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 IDP
        ↳18 IDependencyGraphProof (⇔)
        ↳19 TRUE
    ↳20 IDP
        ↳21 IDPNonInfProof (⇐)
        ↳22 IDP
        ↳23 IDependencyGraphProof (⇔)
        ↳24 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: DivWithoutMinus

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 187 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:
Load645(i66, 0, i82, i78) → Cond_Load645(i82 > 0 && i78 + 1 > 0, i66, 0, i82, i78)
Cond_Load645(TRUE, i66, 0, i82, i78) → Load645(i66, i82, i82, i78 + 1)
Load645(i85, i83, i82, i78) → Cond_Load6451(i83 > 0 && i85 > 0 && i82 > 0, i85, i83, i82, i78)
Cond_Load6451(TRUE, i85, i83, i82, i78) → Load645(i85 + -1, i83 + -1, i82, i78)
The set Q consists of the following terms:
Cond_Load645(TRUE, x0, 0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
  • 0

We removed arguments according to the following replacements:

Cond_Load645(x1, x2, x3, x4, x5) → Cond_Load645(x1, x2, x4, x5)

(6) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load645(i66, 0, i82, i78) → Cond_Load645(i82 > 0 && i78 + 1 > 0, i66, i82, i78)
Cond_Load645(TRUE, i66, i82, i78) → Load645(i66, i82, i82, i78 + 1)
Load645(i85, i83, i82, i78) → Cond_Load6451(i83 > 0 && i85 > 0 && i82 > 0, i85, i83, i82, i78)
Cond_Load6451(TRUE, i85, i83, i82, i78) → Load645(i85 + -1, i83 + -1, i82, i78)
The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load645(i66, 0, i82, i78) → Cond_Load645(i82 > 0 && i78 + 1 > 0, i66, i82, i78)
Cond_Load645(TRUE, i66, i82, i78) → Load645(i66, i82, i82, i78 + 1)
Load645(i85, i83, i82, i78) → Cond_Load6451(i83 > 0 && i85 > 0 && i82 > 0, i85, i83, i82, i78)
Cond_Load6451(TRUE, i85, i83, i82, i78) → Load645(i85 + -1, i83 + -1, i82, i78)

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(2): LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(i83[2] > 0 && i85[2] > 0 && i82[2] > 0, i85[2], i83[2], i82[2], i78[2])
(3): COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(i85[3] + -1, i83[3] + -1, i82[3], i78[3])

(0) -> (1), if ((i66[0] →* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0 →* TRUE)∧(i78[0] →* i78[1])∧(i82[0] →* i82[1]))


(1) -> (0), if ((i78[1] + 1 →* i78[0])∧(i66[1] →* i66[0])∧(i82[1] →* 0)∧(i82[1] →* i82[0]))


(1) -> (2), if ((i66[1] →* i85[2])∧(i82[1] →* i83[2])∧(i82[1] →* i82[2])∧(i78[1] + 1 →* i78[2]))


(2) -> (3), if ((i85[2] →* i85[3])∧(i78[2] →* i78[3])∧(i83[2] > 0 && i85[2] > 0 && i82[2] > 0 →* TRUE)∧(i82[2] →* i82[3])∧(i83[2] →* i83[3]))


(3) -> (0), if ((i83[3] + -1 →* 0)∧(i82[3] →* i82[0])∧(i85[3] + -1 →* i66[0])∧(i78[3] →* i78[0]))


(3) -> (2), if ((i78[3] →* i78[2])∧(i83[3] + -1 →* i83[2])∧(i85[3] + -1 →* i85[2])∧(i82[3] →* i82[2]))



The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(2): LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(i83[2] > 0 && i85[2] > 0 && i82[2] > 0, i85[2], i83[2], i82[2], i78[2])
(3): COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(i85[3] + -1, i83[3] + -1, i82[3], i78[3])

(0) -> (1), if ((i66[0] →* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0 →* TRUE)∧(i78[0] →* i78[1])∧(i82[0] →* i82[1]))


(1) -> (0), if ((i78[1] + 1 →* i78[0])∧(i66[1] →* i66[0])∧(i82[1] →* 0)∧(i82[1] →* i82[0]))


(1) -> (2), if ((i66[1] →* i85[2])∧(i82[1] →* i83[2])∧(i82[1] →* i82[2])∧(i78[1] + 1 →* i78[2]))


(2) -> (3), if ((i85[2] →* i85[3])∧(i78[2] →* i78[3])∧(i83[2] > 0 && i85[2] > 0 && i82[2] > 0 →* TRUE)∧(i82[2] →* i82[3])∧(i83[2] →* i83[3]))


(3) -> (0), if ((i83[3] + -1 →* 0)∧(i82[3] →* i82[0])∧(i85[3] + -1 →* i66[0])∧(i78[3] →* i78[0]))


(3) -> (2), if ((i78[3] →* i78[2])∧(i83[3] + -1 →* i83[2])∧(i85[3] + -1 →* i85[2])∧(i82[3] →* i82[2]))



The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(11) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD645(i66, 0, i82, i78) → COND_LOAD645(&&(>(i82, 0), >(+(i78, 1), 0)), i66, i82, i78) the following chains were created:
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) which results in the following constraint:

    (1)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1] ⇒ LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))



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

    (2)    (>(i82[0], 0)=TRUE∧>(+(i78[0], 1), 0)=TRUE ⇒ LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))



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

    (3)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i66[0] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (4)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i66[0] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (5)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i66[0] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (6)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[bni_21] = 0∧[(-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)



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

    (7)    (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[bni_21] = 0∧[(-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)







For Pair COND_LOAD645(TRUE, i66, i82, i78) → LOAD645(i66, i82, i82, +(i78, 1)) the following chains were created:
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

    (8)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1]∧+(i78[1], 1)=i78[0]1∧i66[1]=i66[0]1∧i82[1]=0∧i82[1]=i82[0]1 ⇒ COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥NonInfC∧COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))∧(UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥))



    We solved constraint (8) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]) which results in the following constraint:

    (9)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1]∧i66[1]=i85[2]∧i82[1]=i83[2]∧i82[1]=i82[2]∧+(i78[1], 1)=i78[2] ⇒ COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥NonInfC∧COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))∧(UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥))



    We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (10)    (>(i82[0], 0)=TRUE∧>(+(i78[0], 1), 0)=TRUE ⇒ COND_LOAD645(TRUE, i66[0], i82[0], i78[0])≥NonInfC∧COND_LOAD645(TRUE, i66[0], i82[0], i78[0])≥LOAD645(i66[0], i82[0], i82[0], +(i78[0], 1))∧(UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥))



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

    (11)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i66[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (12)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i66[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (13)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i66[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (14)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[bni_23] = 0∧[(-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)



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

    (15)    (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[bni_23] = 0∧[(-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)







For Pair LOAD645(i85, i83, i82, i78) → COND_LOAD6451(&&(&&(>(i83, 0), >(i85, 0)), >(i82, 0)), i85, i83, i82, i78) the following chains were created:
  • We consider the chain LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]), COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3]) which results in the following constraint:

    (16)    (i85[2]=i85[3]∧i78[2]=i78[3]∧&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0))=TRUE∧i82[2]=i82[3]∧i83[2]=i83[3] ⇒ LOAD645(i85[2], i83[2], i82[2], i78[2])≥NonInfC∧LOAD645(i85[2], i83[2], i82[2], i78[2])≥COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])∧(UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥))



    We simplified constraint (16) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (17)    (>(i82[2], 0)=TRUE∧>(i83[2], 0)=TRUE∧>(i85[2], 0)=TRUE ⇒ LOAD645(i85[2], i83[2], i82[2], i78[2])≥NonInfC∧LOAD645(i85[2], i83[2], i82[2], i78[2])≥COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])∧(UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥))



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

    (18)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (19)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (20)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (21)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)



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

    (22)    (i82[2] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)



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

    (23)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)



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

    (24)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)







For Pair COND_LOAD6451(TRUE, i85, i83, i82, i78) → LOAD645(+(i85, -1), +(i83, -1), i82, i78) the following chains were created:
  • We consider the chain LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]), COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3]), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

    (25)    (i85[2]=i85[3]∧i78[2]=i78[3]∧&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0))=TRUE∧i82[2]=i82[3]∧i83[2]=i83[3]∧+(i83[3], -1)=0∧i82[3]=i82[0]∧+(i85[3], -1)=i66[0]∧i78[3]=i78[0] ⇒ COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥NonInfC∧COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))



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

    (26)    (+(i83[2], -1)=0∧>(i82[2], 0)=TRUE∧>(i83[2], 0)=TRUE∧>(i85[2], 0)=TRUE ⇒ COND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥NonInfC∧COND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥LOAD645(+(i85[2], -1), +(i83[2], -1), i82[2], i78[2])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))



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

    (27)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (28)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (29)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (30)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



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

    (31)    (i83[2] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



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

    (32)    (i83[2] ≥ 0∧i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



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

    (33)    (i83[2] ≥ 0∧i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



  • We consider the chain LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]), COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3]), LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]) which results in the following constraint:

    (34)    (i85[2]=i85[3]∧i78[2]=i78[3]∧&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0))=TRUE∧i82[2]=i82[3]∧i83[2]=i83[3]∧i78[3]=i78[2]1∧+(i83[3], -1)=i83[2]1∧+(i85[3], -1)=i85[2]1∧i82[3]=i82[2]1 ⇒ COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥NonInfC∧COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))



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

    (35)    (>(i82[2], 0)=TRUE∧>(i83[2], 0)=TRUE∧>(i85[2], 0)=TRUE ⇒ COND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥NonInfC∧COND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥LOAD645(+(i85[2], -1), +(i83[2], -1), i82[2], i78[2])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))



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

    (36)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (37)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (38)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (39)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



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

    (40)    (i82[2] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



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

    (41)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)



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

    (42)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD645(i66, 0, i82, i78) → COND_LOAD645(&&(>(i82, 0), >(+(i78, 1), 0)), i66, i82, i78)
    • (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[bni_21] = 0∧[(-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

  • COND_LOAD645(TRUE, i66, i82, i78) → LOAD645(i66, i82, i82, +(i78, 1))
    • (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[bni_23] = 0∧[(-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

  • LOAD645(i85, i83, i82, i78) → COND_LOAD6451(&&(&&(>(i83, 0), >(i85, 0)), >(i82, 0)), i85, i83, i82, i78)
    • (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

  • COND_LOAD6451(TRUE, i85, i83, i82, i78) → LOAD645(+(i85, -1), +(i83, -1), i82, i78)
    • (i83[2] ≥ 0∧i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)
    • (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 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) = [1]   
POL(FALSE) = [1]   
POL(LOAD645(x1, x2, x3, x4)) = x1   
POL(0) = 0   
POL(COND_LOAD645(x1, x2, x3, x4)) = [1] + x2 + [-1]x1   
POL(&&(x1, x2)) = [1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(COND_LOAD6451(x1, x2, x3, x4, x5)) = [-1] + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])

The following pairs are in Pbound:

LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])
COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])

The following pairs are in P≥:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))
COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])

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

&&(TRUE, TRUE)1 ↔ TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
&&(FALSE, FALSE)1 ↔ FALSE1

(12) 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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(3): COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(i85[3] + -1, i83[3] + -1, i82[3], i78[3])

(1) -> (0), if ((i78[1] + 1 →* i78[0])∧(i66[1] →* i66[0])∧(i82[1] →* 0)∧(i82[1] →* i82[0]))


(3) -> (0), if ((i83[3] + -1 →* 0)∧(i82[3] →* i82[0])∧(i85[3] + -1 →* i66[0])∧(i78[3] →* i78[0]))


(0) -> (1), if ((i66[0] →* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0 →* TRUE)∧(i78[0] →* i78[1])∧(i82[0] →* i82[1]))



The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])

(1) -> (0), if ((i78[1] + 1 →* i78[0])∧(i66[1] →* i66[0])∧(i82[1] →* 0)∧(i82[1] →* i82[0]))


(0) -> (1), if ((i66[0] →* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0 →* TRUE)∧(i78[0] →* i78[1])∧(i82[0] →* i82[1]))



The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(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_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) the following chains were created:
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

    (1)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1]∧+(i78[1], 1)=i78[0]1∧i66[1]=i66[0]1∧i82[1]=0∧i82[1]=i82[0]1 ⇒ COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥NonInfC∧COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))∧(UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥))



    We solved constraint (1) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).




For Pair LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) the following chains were created:
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) which results in the following constraint:

    (2)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1] ⇒ LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))



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

    (3)    (&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE ⇒ LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)



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

    (7)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))

  • LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
    • (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_LOAD645(x1, x2, x3, x4)) = x4 + x3 + x2   
POL(LOAD645(x1, x2, x3, x4)) = [2] + [2]x4 + [2]x3 + [2]x1   
POL(+(x1, x2)) = 0   
POL(1) = 0   
POL(0) = 0   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = 0   

The following pairs are in P>:

COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))
LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])

The following pairs are in Pbound:

COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))
LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])

The following pairs are in P≥:
none

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

&&(TRUE, TRUE)1 → TRUE1
&&(TRUE, FALSE)1 → FALSE1
&&(FALSE, TRUE)1 → FALSE1
&&(FALSE, FALSE)1 → FALSE1

(17) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)

(1) -> (0), if ((i78[1] + 1 →* i78[0])∧(i66[1] →* i66[0])∧(i82[1] →* 0)∧(i82[1] →* i82[0]))


(0) -> (1), if ((i66[0] →* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0 →* TRUE)∧(i78[0] →* i78[1])∧(i82[0] →* i82[1]))



The set Q consists of the following terms:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(21) 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 LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) the following chains were created:
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) which results in the following constraint:

    (1)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1] ⇒ LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))



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

    (2)    (&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE ⇒ LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 0)







For Pair COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) the following chains were created:
  • We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

    (7)    (i66[0]=i66[1]∧&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUE∧i78[0]=i78[1]∧i82[0]=i82[1]∧+(i78[1], 1)=i78[0]1∧i66[1]=i66[0]1∧i82[1]=0∧i82[1]=i82[0]1 ⇒ COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥NonInfC∧COND_LOAD645(TRUE, i66[1], i82[1], i78[1])≥LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))∧(UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥))



    We solved constraint (7) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).




To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
    • (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 0)

  • COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))




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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(LOAD645(x1, x2, x3, x4)) = [2] + [2]x4 + [2]x3 + [2]x1   
POL(0) = 0   
POL(COND_LOAD645(x1, x2, x3, x4)) = x4 + x3 + x2   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = 0   
POL(+(x1, x2)) = 0   
POL(1) = 0   

The following pairs are in P>:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))

The following pairs are in Pbound:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))

The following pairs are in P≥:
none

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

&&(TRUE, TRUE)1 → TRUE1
&&(TRUE, FALSE)1 → FALSE1
&&(FALSE, TRUE)1 → FALSE1
&&(FALSE, FALSE)1 → FALSE1

(22) 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:
Cond_Load645(TRUE, x0, x1, x2)
Load645(x0, x1, x2, x3)
Cond_Load6451(TRUE, x0, x1, x2, x3)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE