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: PastaB12

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 173 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:
Load344(0, i50) → Cond_Load344(i50 > 0, 0, i50)
Cond_Load344(TRUE, 0, i50) → Load344(0, i50 + -1)
Load344(i43, i33) → Cond_Load3441(i43 > 0, i43, i33)
Cond_Load3441(TRUE, i43, i33) → Load344(i43 + -1, i33)
The set Q consists of the following terms:
Cond_Load344(TRUE, 0, x0)
Load344(x0, x1)
Cond_Load3441(TRUE, x0, x1)

(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_Load344(x1, x2, x3) → Cond_Load344(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:
Load344(0, i50) → Cond_Load344(i50 > 0, i50)
Cond_Load344(TRUE, i50) → Load344(0, i50 + -1)
Load344(i43, i33) → Cond_Load3441(i43 > 0, i43, i33)
Cond_Load3441(TRUE, i43, i33) → Load344(i43 + -1, i33)
The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
Load344(0, i50) → Cond_Load344(i50 > 0, i50)
Cond_Load344(TRUE, i50) → Load344(0, i50 + -1)
Load344(i43, i33) → Cond_Load3441(i43 > 0, i43, i33)
Cond_Load3441(TRUE, i43, i33) → Load344(i43 + -1, i33)

The integer pair graph contains the following rules and edges:
(0): LOAD344(0, i50[0]) → COND_LOAD344(i50[0] > 0, i50[0])
(1): COND_LOAD344(TRUE, i50[1]) → LOAD344(0, i50[1] + -1)
(2): LOAD344(i43[2], i33[2]) → COND_LOAD3441(i43[2] > 0, i43[2], i33[2])
(3): COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(i43[3] + -1, i33[3])

(0) -> (1), if ((i50[0] →* i50[1])∧(i50[0] > 0 →* TRUE))


(1) -> (0), if (i50[1] + -1 →* i50[0])


(1) -> (2), if ((i50[1] + -1 →* i33[2])∧(0 →* i43[2]))


(2) -> (3), if ((i43[2] > 0 →* TRUE)∧(i33[2] →* i33[3])∧(i43[2] →* i43[3]))


(3) -> (0), if ((i33[3] →* i50[0])∧(i43[3] + -1 →* 0))


(3) -> (2), if ((i33[3] →* i33[2])∧(i43[3] + -1 →* i43[2]))



The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(TRUE, x0, x1)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD344(0, i50[0]) → COND_LOAD344(i50[0] > 0, i50[0])
(1): COND_LOAD344(TRUE, i50[1]) → LOAD344(0, i50[1] + -1)
(2): LOAD344(i43[2], i33[2]) → COND_LOAD3441(i43[2] > 0, i43[2], i33[2])
(3): COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(i43[3] + -1, i33[3])

(0) -> (1), if ((i50[0] →* i50[1])∧(i50[0] > 0 →* TRUE))


(1) -> (0), if (i50[1] + -1 →* i50[0])


(1) -> (2), if ((i50[1] + -1 →* i33[2])∧(0 →* i43[2]))


(2) -> (3), if ((i43[2] > 0 →* TRUE)∧(i33[2] →* i33[3])∧(i43[2] →* i43[3]))


(3) -> (0), if ((i33[3] →* i50[0])∧(i43[3] + -1 →* 0))


(3) -> (2), if ((i33[3] →* i33[2])∧(i43[3] + -1 →* i43[2]))



The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(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 LOAD344(0, i50) → COND_LOAD344(>(i50, 0), i50) the following chains were created:
  • We consider the chain LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0]), COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) which results in the following constraint:

    (1)    (i50[0]=i50[1]∧>(i50[0], 0)=TRUE ⇒ LOAD344(0, i50[0])≥NonInfC∧LOAD344(0, i50[0])≥COND_LOAD344(>(i50[0], 0), i50[0])∧(UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥))



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

    (2)    (>(i50[0], 0)=TRUE ⇒ LOAD344(0, i50[0])≥NonInfC∧LOAD344(0, i50[0])≥COND_LOAD344(>(i50[0], 0), i50[0])∧(UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥))



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

    (3)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (4)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (5)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (6)    (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)







For Pair COND_LOAD344(TRUE, i50) → LOAD344(0, +(i50, -1)) the following chains were created:
  • We consider the chain COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) which results in the following constraint:

    (7)    (COND_LOAD344(TRUE, i50[1])≥NonInfC∧COND_LOAD344(TRUE, i50[1])≥LOAD344(0, +(i50[1], -1))∧(UIncreasing(LOAD344(0, +(i50[1], -1))), ≥))



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

    (8)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)



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

    (9)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[(-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)







For Pair LOAD344(i43, i33) → COND_LOAD3441(>(i43, 0), i43, i33) the following chains were created:
  • We consider the chain LOAD344(i43[2], i33[2]) → COND_LOAD3441(>(i43[2], 0), i43[2], i33[2]), COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(+(i43[3], -1), i33[3]) which results in the following constraint:

    (12)    (>(i43[2], 0)=TRUE∧i33[2]=i33[3]∧i43[2]=i43[3] ⇒ LOAD344(i43[2], i33[2])≥NonInfC∧LOAD344(i43[2], i33[2])≥COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])∧(UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥))



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

    (13)    (>(i43[2], 0)=TRUE ⇒ LOAD344(i43[2], i33[2])≥NonInfC∧LOAD344(i43[2], i33[2])≥COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])∧(UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥))



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

    (14)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (15)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (16)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (17)    (i43[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)



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

    (18)    (i43[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)







For Pair COND_LOAD3441(TRUE, i43, i33) → LOAD344(+(i43, -1), i33) the following chains were created:
  • We consider the chain COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(+(i43[3], -1), i33[3]) which results in the following constraint:

    (19)    (COND_LOAD3441(TRUE, i43[3], i33[3])≥NonInfC∧COND_LOAD3441(TRUE, i43[3], i33[3])≥LOAD344(+(i43[3], -1), i33[3])∧(UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥))



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

    (20)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧[1 + (-1)bso_17] ≥ 0)



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

    (21)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧[1 + (-1)bso_17] ≥ 0)



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

    (22)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧[1 + (-1)bso_17] ≥ 0)



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

    (23)    ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD344(0, i50) → COND_LOAD344(>(i50, 0), i50)
    • (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

  • COND_LOAD344(TRUE, i50) → LOAD344(0, +(i50, -1))
    • ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

  • LOAD344(i43, i33) → COND_LOAD3441(>(i43, 0), i43, i33)
    • (i43[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i43[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

  • COND_LOAD3441(TRUE, i43, i33) → LOAD344(+(i43, -1), i33)
    • ((UIncreasing(LOAD344(+(i43[3], -1), i33[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 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(LOAD344(x1, x2)) = [-1] + x1   
POL(0) = 0   
POL(COND_LOAD344(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_LOAD3441(x1, x2, x3)) = [-1] + x2   

The following pairs are in P>:

COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(+(i43[3], -1), i33[3])

The following pairs are in Pbound:

LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0])
LOAD344(i43[2], i33[2]) → COND_LOAD3441(>(i43[2], 0), i43[2], i33[2])

The following pairs are in P≥:

LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0])
COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1))
LOAD344(i43[2], i33[2]) → COND_LOAD3441(>(i43[2], 0), i43[2], i33[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


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD344(0, i50[0]) → COND_LOAD344(i50[0] > 0, i50[0])
(1): COND_LOAD344(TRUE, i50[1]) → LOAD344(0, i50[1] + -1)
(2): LOAD344(i43[2], i33[2]) → COND_LOAD3441(i43[2] > 0, i43[2], i33[2])

(1) -> (0), if (i50[1] + -1 →* i50[0])


(0) -> (1), if ((i50[0] →* i50[1])∧(i50[0] > 0 →* TRUE))


(1) -> (2), if ((i50[1] + -1 →* i33[2])∧(0 →* i43[2]))



The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(TRUE, x0, x1)

(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


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD344(TRUE, i50[1]) → LOAD344(0, i50[1] + -1)
(0): LOAD344(0, i50[0]) → COND_LOAD344(i50[0] > 0, i50[0])

(1) -> (0), if (i50[1] + -1 →* i50[0])


(0) -> (1), if ((i50[0] →* i50[1])∧(i50[0] > 0 →* TRUE))



The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(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_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) the following chains were created:
  • We consider the chain COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) which results in the following constraint:

    (1)    (COND_LOAD344(TRUE, i50[1])≥NonInfC∧COND_LOAD344(TRUE, i50[1])≥LOAD344(0, +(i50[1], -1))∧(UIncreasing(LOAD344(0, +(i50[1], -1))), ≥))



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

    (2)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (3)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (4)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (5)    ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)







For Pair LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0]) the following chains were created:
  • We consider the chain LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0]), COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1)) which results in the following constraint:

    (6)    (i50[0]=i50[1]∧>(i50[0], 0)=TRUE ⇒ LOAD344(0, i50[0])≥NonInfC∧LOAD344(0, i50[0])≥COND_LOAD344(>(i50[0], 0), i50[0])∧(UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥))



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

    (7)    (>(i50[0], 0)=TRUE ⇒ LOAD344(0, i50[0])≥NonInfC∧LOAD344(0, i50[0])≥COND_LOAD344(>(i50[0], 0), i50[0])∧(UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥))



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

    (8)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (9)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (10)    (i50[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (11)    (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1))
    • ((UIncreasing(LOAD344(0, +(i50[1], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

  • LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0])
    • (i50[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD344(>(i50[0], 0), i50[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i50[0] ≥ 0∧[(-1)bso_9] ≥ 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_LOAD344(x1, x2)) = [1] + x2   
POL(LOAD344(x1, x2)) = [1] + x2   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD344(TRUE, i50[1]) → LOAD344(0, +(i50[1], -1))

The following pairs are in Pbound:

LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0])

The following pairs are in P≥:

LOAD344(0, i50[0]) → COND_LOAD344(>(i50[0], 0), i50[0])

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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD344(0, i50[0]) → COND_LOAD344(i50[0] > 0, i50[0])


The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(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:
(1): COND_LOAD344(TRUE, i50[1]) → LOAD344(0, i50[1] + -1)


The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(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_LOAD344(TRUE, i50[1]) → LOAD344(0, i50[1] + -1)
(3): COND_LOAD3441(TRUE, i43[3], i33[3]) → LOAD344(i43[3] + -1, i33[3])


The set Q consists of the following terms:
Cond_Load344(TRUE, x0)
Load344(x0, x1)
Cond_Load3441(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE