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: PastaB10

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 174 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:
Load431(0, 0) → Cond_Load431(0 > 0, 0, 0)
Cond_Load431(TRUE, 0, 0) → Load431(0, 0)
Load431(0, i92) → Cond_Load4311(i92 > 0, 0, i92)
Cond_Load4311(TRUE, 0, i92) → Load431(0, i92 + -1)
Load431(i84, i48) → Cond_Load4312(i84 > 0 && i84 + i48 > 0, i84, i48)
Cond_Load4312(TRUE, i84, i48) → Load431(i84 + -1, i48)
The set Q consists of the following terms:
Cond_Load431(TRUE, 0, 0)
Cond_Load4311(TRUE, 0, x0)
Load431(x0, x1)
Cond_Load4312(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_Load4311(x1, x2, x3) → Cond_Load4311(x1, x3)
Cond_Load431(x1, x2, x3) → Cond_Load431(x1)

(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:
Load431(0, 0) → Cond_Load431(0 > 0)
Cond_Load431(TRUE) → Load431(0, 0)
Load431(0, i92) → Cond_Load4311(i92 > 0, i92)
Cond_Load4311(TRUE, i92) → Load431(0, i92 + -1)
Load431(i84, i48) → Cond_Load4312(i84 > 0 && i84 + i48 > 0, i84, i48)
Cond_Load4312(TRUE, i84, i48) → Load431(i84 + -1, i48)
The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(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, Boolean


The ITRS R consists of the following rules:
Load431(0, 0) → Cond_Load431(0 > 0)
Cond_Load431(TRUE) → Load431(0, 0)
Load431(0, i92) → Cond_Load4311(i92 > 0, i92)
Cond_Load4311(TRUE, i92) → Load431(0, i92 + -1)
Load431(i84, i48) → Cond_Load4312(i84 > 0 && i84 + i48 > 0, i84, i48)
Cond_Load4312(TRUE, i84, i48) → Load431(i84 + -1, i48)

The integer pair graph contains the following rules and edges:
(0): LOAD431(0, 0) → COND_LOAD431(0 > 0)
(1): COND_LOAD431(TRUE) → LOAD431(0, 0)
(2): LOAD431(0, i92[2]) → COND_LOAD4311(i92[2] > 0, i92[2])
(3): COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, i92[3] + -1)
(4): LOAD431(i84[4], i48[4]) → COND_LOAD4312(i84[4] > 0 && i84[4] + i48[4] > 0, i84[4], i48[4])
(5): COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(i84[5] + -1, i48[5])

(0) -> (1), if ((0 > 0 →* TRUE))


(1) -> (0), if true


(1) -> (2), if (0 →* i92[2])


(1) -> (4), if ((0 →* i84[4])∧(0 →* i48[4]))


(2) -> (3), if ((i92[2] →* i92[3])∧(i92[2] > 0 →* TRUE))


(3) -> (0), if (i92[3] + -1 →* 0)


(3) -> (2), if (i92[3] + -1 →* i92[2])


(3) -> (4), if ((i92[3] + -1 →* i48[4])∧(0 →* i84[4]))


(4) -> (5), if ((i84[4] →* i84[5])∧(i84[4] > 0 && i84[4] + i48[4] > 0 →* TRUE)∧(i48[4] →* i48[5]))


(5) -> (0), if ((i48[5] →* 0)∧(i84[5] + -1 →* 0))


(5) -> (2), if ((i84[5] + -1 →* 0)∧(i48[5] →* i92[2]))


(5) -> (4), if ((i48[5] →* i48[4])∧(i84[5] + -1 →* i84[4]))



The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD431(0, 0) → COND_LOAD431(0 > 0)
(1): COND_LOAD431(TRUE) → LOAD431(0, 0)
(2): LOAD431(0, i92[2]) → COND_LOAD4311(i92[2] > 0, i92[2])
(3): COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, i92[3] + -1)
(4): LOAD431(i84[4], i48[4]) → COND_LOAD4312(i84[4] > 0 && i84[4] + i48[4] > 0, i84[4], i48[4])
(5): COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(i84[5] + -1, i48[5])

(0) -> (1), if ((0 > 0 →* TRUE))


(1) -> (0), if true


(1) -> (2), if (0 →* i92[2])


(1) -> (4), if ((0 →* i84[4])∧(0 →* i48[4]))


(2) -> (3), if ((i92[2] →* i92[3])∧(i92[2] > 0 →* TRUE))


(3) -> (0), if (i92[3] + -1 →* 0)


(3) -> (2), if (i92[3] + -1 →* i92[2])


(3) -> (4), if ((i92[3] + -1 →* i48[4])∧(0 →* i84[4]))


(4) -> (5), if ((i84[4] →* i84[5])∧(i84[4] > 0 && i84[4] + i48[4] > 0 →* TRUE)∧(i48[4] →* i48[5]))


(5) -> (0), if ((i48[5] →* 0)∧(i84[5] + -1 →* 0))


(5) -> (2), if ((i84[5] + -1 →* 0)∧(i48[5] →* i92[2]))


(5) -> (4), if ((i48[5] →* i48[4])∧(i84[5] + -1 →* i84[4]))



The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(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 LOAD431(0, 0) → COND_LOAD431(>(0, 0)) the following chains were created:
  • We consider the chain LOAD431(0, 0) → COND_LOAD431(>(0, 0)), COND_LOAD431(TRUE) → LOAD431(0, 0) which results in the following constraint:

    (1)    (>(0, 0)=TRUE ⇒ LOAD431(0, 0)≥NonInfC∧LOAD431(0, 0)≥COND_LOAD431(>(0, 0))∧(UIncreasing(COND_LOAD431(>(0, 0))), ≥))



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




For Pair COND_LOAD431(TRUE) → LOAD431(0, 0) the following chains were created:
  • We consider the chain COND_LOAD431(TRUE) → LOAD431(0, 0) which results in the following constraint:

    (2)    (COND_LOAD431(TRUE)≥NonInfC∧COND_LOAD431(TRUE)≥LOAD431(0, 0)∧(UIncreasing(LOAD431(0, 0)), ≥))



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

    (3)    ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)



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

    (4)    ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)



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

    (5)    ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)







For Pair LOAD431(0, i92) → COND_LOAD4311(>(i92, 0), i92) the following chains were created:
  • We consider the chain LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2]), COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

    (6)    (i92[2]=i92[3]∧>(i92[2], 0)=TRUE ⇒ LOAD431(0, i92[2])≥NonInfC∧LOAD431(0, i92[2])≥COND_LOAD4311(>(i92[2], 0), i92[2])∧(UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥))



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

    (7)    (>(i92[2], 0)=TRUE ⇒ LOAD431(0, i92[2])≥NonInfC∧LOAD431(0, i92[2])≥COND_LOAD4311(>(i92[2], 0), i92[2])∧(UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥))



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

    (8)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (9)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (10)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (11)    (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)







For Pair COND_LOAD4311(TRUE, i92) → LOAD431(0, +(i92, -1)) the following chains were created:
  • We consider the chain COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

    (12)    (COND_LOAD4311(TRUE, i92[3])≥NonInfC∧COND_LOAD4311(TRUE, i92[3])≥LOAD431(0, +(i92[3], -1))∧(UIncreasing(LOAD431(0, +(i92[3], -1))), ≥))



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

    (13)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (14)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (15)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (16)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair LOAD431(i84, i48) → COND_LOAD4312(&&(>(i84, 0), >(+(i84, i48), 0)), i84, i48) the following chains were created:
  • We consider the chain LOAD431(i84[4], i48[4]) → COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4]), COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(+(i84[5], -1), i48[5]) which results in the following constraint:

    (17)    (i84[4]=i84[5]∧&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0))=TRUE∧i48[4]=i48[5] ⇒ LOAD431(i84[4], i48[4])≥NonInfC∧LOAD431(i84[4], i48[4])≥COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])∧(UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥))



    We simplified constraint (17) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (18)    (>(i84[4], 0)=TRUE∧>(+(i84[4], i48[4]), 0)=TRUE ⇒ LOAD431(i84[4], i48[4])≥NonInfC∧LOAD431(i84[4], i48[4])≥COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])∧(UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥))



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

    (19)    (i84[4] + [-1] ≥ 0∧i84[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (20)    (i84[4] + [-1] ≥ 0∧i84[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (21)    (i84[4] + [-1] ≥ 0∧i84[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (22)    (i84[4] ≥ 0∧i84[4] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (23)    (i84[4] ≥ 0∧i84[4] + i48[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)


    (24)    (i84[4] ≥ 0∧i84[4] + [-1]i48[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (25)    (i48[4] + i84[4] ≥ 0∧i84[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i48[4] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_LOAD4312(TRUE, i84, i48) → LOAD431(+(i84, -1), i48) the following chains were created:
  • We consider the chain COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(+(i84[5], -1), i48[5]) which results in the following constraint:

    (26)    (COND_LOAD4312(TRUE, i84[5], i48[5])≥NonInfC∧COND_LOAD4312(TRUE, i84[5], i48[5])≥LOAD431(+(i84[5], -1), i48[5])∧(UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥))



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

    (27)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧[1 + (-1)bso_21] ≥ 0)



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

    (28)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧[1 + (-1)bso_21] ≥ 0)



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

    (29)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧[1 + (-1)bso_21] ≥ 0)



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

    (30)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD431(0, 0) → COND_LOAD431(>(0, 0))

  • COND_LOAD431(TRUE) → LOAD431(0, 0)
    • ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)

  • LOAD431(0, i92) → COND_LOAD4311(>(i92, 0), i92)
    • (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)

  • COND_LOAD4311(TRUE, i92) → LOAD431(0, +(i92, -1))
    • ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[(-1)bso_17] ≥ 0)

  • LOAD431(i84, i48) → COND_LOAD4312(&&(>(i84, 0), >(+(i84, i48), 0)), i84, i48)
    • (i84[4] ≥ 0∧i84[4] + i48[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)
    • (i48[4] + i84[4] ≥ 0∧i84[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i48[4] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_LOAD4312(TRUE, i84, i48) → LOAD431(+(i84, -1), i48)
    • ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 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(LOAD431(x1, x2)) = [-1] + x1   
POL(0) = 0   
POL(COND_LOAD431(x1)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(COND_LOAD4311(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_LOAD4312(x1, x2, x3)) = [-1] + x2   
POL(&&(x1, x2)) = [-1]   

The following pairs are in P>:

LOAD431(0, 0) → COND_LOAD431(>(0, 0))
COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(+(i84[5], -1), i48[5])

The following pairs are in Pbound:

LOAD431(0, 0) → COND_LOAD431(>(0, 0))
LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2])
LOAD431(i84[4], i48[4]) → COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])

The following pairs are in P≥:

COND_LOAD431(TRUE) → LOAD431(0, 0)
LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2])
COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1))
LOAD431(i84[4], i48[4]) → COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])

There are no usable rules.

(12) Complex Obligation (AND)

(13) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD431(TRUE) → LOAD431(0, 0)
(2): LOAD431(0, i92[2]) → COND_LOAD4311(i92[2] > 0, i92[2])
(3): COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, i92[3] + -1)
(4): LOAD431(i84[4], i48[4]) → COND_LOAD4312(i84[4] > 0 && i84[4] + i48[4] > 0, i84[4], i48[4])

(1) -> (2), if (0 →* i92[2])


(3) -> (2), if (i92[3] + -1 →* i92[2])


(2) -> (3), if ((i92[2] →* i92[3])∧(i92[2] > 0 →* TRUE))


(1) -> (4), if ((0 →* i84[4])∧(0 →* i48[4]))


(3) -> (4), if ((i92[3] + -1 →* i48[4])∧(0 →* i84[4]))



The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(TRUE, x0, x1)

(14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(15) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, i92[3] + -1)
(2): LOAD431(0, i92[2]) → COND_LOAD4311(i92[2] > 0, i92[2])

(3) -> (2), if (i92[3] + -1 →* i92[2])


(2) -> (3), if ((i92[2] →* i92[3])∧(i92[2] > 0 →* TRUE))



The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(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_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) the following chains were created:
  • We consider the chain COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

    (1)    (COND_LOAD4311(TRUE, i92[3])≥NonInfC∧COND_LOAD4311(TRUE, i92[3])≥LOAD431(0, +(i92[3], -1))∧(UIncreasing(LOAD431(0, +(i92[3], -1))), ≥))



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

    (2)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (3)    ((UIncreasing(LOAD431(0, +(i92[3], -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(LOAD431(0, +(i92[3], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (5)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)







For Pair LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2]) the following chains were created:
  • We consider the chain LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2]), COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

    (6)    (i92[2]=i92[3]∧>(i92[2], 0)=TRUE ⇒ LOAD431(0, i92[2])≥NonInfC∧LOAD431(0, i92[2])≥COND_LOAD4311(>(i92[2], 0), i92[2])∧(UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥))



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

    (7)    (>(i92[2], 0)=TRUE ⇒ LOAD431(0, i92[2])≥NonInfC∧LOAD431(0, i92[2])≥COND_LOAD4311(>(i92[2], 0), i92[2])∧(UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥))



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

    (8)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (9)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (10)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (11)    (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1))
    • ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

  • LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2])
    • (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 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_LOAD4311(x1, x2)) = [1] + x2   
POL(LOAD431(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_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1))

The following pairs are in Pbound:

LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2])

The following pairs are in P≥:

LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2])

There are no usable rules.

(17) Complex Obligation (AND)

(18) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD431(0, i92[2]) → COND_LOAD4311(i92[2] > 0, i92[2])


The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(TRUE, x0, x1)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, i92[3] + -1)


The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(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_LOAD431(TRUE) → LOAD431(0, 0)
(3): COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, i92[3] + -1)
(5): COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(i84[5] + -1, i48[5])


The set Q consists of the following terms:
Cond_Load431(TRUE)
Cond_Load4311(TRUE, x0)
Load431(x0, x1)
Cond_Load4312(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE