, i120, i123)
Cond_Load759(TRUE, i120, i123) → Load759(i120 + -1, i123 + -1)
Load759(i120, 0) → Load661(i120, 0)
The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Cond_Load661(x1, x2, x3) → Cond_Load661(x1, x3)

(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:
Load661(i104, i104) → Cond_Load661(i104 > 0, i104)
Cond_Load661(TRUE, i104) → Load759(i104, i104)
Load759(i120, i123) → Cond_Load759(i123 > 0, i120, i123)
Cond_Load759(TRUE, i120, i123) → Load759(i120 + -1, i123 + -1)
Load759(i120, 0) → Load661(i120, 0)
The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
Load661(i104, i104) → Cond_Load661(i104 > 0, i104)
Cond_Load661(TRUE, i104) → Load759(i104, i104)
Load759(i120, i123) → Cond_Load759(i123 > 0, i120, i123)
Cond_Load759(TRUE, i120, i123) → Load759(i120 + -1, i123 + -1)
Load759(i120, 0) → Load661(i120, 0)

The integer pair graph contains the following rules and edges:
(0): LOAD661(i104[0], i104[0]) → COND_LOAD661(i104[0] > 0, i104[0])
(1): COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1])
(2): LOAD759(i120[2], i123[2]) → COND_LOAD759(i123[2] > 0, i120[2], i123[2])
(3): COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(i120[3] + -1, i123[3] + -1)
(4): LOAD759(i120[4], 0) → LOAD661(i120[4], 0)

(0) -> (1), if ((i104[0] > 0 →* TRUE)∧(i104[0] →* i104[1]))


(1) -> (2), if ((i104[1] →* i123[2])∧(i104[1] →* i120[2]))


(1) -> (4), if ((i104[1] →* i120[4])∧(i104[1] →* 0))


(2) -> (3), if ((i123[2] →* i123[3])∧(i123[2] > 0 →* TRUE)∧(i120[2] →* i120[3]))


(3) -> (2), if ((i123[3] + -1 →* i123[2])∧(i120[3] + -1 →* i120[2]))


(3) -> (4), if ((i120[3] + -1 →* i120[4])∧(i123[3] + -1 →* 0))


(4) -> (0), if ((i120[4] →* i104[0])∧(0 →* i104[0]))



The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD661(i104[0], i104[0]) → COND_LOAD661(i104[0] > 0, i104[0])
(1): COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1])
(2): LOAD759(i120[2], i123[2]) → COND_LOAD759(i123[2] > 0, i120[2], i123[2])
(3): COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(i120[3] + -1, i123[3] + -1)
(4): LOAD759(i120[4], 0) → LOAD661(i120[4], 0)

(0) -> (1), if ((i104[0] > 0 →* TRUE)∧(i104[0] →* i104[1]))


(1) -> (2), if ((i104[1] →* i123[2])∧(i104[1] →* i120[2]))


(1) -> (4), if ((i104[1] →* i120[4])∧(i104[1] →* 0))


(2) -> (3), if ((i123[2] →* i123[3])∧(i123[2] > 0 →* TRUE)∧(i120[2] →* i120[3]))


(3) -> (2), if ((i123[3] + -1 →* i123[2])∧(i120[3] + -1 →* i120[2]))


(3) -> (4), if ((i120[3] + -1 →* i120[4])∧(i123[3] + -1 →* 0))


(4) -> (0), if ((i120[4] →* i104[0])∧(0 →* i104[0]))



The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(11) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD661(i104, i104) → COND_LOAD661(>(i104, 0), i104) the following chains were created:
  • We consider the chain LOAD661(i104[0], i104[0]) → COND_LOAD661(>(i104[0], 0), i104[0]), COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1]) which results in the following constraint:

    (1)    (>(i104[0], 0)=TRUE∧i104[0]=i104[1] ⇒ LOAD661(i104[0], i104[0])≥NonInfC∧LOAD661(i104[0], i104[0])≥COND_LOAD661(>(i104[0], 0), i104[0])∧(UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥))



    We simplified constraint (1) using rule (IV) which results in the following new constraint:

    (2)    (>(i104[0], 0)=TRUE ⇒ LOAD661(i104[0], i104[0])≥NonInfC∧LOAD661(i104[0], i104[0])≥COND_LOAD661(>(i104[0], 0), i104[0])∧(UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥))



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

    (3)    (i104[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[(-1)bso_18] + i104[0] ≥ 0)



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

    (4)    (i104[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[(-1)bso_18] + i104[0] ≥ 0)



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

    (5)    (i104[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[(-1)bso_18] + i104[0] ≥ 0)



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

    (6)    (i104[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[1 + (-1)bso_18] + i104[0] ≥ 0)







For Pair COND_LOAD661(TRUE, i104) → LOAD759(i104, i104) the following chains were created:
  • We consider the chain COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1]), LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]) which results in the following constraint:

    (7)    (i104[1]=i123[2]∧i104[1]=i120[2] ⇒ COND_LOAD661(TRUE, i104[1])≥NonInfC∧COND_LOAD661(TRUE, i104[1])≥LOAD759(i104[1], i104[1])∧(UIncreasing(LOAD759(i104[1], i104[1])), ≥))



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

    (8)    (COND_LOAD661(TRUE, i104[1])≥NonInfC∧COND_LOAD661(TRUE, i104[1])≥LOAD759(i104[1], i104[1])∧(UIncreasing(LOAD759(i104[1], i104[1])), ≥))



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

    (9)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (10)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (11)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (12)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)



  • We consider the chain COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1]), LOAD759(i120[4], 0) → LOAD661(i120[4], 0) which results in the following constraint:

    (13)    (i104[1]=i120[4]∧i104[1]=0 ⇒ COND_LOAD661(TRUE, i104[1])≥NonInfC∧COND_LOAD661(TRUE, i104[1])≥LOAD759(i104[1], i104[1])∧(UIncreasing(LOAD759(i104[1], i104[1])), ≥))



    We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

    (14)    (COND_LOAD661(TRUE, 0)≥NonInfC∧COND_LOAD661(TRUE, 0)≥LOAD759(0, 0)∧(UIncreasing(LOAD759(i104[1], i104[1])), ≥))



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

    (15)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (16)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (17)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)







For Pair LOAD759(i120, i123) → COND_LOAD759(>(i123, 0), i120, i123) the following chains were created:
  • We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)) which results in the following constraint:

    (18)    (i123[2]=i123[3]∧>(i123[2], 0)=TRUE∧i120[2]=i120[3] ⇒ LOAD759(i120[2], i123[2])≥NonInfC∧LOAD759(i120[2], i123[2])≥COND_LOAD759(>(i123[2], 0), i120[2], i123[2])∧(UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥))



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

    (19)    (>(i123[2], 0)=TRUE ⇒ LOAD759(i120[2], i123[2])≥NonInfC∧LOAD759(i120[2], i123[2])≥COND_LOAD759(>(i123[2], 0), i120[2], i123[2])∧(UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥))



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

    (20)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] + [(2)bni_21]i120[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (21)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] + [(2)bni_21]i120[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (22)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] + [(2)bni_21]i120[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (23)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21] = 0∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)



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

    (24)    (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21] = 0∧[bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)







For Pair COND_LOAD759(TRUE, i120, i123) → LOAD759(+(i120, -1), +(i123, -1)) the following chains were created:
  • We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)), LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]) which results in the following constraint:

    (25)    (i123[2]=i123[3]∧>(i123[2], 0)=TRUE∧i120[2]=i120[3]∧+(i123[3], -1)=i123[2]1∧+(i120[3], -1)=i120[2]1 ⇒ COND_LOAD759(TRUE, i120[3], i123[3])≥NonInfC∧COND_LOAD759(TRUE, i120[3], i123[3])≥LOAD759(+(i120[3], -1), +(i123[3], -1))∧(UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥))



    We simplified constraint (25) using rules (III), (IV) which results in the following new constraint:

    (26)    (>(i123[2], 0)=TRUE ⇒ COND_LOAD759(TRUE, i120[2], i123[2])≥NonInfC∧COND_LOAD759(TRUE, i120[2], i123[2])≥LOAD759(+(i120[2], -1), +(i123[2], -1))∧(UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥))



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

    (27)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (28)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (29)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (30)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)



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

    (31)    (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)



  • We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)), LOAD759(i120[4], 0) → LOAD661(i120[4], 0) which results in the following constraint:

    (32)    (i123[2]=i123[3]∧>(i123[2], 0)=TRUE∧i120[2]=i120[3]∧+(i120[3], -1)=i120[4]∧+(i123[3], -1)=0 ⇒ COND_LOAD759(TRUE, i120[3], i123[3])≥NonInfC∧COND_LOAD759(TRUE, i120[3], i123[3])≥LOAD759(+(i120[3], -1), +(i123[3], -1))∧(UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥))



    We simplified constraint (32) using rules (III), (IV) which results in the following new constraint:

    (33)    (>(i123[2], 0)=TRUE∧+(i123[2], -1)=0 ⇒ COND_LOAD759(TRUE, i120[2], i123[2])≥NonInfC∧COND_LOAD759(TRUE, i120[2], i123[2])≥LOAD759(+(i120[2], -1), +(i123[2], -1))∧(UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥))



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

    (34)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (35)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (36)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (37)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)



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

    (38)    (i123[2] ≥ 0∧i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)







For Pair LOAD759(i120, 0) → LOAD661(i120, 0) the following chains were created:
  • We consider the chain LOAD759(i120[4], 0) → LOAD661(i120[4], 0), LOAD661(i104[0], i104[0]) → COND_LOAD661(>(i104[0], 0), i104[0]) which results in the following constraint:

    (39)    (i120[4]=i104[0]∧0=i104[0] ⇒ LOAD759(i120[4], 0)≥NonInfC∧LOAD759(i120[4], 0)≥LOAD661(i120[4], 0)∧(UIncreasing(LOAD661(i120[4], 0)), ≥))



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

    (40)    (LOAD759(0, 0)≥NonInfC∧LOAD759(0, 0)≥LOAD661(0, 0)∧(UIncreasing(LOAD661(i120[4], 0)), ≥))



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

    (41)    ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 0)



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

    (42)    ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 0)



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

    (43)    ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD661(i104, i104) → COND_LOAD661(>(i104, 0), i104)
    • (i104[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[1 + (-1)bso_18] + i104[0] ≥ 0)

  • COND_LOAD661(TRUE, i104) → LOAD759(i104, i104)
    • ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)
    • ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

  • LOAD759(i120, i123) → COND_LOAD759(>(i123, 0), i120, i123)
    • (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21] = 0∧[bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

  • COND_LOAD759(TRUE, i120, i123) → LOAD759(+(i120, -1), +(i123, -1))
    • (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)
    • (i123[2] ≥ 0∧i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

  • LOAD759(i120, 0) → LOAD661(i120, 0)
    • ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 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(LOAD661(x1, x2)) = [2] + x2 + x1   
POL(COND_LOAD661(x1, x2)) = [2] + x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(LOAD759(x1, x2)) = [2] + [-1]x2 + [2]x1   
POL(COND_LOAD759(x1, x2, x3)) = [2] + [-1]x3 + [2]x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD661(i104[0], i104[0]) → COND_LOAD661(>(i104[0], 0), i104[0])
COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1))

The following pairs are in Pbound:

LOAD661(i104[0], i104[0]) → COND_LOAD661(>(i104[0], 0), i104[0])

The following pairs are in P≥:

COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1])
LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2])
LOAD759(i120[4], 0) → LOAD661(i120[4], 0)

There are no usable rules.

(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_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1])
(2): LOAD759(i120[2], i123[2]) → COND_LOAD759(i123[2] > 0, i120[2], i123[2])
(4): LOAD759(i120[4], 0) → LOAD661(i120[4], 0)

(1) -> (2), if ((i104[1] →* i123[2])∧(i104[1] →* i120[2]))


(1) -> (4), if ((i104[1] →* i120[4])∧(i104[1] →* 0))



The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(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_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1])
(2): LOAD759(i120[2], i123[2]) → COND_LOAD759(i123[2] > 0, i120[2], i123[2])
(3): COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(i120[3] + -1, i123[3] + -1)
(4): LOAD759(i120[4], 0) → LOAD661(i120[4], 0)

(1) -> (2), if ((i104[1] →* i123[2])∧(i104[1] →* i120[2]))


(3) -> (2), if ((i123[3] + -1 →* i123[2])∧(i120[3] + -1 →* i120[2]))


(2) -> (3), if ((i123[2] →* i123[3])∧(i123[2] > 0 →* TRUE)∧(i120[2] →* i120[3]))


(1) -> (4), if ((i104[1] →* i120[4])∧(i104[1] →* 0))


(3) -> (4), if ((i120[3] + -1 →* i120[4])∧(i123[3] + -1 →* 0))



The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(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_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(i120[3] + -1, i123[3] + -1)
(2): LOAD759(i120[2], i123[2]) → COND_LOAD759(i123[2] > 0, i120[2], i123[2])

(3) -> (2), if ((i123[3] + -1 →* i123[2])∧(i120[3] + -1 →* i120[2]))


(2) -> (3), if ((i123[2] →* i123[3])∧(i123[2] > 0 →* TRUE)∧(i120[2] →* i120[3]))



The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(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_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)) the following chains were created:
  • We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)), LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]) which results in the following constraint:

    (1)    (i123[2]=i123[3]∧>(i123[2], 0)=TRUE∧i120[2]=i120[3]∧+(i123[3], -1)=i123[2]1∧+(i120[3], -1)=i120[2]1 ⇒ COND_LOAD759(TRUE, i120[3], i123[3])≥NonInfC∧COND_LOAD759(TRUE, i120[3], i123[3])≥LOAD759(+(i120[3], -1), +(i123[3], -1))∧(UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥))



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

    (2)    (>(i123[2], 0)=TRUE ⇒ COND_LOAD759(TRUE, i120[2], i123[2])≥NonInfC∧COND_LOAD759(TRUE, i120[2], i123[2])≥LOAD759(+(i120[2], -1), +(i123[2], -1))∧(UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥))



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

    (3)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]) the following chains were created:
  • We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)) which results in the following constraint:

    (8)    (i123[2]=i123[3]∧>(i123[2], 0)=TRUE∧i120[2]=i120[3] ⇒ LOAD759(i120[2], i123[2])≥NonInfC∧LOAD759(i120[2], i123[2])≥COND_LOAD759(>(i123[2], 0), i120[2], i123[2])∧(UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥))



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

    (9)    (>(i123[2], 0)=TRUE ⇒ LOAD759(i120[2], i123[2])≥NonInfC∧LOAD759(i120[2], i123[2])≥COND_LOAD759(>(i123[2], 0), i120[2], i123[2])∧(UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥))



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

    (10)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (13)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)



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

    (14)    (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1))
    • (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2])
    • (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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_LOAD759(x1, x2, x3)) = [-1] + x3   
POL(LOAD759(x1, x2)) = [-1] + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [2]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1))

The following pairs are in Pbound:

COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1))
LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2])

The following pairs are in P≥:

LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[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): LOAD759(i120[2], i123[2]) → COND_LOAD759(i123[2] > 0, i120[2], i123[2])


The set Q consists of the following terms:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(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:
Load661(x0, x0)
Cond_Load661(TRUE, x0)
Load759(x0, x1)
Cond_Load759(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE