Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 ITRStoIDPProof (⇔)
↳6 IDP
↳7 UsableRulesProof (⇔)
↳8 IDP
↳9 IDPNonInfProof (⇐)
↳10 AND
    ↳11 IDP
        ↳12 IDependencyGraphProof (⇔)
        ↳13 TRUE
    ↳14 IDP
        ↳15 IDependencyGraphProof (⇔)
        ↳16 IDP
        ↳17 IDPNonInfProof (⇐)
        ↳18 IDP
        ↳19 IDependencyGraphProof (⇔)
        ↳20 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: PastaC3

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 230 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:
Load875(i12, i35, i76) → Cond_Load875(i76 >= 0 && i12 >= i76 && i12 < i35 && i76 + 1 > 0, i12, i35, i76)
Cond_Load875(TRUE, i12, i35, i76) → Load875(i12, i35, i76 + 1)
Load875(i12, i35, i76) → Cond_Load8751(i12 >= 0 && i12 < i76 && i12 < i35 && i12 + 1 > 0, i12, i35, i76)
Cond_Load8751(TRUE, i12, i35, i76) → Load875(i12 + 1, i35, i76)
The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load875(i12, i35, i76) → Cond_Load875(i76 >= 0 && i12 >= i76 && i12 < i35 && i76 + 1 > 0, i12, i35, i76)
Cond_Load875(TRUE, i12, i35, i76) → Load875(i12, i35, i76 + 1)
Load875(i12, i35, i76) → Cond_Load8751(i12 >= 0 && i12 < i76 && i12 < i35 && i12 + 1 > 0, i12, i35, i76)
Cond_Load8751(TRUE, i12, i35, i76) → Load875(i12 + 1, i35, i76)

The integer pair graph contains the following rules and edges:
(0): LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0, i12[0], i35[0], i76[0])
(1): COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], i76[1] + 1)
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])
(3): COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(i12[3] + 1, i35[3], i76[3])

(0) -> (1), if ((i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0 →* TRUE)∧(i76[0] →* i76[1])∧(i12[0] →* i12[1])∧(i35[0] →* i35[1]))


(1) -> (0), if ((i35[1] →* i35[0])∧(i12[1] →* i12[0])∧(i76[1] + 1 →* i76[0]))


(1) -> (2), if ((i12[1] →* i12[2])∧(i35[1] →* i35[2])∧(i76[1] + 1 →* i76[2]))


(2) -> (3), if ((i35[2] →* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0 →* TRUE)∧(i76[2] →* i76[3])∧(i12[2] →* i12[3]))


(3) -> (0), if ((i35[3] →* i35[0])∧(i76[3] →* i76[0])∧(i12[3] + 1 →* i12[0]))


(3) -> (2), if ((i35[3] →* i35[2])∧(i76[3] →* i76[2])∧(i12[3] + 1 →* i12[2]))



The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0, i12[0], i35[0], i76[0])
(1): COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], i76[1] + 1)
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])
(3): COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(i12[3] + 1, i35[3], i76[3])

(0) -> (1), if ((i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0 →* TRUE)∧(i76[0] →* i76[1])∧(i12[0] →* i12[1])∧(i35[0] →* i35[1]))


(1) -> (0), if ((i35[1] →* i35[0])∧(i12[1] →* i12[0])∧(i76[1] + 1 →* i76[0]))


(1) -> (2), if ((i12[1] →* i12[2])∧(i35[1] →* i35[2])∧(i76[1] + 1 →* i76[2]))


(2) -> (3), if ((i35[2] →* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0 →* TRUE)∧(i76[2] →* i76[3])∧(i12[2] →* i12[3]))


(3) -> (0), if ((i35[3] →* i35[0])∧(i76[3] →* i76[0])∧(i12[3] + 1 →* i12[0]))


(3) -> (2), if ((i35[3] →* i35[2])∧(i76[3] →* i76[2])∧(i12[3] + 1 →* i12[2]))



The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(9) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD875(i12, i35, i76) → COND_LOAD875(&&(&&(&&(>=(i76, 0), >=(i12, i76)), <(i12, i35)), >(+(i76, 1), 0)), i12, i35, i76) the following chains were created:
  • We consider the chain LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0]), COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], +(i76[1], 1)) which results in the following constraint:

    (1)    (&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0))=TRUE∧i76[0]=i76[1]∧i12[0]=i12[1]∧i35[0]=i35[1] ⇒ LOAD875(i12[0], i35[0], i76[0])≥NonInfC∧LOAD875(i12[0], i35[0], i76[0])≥COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])∧(UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥))



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

    (2)    (>(+(i76[0], 1), 0)=TRUE∧<(i12[0], i35[0])=TRUE∧>=(i76[0], 0)=TRUE∧>=(i12[0], i76[0])=TRUE ⇒ LOAD875(i12[0], i35[0], i76[0])≥NonInfC∧LOAD875(i12[0], i35[0], i76[0])≥COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])∧(UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥))



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

    (3)    (i76[0] ≥ 0∧i35[0] + [-1] + [-1]i12[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [(2)bni_13]i35[0] + [(-1)bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    (i76[0] ≥ 0∧i35[0] + [-1] + [-1]i12[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [(2)bni_13]i35[0] + [(-1)bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    (i76[0] ≥ 0∧i35[0] + [-1] + [-1]i12[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [(2)bni_13]i35[0] + [(-1)bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    (i76[0] ≥ 0∧i35[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(4)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [bni_13]i12[0] + [(2)bni_13]i35[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (7)    (i76[0] ≥ 0∧i35[0] ≥ 0∧i76[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(4)bni_13 + (-1)Bound*bni_13] + [bni_13]i12[0] + [(2)bni_13]i35[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_LOAD875(TRUE, i12, i35, i76) → LOAD875(i12, i35, +(i76, 1)) the following chains were created:
  • We consider the chain COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], +(i76[1], 1)) which results in the following constraint:

    (8)    (COND_LOAD875(TRUE, i12[1], i35[1], i76[1])≥NonInfC∧COND_LOAD875(TRUE, i12[1], i35[1], i76[1])≥LOAD875(i12[1], i35[1], +(i76[1], 1))∧(UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥))



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

    (9)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)



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

    (10)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)



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

    (11)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)



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

    (12)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)







For Pair LOAD875(i12, i35, i76) → COND_LOAD8751(&&(&&(&&(>=(i12, 0), <(i12, i76)), <(i12, i35)), >(+(i12, 1), 0)), i12, i35, i76) the following chains were created:
  • We consider the chain LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2]), COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

    (13)    (i35[2]=i35[3]∧&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0))=TRUE∧i76[2]=i76[3]∧i12[2]=i12[3] ⇒ LOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))



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

    (14)    (>(+(i12[2], 1), 0)=TRUE∧<(i12[2], i35[2])=TRUE∧>=(i12[2], 0)=TRUE∧<(i12[2], i76[2])=TRUE ⇒ LOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))



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

    (15)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] + [(-1)bni_17]i12[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (16)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] + [(-1)bni_17]i12[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (17)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] + [(-1)bni_17]i12[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (18)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [bni_17]i12[2] + [(2)bni_17]i35[2] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (19)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_LOAD8751(TRUE, i12, i35, i76) → LOAD875(+(i12, 1), i35, i76) the following chains were created:
  • We consider the chain COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

    (20)    (COND_LOAD8751(TRUE, i12[3], i35[3], i76[3])≥NonInfC∧COND_LOAD8751(TRUE, i12[3], i35[3], i76[3])≥LOAD875(+(i12[3], 1), i35[3], i76[3])∧(UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥))



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

    (21)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)



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

    (22)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)



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

    (23)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)



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

    (24)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD875(i12, i35, i76) → COND_LOAD875(&&(&&(&&(>=(i76, 0), >=(i12, i76)), <(i12, i35)), >(+(i76, 1), 0)), i12, i35, i76)
    • (i76[0] ≥ 0∧i35[0] ≥ 0∧i76[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(4)bni_13 + (-1)Bound*bni_13] + [bni_13]i12[0] + [(2)bni_13]i35[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_LOAD875(TRUE, i12, i35, i76) → LOAD875(i12, i35, +(i76, 1))
    • ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

  • LOAD875(i12, i35, i76) → COND_LOAD8751(&&(&&(&&(>=(i12, 0), <(i12, i76)), <(i12, i35)), >(+(i12, 1), 0)), i12, i35, i76)
    • (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_LOAD8751(TRUE, i12, i35, i76) → LOAD875(+(i12, 1), i35, i76)
    • ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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(LOAD875(x1, x2, x3)) = [2] + [-1]x3 + [2]x2 + [-1]x1   
POL(COND_LOAD875(x1, x2, x3, x4)) = [2] + [-1]x4 + [2]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(COND_LOAD8751(x1, x2, x3, x4)) = [2] + [-1]x4 + [2]x3 + [-1]x2   

The following pairs are in P>:

COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], +(i76[1], 1))
COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3])

The following pairs are in Pbound:

LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])

The following pairs are in P≥:

LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])
LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])

There are no usable rules.

(10) Complex Obligation (AND)

(11) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0, i12[0], i35[0], i76[0])
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])


The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(13) TRUE

(14) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], i76[1] + 1)
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])
(3): COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(i12[3] + 1, i35[3], i76[3])

(1) -> (2), if ((i12[1] →* i12[2])∧(i35[1] →* i35[2])∧(i76[1] + 1 →* i76[2]))


(3) -> (2), if ((i35[3] →* i35[2])∧(i76[3] →* i76[2])∧(i12[3] + 1 →* i12[2]))


(2) -> (3), if ((i35[2] →* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0 →* TRUE)∧(i76[2] →* i76[3])∧(i12[2] →* i12[3]))



The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(i12[3] + 1, i35[3], i76[3])
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])

(3) -> (2), if ((i35[3] →* i35[2])∧(i76[3] →* i76[2])∧(i12[3] + 1 →* i12[2]))


(2) -> (3), if ((i35[2] →* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0 →* TRUE)∧(i76[2] →* i76[3])∧(i12[2] →* i12[3]))



The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(17) 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_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) the following chains were created:
  • We consider the chain COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

    (1)    (COND_LOAD8751(TRUE, i12[3], i35[3], i76[3])≥NonInfC∧COND_LOAD8751(TRUE, i12[3], i35[3], i76[3])≥LOAD875(+(i12[3], 1), i35[3], i76[3])∧(UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥))



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

    (2)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[(-1)bso_13] ≥ 0)



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

    (3)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[(-1)bso_13] ≥ 0)



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

    (4)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[(-1)bso_13] ≥ 0)



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

    (5)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)







For Pair LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2]) the following chains were created:
  • We consider the chain LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2]), COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

    (6)    (i35[2]=i35[3]∧&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0))=TRUE∧i76[2]=i76[3]∧i12[2]=i12[3] ⇒ LOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))



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

    (7)    (>(+(i12[2], 1), 0)=TRUE∧<(i12[2], i35[2])=TRUE∧>=(i12[2], 0)=TRUE∧<(i12[2], i76[2])=TRUE ⇒ LOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))



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

    (8)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (9)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (10)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (11)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (12)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14 + (3)bni_14] + [(2)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3])
    • ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

  • LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])
    • (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14 + (3)bni_14] + [(2)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 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_LOAD8751(x1, x2, x3, x4)) = [-1] + x4 + [2]x3 + [-1]x2   
POL(LOAD875(x1, x2, x3)) = [-1]x1 + x3 + [2]x2   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   

The following pairs are in P>:

LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])

The following pairs are in Pbound:

LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])

The following pairs are in P≥:

COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3])

There are no usable rules.

(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_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(i12[3] + 1, i35[3], i76[3])


The set Q consists of the following terms:
Load875(x0, x1, x2)
Cond_Load875(TRUE, x0, x1, x2)
Cond_Load8751(TRUE, x0, x1, x2)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE