Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 GroundTermsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 IDP
        ↳16 IDPNonInfProof (⇐)
        ↳17 AND
            ↳18 IDP
                ↳19 IDependencyGraphProof (⇔)
                ↳20 TRUE
            ↳21 IDP
                ↳22 IDependencyGraphProof (⇔)
                ↳23 TRUE
    ↳24 IDP
        ↳25 IDependencyGraphProof (⇔)
        ↳26 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
No human-readable program information known.

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC2

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 118 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:
Load373(1, i38) → Cond_Load373(i38 >= 0 && i38 + 1 > 0, 1, i38)
Cond_Load373(TRUE, 1, i38) → Load439(1, i38 + 1, 1)
Load439(1, i40, i45) → Cond_Load439(i45 > 0 && i40 >= i45, 1, i40, i45)
Cond_Load439(TRUE, 1, i40, i45) → Load439(1, i40, i45 + 1)
Load439(1, i40, i45) → Cond_Load4391(i40 > 0 && i40 < i45, 1, i40, i45)
Cond_Load4391(TRUE, 1, i40, i45) → Load373(1, i40 - 2)
The set Q consists of the following terms:
Load373(1, x0)
Cond_Load373(TRUE, 1, x0)
Load439(1, x0, x1)
Cond_Load439(TRUE, 1, x0, x1)
Cond_Load4391(TRUE, 1, x0, x1)

(5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
  • 1

We removed arguments according to the following replacements:

Load373(x1, x2) → Load373(x2)
Cond_Load4391(x1, x2, x3, x4) → Cond_Load4391(x1, x3, x4)
Load439(x1, x2, x3) → Load439(x2, x3)
Cond_Load439(x1, x2, x3, x4) → Cond_Load439(x1, x3, x4)
Cond_Load373(x1, x2, x3) → Cond_Load373(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:
Load373(i38) → Cond_Load373(i38 >= 0 && i38 + 1 > 0, i38)
Cond_Load373(TRUE, i38) → Load439(i38 + 1, 1)
Load439(i40, i45) → Cond_Load439(i45 > 0 && i40 >= i45, i40, i45)
Cond_Load439(TRUE, i40, i45) → Load439(i40, i45 + 1)
Load439(i40, i45) → Cond_Load4391(i40 > 0 && i40 < i45, i40, i45)
Cond_Load4391(TRUE, i40, i45) → Load373(i40 - 2)
The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load373(i38) → Cond_Load373(i38 >= 0 && i38 + 1 > 0, i38)
Cond_Load373(TRUE, i38) → Load439(i38 + 1, 1)
Load439(i40, i45) → Cond_Load439(i45 > 0 && i40 >= i45, i40, i45)
Cond_Load439(TRUE, i40, i45) → Load439(i40, i45 + 1)
Load439(i40, i45) → Cond_Load4391(i40 > 0 && i40 < i45, i40, i45)
Cond_Load4391(TRUE, i40, i45) → Load373(i40 - 2)

The integer pair graph contains the following rules and edges:
(0): LOAD373(i38[0]) → COND_LOAD373(i38[0] >= 0 && i38[0] + 1 > 0, i38[0])
(1): COND_LOAD373(TRUE, i38[1]) → LOAD439(i38[1] + 1, 1)
(2): LOAD439(i40[2], i45[2]) → COND_LOAD439(i45[2] > 0 && i40[2] >= i45[2], i40[2], i45[2])
(3): COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], i45[3] + 1)
(4): LOAD439(i40[4], i45[4]) → COND_LOAD4391(i40[4] > 0 && i40[4] < i45[4], i40[4], i45[4])
(5): COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(i40[5] - 2)

(0) -> (1), if ((i38[0] →* i38[1])∧(i38[0] >= 0 && i38[0] + 1 > 0 →* TRUE))


(1) -> (2), if ((i38[1] + 1 →* i40[2])∧(1 →* i45[2]))


(1) -> (4), if ((i38[1] + 1 →* i40[4])∧(1 →* i45[4]))


(2) -> (3), if ((i45[2] > 0 && i40[2] >= i45[2] →* TRUE)∧(i45[2] →* i45[3])∧(i40[2] →* i40[3]))


(3) -> (2), if ((i40[3] →* i40[2])∧(i45[3] + 1 →* i45[2]))


(3) -> (4), if ((i40[3] →* i40[4])∧(i45[3] + 1 →* i45[4]))


(4) -> (5), if ((i40[4] →* i40[5])∧(i40[4] > 0 && i40[4] < i45[4] →* TRUE)∧(i45[4] →* i45[5]))


(5) -> (0), if ((i40[5] - 2 →* i38[0]))



The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD373(i38[0]) → COND_LOAD373(i38[0] >= 0 && i38[0] + 1 > 0, i38[0])
(1): COND_LOAD373(TRUE, i38[1]) → LOAD439(i38[1] + 1, 1)
(2): LOAD439(i40[2], i45[2]) → COND_LOAD439(i45[2] > 0 && i40[2] >= i45[2], i40[2], i45[2])
(3): COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], i45[3] + 1)
(4): LOAD439(i40[4], i45[4]) → COND_LOAD4391(i40[4] > 0 && i40[4] < i45[4], i40[4], i45[4])
(5): COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(i40[5] - 2)

(0) -> (1), if ((i38[0] →* i38[1])∧(i38[0] >= 0 && i38[0] + 1 > 0 →* TRUE))


(1) -> (2), if ((i38[1] + 1 →* i40[2])∧(1 →* i45[2]))


(1) -> (4), if ((i38[1] + 1 →* i40[4])∧(1 →* i45[4]))


(2) -> (3), if ((i45[2] > 0 && i40[2] >= i45[2] →* TRUE)∧(i45[2] →* i45[3])∧(i40[2] →* i40[3]))


(3) -> (2), if ((i40[3] →* i40[2])∧(i45[3] + 1 →* i45[2]))


(3) -> (4), if ((i40[3] →* i40[4])∧(i45[3] + 1 →* i45[4]))


(4) -> (5), if ((i40[4] →* i40[5])∧(i40[4] > 0 && i40[4] < i45[4] →* TRUE)∧(i45[4] →* i45[5]))


(5) -> (0), if ((i40[5] - 2 →* i38[0]))



The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(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 LOAD373(i38) → COND_LOAD373(&&(>=(i38, 0), >(+(i38, 1), 0)), i38) the following chains were created:
  • We consider the chain LOAD373(i38[0]) → COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0]), COND_LOAD373(TRUE, i38[1]) → LOAD439(+(i38[1], 1), 1) which results in the following constraint:

    (1)    (i38[0]=i38[1]∧&&(>=(i38[0], 0), >(+(i38[0], 1), 0))=TRUE ⇒ LOAD373(i38[0])≥NonInfC∧LOAD373(i38[0])≥COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])∧(UIncreasing(COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])), ≥))



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

    (2)    (>=(i38[0], 0)=TRUE∧>(+(i38[0], 1), 0)=TRUE ⇒ LOAD373(i38[0])≥NonInfC∧LOAD373(i38[0])≥COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])∧(UIncreasing(COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])), ≥))



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

    (3)    (i38[0] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]i38[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (4)    (i38[0] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]i38[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (5)    (i38[0] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]i38[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)







For Pair COND_LOAD373(TRUE, i38) → LOAD439(+(i38, 1), 1) the following chains were created:
  • We consider the chain COND_LOAD373(TRUE, i38[1]) → LOAD439(+(i38[1], 1), 1) which results in the following constraint:

    (6)    (COND_LOAD373(TRUE, i38[1])≥NonInfC∧COND_LOAD373(TRUE, i38[1])≥LOAD439(+(i38[1], 1), 1)∧(UIncreasing(LOAD439(+(i38[1], 1), 1)), ≥))



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

    (7)    ((UIncreasing(LOAD439(+(i38[1], 1), 1)), ≥)∧[(-1)bso_19] ≥ 0)



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

    (8)    ((UIncreasing(LOAD439(+(i38[1], 1), 1)), ≥)∧[(-1)bso_19] ≥ 0)



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

    (9)    ((UIncreasing(LOAD439(+(i38[1], 1), 1)), ≥)∧[(-1)bso_19] ≥ 0)



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

    (10)    ((UIncreasing(LOAD439(+(i38[1], 1), 1)), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair LOAD439(i40, i45) → COND_LOAD439(&&(>(i45, 0), >=(i40, i45)), i40, i45) the following chains were created:
  • We consider the chain LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2]), COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1)) which results in the following constraint:

    (11)    (&&(>(i45[2], 0), >=(i40[2], i45[2]))=TRUE∧i45[2]=i45[3]∧i40[2]=i40[3] ⇒ LOAD439(i40[2], i45[2])≥NonInfC∧LOAD439(i40[2], i45[2])≥COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])∧(UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥))



    We simplified constraint (11) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (12)    (>(i45[2], 0)=TRUE∧>=(i40[2], i45[2])=TRUE ⇒ LOAD439(i40[2], i45[2])≥NonInfC∧LOAD439(i40[2], i45[2])≥COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])∧(UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥))



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

    (13)    (i45[2] + [-1] ≥ 0∧i40[2] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i40[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (14)    (i45[2] + [-1] ≥ 0∧i40[2] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i40[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (15)    (i45[2] + [-1] ≥ 0∧i40[2] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i40[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (16)    (i45[2] ≥ 0∧i40[2] + [-1] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i40[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (17)    (i45[2] ≥ 0∧i40[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i45[2] + [bni_20]i40[2] ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair COND_LOAD439(TRUE, i40, i45) → LOAD439(i40, +(i45, 1)) the following chains were created:
  • We consider the chain COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1)) which results in the following constraint:

    (18)    (COND_LOAD439(TRUE, i40[3], i45[3])≥NonInfC∧COND_LOAD439(TRUE, i40[3], i45[3])≥LOAD439(i40[3], +(i45[3], 1))∧(UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥))



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

    (19)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧[(-1)bso_23] ≥ 0)



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

    (20)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧[(-1)bso_23] ≥ 0)



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

    (21)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧[(-1)bso_23] ≥ 0)



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

    (22)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)







For Pair LOAD439(i40, i45) → COND_LOAD4391(&&(>(i40, 0), <(i40, i45)), i40, i45) the following chains were created:
  • We consider the chain LOAD439(i40[4], i45[4]) → COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4]), COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(-(i40[5], 2)) which results in the following constraint:

    (23)    (i40[4]=i40[5]∧&&(>(i40[4], 0), <(i40[4], i45[4]))=TRUE∧i45[4]=i45[5] ⇒ LOAD439(i40[4], i45[4])≥NonInfC∧LOAD439(i40[4], i45[4])≥COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])∧(UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥))



    We simplified constraint (23) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (24)    (>(i40[4], 0)=TRUE∧<(i40[4], i45[4])=TRUE ⇒ LOAD439(i40[4], i45[4])≥NonInfC∧LOAD439(i40[4], i45[4])≥COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])∧(UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥))



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

    (25)    (i40[4] + [-1] ≥ 0∧i45[4] + [-1] + [-1]i40[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i40[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (26)    (i40[4] + [-1] ≥ 0∧i45[4] + [-1] + [-1]i40[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i40[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (27)    (i40[4] + [-1] ≥ 0∧i45[4] + [-1] + [-1]i40[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i40[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (28)    (i40[4] ≥ 0∧i45[4] + [-2] + [-1]i40[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i40[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (29)    (i40[4] ≥ 0∧i45[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i40[4] ≥ 0∧[(-1)bso_25] ≥ 0)







For Pair COND_LOAD4391(TRUE, i40, i45) → LOAD373(-(i40, 2)) the following chains were created:
  • We consider the chain COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(-(i40[5], 2)) which results in the following constraint:

    (30)    (COND_LOAD4391(TRUE, i40[5], i45[5])≥NonInfC∧COND_LOAD4391(TRUE, i40[5], i45[5])≥LOAD373(-(i40[5], 2))∧(UIncreasing(LOAD373(-(i40[5], 2))), ≥))



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

    (31)    ((UIncreasing(LOAD373(-(i40[5], 2))), ≥)∧[(-1)bso_27] ≥ 0)



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

    (32)    ((UIncreasing(LOAD373(-(i40[5], 2))), ≥)∧[(-1)bso_27] ≥ 0)



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

    (33)    ((UIncreasing(LOAD373(-(i40[5], 2))), ≥)∧[(-1)bso_27] ≥ 0)



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

    (34)    ((UIncreasing(LOAD373(-(i40[5], 2))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD373(i38) → COND_LOAD373(&&(>=(i38, 0), >(+(i38, 1), 0)), i38)
    • (i38[0] ≥ 0∧i38[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]i38[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

  • COND_LOAD373(TRUE, i38) → LOAD439(+(i38, 1), 1)
    • ((UIncreasing(LOAD439(+(i38[1], 1), 1)), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

  • LOAD439(i40, i45) → COND_LOAD439(&&(>(i45, 0), >=(i40, i45)), i40, i45)
    • (i45[2] ≥ 0∧i40[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i45[2] + [bni_20]i40[2] ≥ 0∧[(-1)bso_21] ≥ 0)

  • COND_LOAD439(TRUE, i40, i45) → LOAD439(i40, +(i45, 1))
    • ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

  • LOAD439(i40, i45) → COND_LOAD4391(&&(>(i40, 0), <(i40, i45)), i40, i45)
    • (i40[4] ≥ 0∧i45[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i40[4] ≥ 0∧[(-1)bso_25] ≥ 0)

  • COND_LOAD4391(TRUE, i40, i45) → LOAD373(-(i40, 2))
    • ((UIncreasing(LOAD373(-(i40[5], 2))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 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(LOAD373(x1)) = [1] + x1   
POL(COND_LOAD373(x1, x2)) = x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(LOAD439(x1, x2)) = [-1] + x1   
POL(COND_LOAD439(x1, x2, x3)) = [-1] + x2   
POL(COND_LOAD4391(x1, x2, x3)) = [-1] + x2   
POL(<(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(2) = [2]   

The following pairs are in P>:

LOAD373(i38[0]) → COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])

The following pairs are in Pbound:

LOAD373(i38[0]) → COND_LOAD373(&&(>=(i38[0], 0), >(+(i38[0], 1), 0)), i38[0])
LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])
LOAD439(i40[4], i45[4]) → COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])

The following pairs are in P≥:

COND_LOAD373(TRUE, i38[1]) → LOAD439(+(i38[1], 1), 1)
LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])
COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1))
LOAD439(i40[4], i45[4]) → COND_LOAD4391(&&(>(i40[4], 0), <(i40[4], i45[4])), i40[4], i45[4])
COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(-(i40[5], 2))

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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD373(TRUE, i38[1]) → LOAD439(i38[1] + 1, 1)
(2): LOAD439(i40[2], i45[2]) → COND_LOAD439(i45[2] > 0 && i40[2] >= i45[2], i40[2], i45[2])
(3): COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], i45[3] + 1)
(4): LOAD439(i40[4], i45[4]) → COND_LOAD4391(i40[4] > 0 && i40[4] < i45[4], i40[4], i45[4])
(5): COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(i40[5] - 2)

(1) -> (2), if ((i38[1] + 1 →* i40[2])∧(1 →* i45[2]))


(3) -> (2), if ((i40[3] →* i40[2])∧(i45[3] + 1 →* i45[2]))


(2) -> (3), if ((i45[2] > 0 && i40[2] >= i45[2] →* TRUE)∧(i45[2] →* i45[3])∧(i40[2] →* i40[3]))


(1) -> (4), if ((i38[1] + 1 →* i40[4])∧(1 →* i45[4]))


(3) -> (4), if ((i40[3] →* i40[4])∧(i45[3] + 1 →* i45[4]))


(4) -> (5), if ((i40[4] →* i40[5])∧(i40[4] > 0 && i40[4] < i45[4] →* TRUE)∧(i45[4] →* i45[5]))



The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(15) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], i45[3] + 1)
(2): LOAD439(i40[2], i45[2]) → COND_LOAD439(i45[2] > 0 && i40[2] >= i45[2], i40[2], i45[2])

(3) -> (2), if ((i40[3] →* i40[2])∧(i45[3] + 1 →* i45[2]))


(2) -> (3), if ((i45[2] > 0 && i40[2] >= i45[2] →* TRUE)∧(i45[2] →* i45[3])∧(i40[2] →* i40[3]))



The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(16) IDPNonInfProof (SOUND transformation)

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


For Pair COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1)) the following chains were created:
  • We consider the chain COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1)) which results in the following constraint:

    (1)    (COND_LOAD439(TRUE, i40[3], i45[3])≥NonInfC∧COND_LOAD439(TRUE, i40[3], i45[3])≥LOAD439(i40[3], +(i45[3], 1))∧(UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥))



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

    (2)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧[1 + (-1)bso_10] ≥ 0)



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

    (3)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧[1 + (-1)bso_10] ≥ 0)



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

    (4)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧[1 + (-1)bso_10] ≥ 0)



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

    (5)    ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)







For Pair LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2]) the following chains were created:
  • We consider the chain LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2]), COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1)) which results in the following constraint:

    (6)    (&&(>(i45[2], 0), >=(i40[2], i45[2]))=TRUE∧i45[2]=i45[3]∧i40[2]=i40[3] ⇒ LOAD439(i40[2], i45[2])≥NonInfC∧LOAD439(i40[2], i45[2])≥COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])∧(UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥))



    We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (7)    (>(i45[2], 0)=TRUE∧>=(i40[2], i45[2])=TRUE ⇒ LOAD439(i40[2], i45[2])≥NonInfC∧LOAD439(i40[2], i45[2])≥COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])∧(UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥))



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

    (8)    (i45[2] + [-1] ≥ 0∧i40[2] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]i45[2] + [bni_11]i40[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (9)    (i45[2] + [-1] ≥ 0∧i40[2] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]i45[2] + [bni_11]i40[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (10)    (i45[2] + [-1] ≥ 0∧i40[2] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]i45[2] + [bni_11]i40[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (11)    (i45[2] ≥ 0∧i40[2] + [-1] + [-1]i45[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-2)bni_11 + (-1)Bound*bni_11] + [(-1)bni_11]i45[2] + [bni_11]i40[2] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (12)    (i45[2] ≥ 0∧i40[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i40[2] ≥ 0∧[(-1)bso_12] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1))
    • ((UIncreasing(LOAD439(i40[3], +(i45[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_10] ≥ 0)

  • LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])
    • (i45[2] ≥ 0∧i40[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i40[2] ≥ 0∧[(-1)bso_12] ≥ 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_LOAD439(x1, x2, x3)) = [-1] + x2 + [-1]x3   
POL(LOAD439(x1, x2)) = [-1] + [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(>=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], +(i45[3], 1))

The following pairs are in Pbound:

LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])

The following pairs are in P≥:

LOAD439(i40[2], i45[2]) → COND_LOAD439(&&(>(i45[2], 0), >=(i40[2], i45[2])), i40[2], i45[2])

There are no usable rules.

(17) Complex Obligation (AND)

(18) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD439(i40[2], i45[2]) → COND_LOAD439(i45[2] > 0 && i40[2] >= i45[2], i40[2], i45[2])


The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], i45[3] + 1)


The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD373(TRUE, i38[1]) → LOAD439(i38[1] + 1, 1)
(3): COND_LOAD439(TRUE, i40[3], i45[3]) → LOAD439(i40[3], i45[3] + 1)
(5): COND_LOAD4391(TRUE, i40[5], i45[5]) → LOAD373(i40[5] - 2)


The set Q consists of the following terms:
Load373(x0)
Cond_Load373(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE