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: PastaC1

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 114 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:
Load299(1, i26) → Cond_Load299(i26 >= 0, 1, i26)
Cond_Load299(TRUE, 1, i26) → Load439(1, i26, 1)
Load439(1, i36, i37) → Cond_Load439(i37 > 0 && i36 > i37, 1, i36, i37)
Cond_Load439(TRUE, 1, i36, i37) → Load439(1, i36, 2 * i37)
Load439(1, i36, i37) → Cond_Load4391(i36 >= 0 && i37 > 0 && i36 <= i37, 1, i36, i37)
Cond_Load4391(TRUE, 1, i36, i37) → Load299(1, i36 + -1)
The set Q consists of the following terms:
Load299(1, x0)
Cond_Load299(TRUE, 1, x0)
Load439(1, x0, x1)
Cond_Load439(TRUE, 1, x0, x1)
Cond_Load4391(TRUE, 1, x0, x1)

(5) GroundTermsRemoverProof (EQUIVALENT transformation)

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

We removed arguments according to the following replacements:

Load299(x1, x2) → Load299(x2)
Cond_Load4391(x1, x2, x3, x4) → Cond_Load4391(x1, x3, x4)
Load439(x1, x2, x3) → Load439(x2, x3)
Cond_Load439(x1, x2, x3, x4) → Cond_Load439(x1, x3, x4)
Cond_Load299(x1, x2, x3) → Cond_Load299(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:
Load299(i26) → Cond_Load299(i26 >= 0, i26)
Cond_Load299(TRUE, i26) → Load439(i26, 1)
Load439(i36, i37) → Cond_Load439(i37 > 0 && i36 > i37, i36, i37)
Cond_Load439(TRUE, i36, i37) → Load439(i36, 2 * i37)
Load439(i36, i37) → Cond_Load4391(i36 >= 0 && i37 > 0 && i36 <= i37, i36, i37)
Cond_Load4391(TRUE, i36, i37) → Load299(i36 + -1)
The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Integer, Boolean


The ITRS R consists of the following rules:
Load299(i26) → Cond_Load299(i26 >= 0, i26)
Cond_Load299(TRUE, i26) → Load439(i26, 1)
Load439(i36, i37) → Cond_Load439(i37 > 0 && i36 > i37, i36, i37)
Cond_Load439(TRUE, i36, i37) → Load439(i36, 2 * i37)
Load439(i36, i37) → Cond_Load4391(i36 >= 0 && i37 > 0 && i36 <= i37, i36, i37)
Cond_Load4391(TRUE, i36, i37) → Load299(i36 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD299(i26[0]) → COND_LOAD299(i26[0] >= 0, i26[0])
(1): COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1)
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])
(3): COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], 2 * i37[3])
(4): LOAD439(i36[4], i37[4]) → COND_LOAD4391(i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4], i36[4], i37[4])
(5): COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(i36[5] + -1)

(0) -> (1), if ((i26[0] →* i26[1])∧(i26[0] >= 0 →* TRUE))


(1) -> (2), if ((1 →* i37[2])∧(i26[1] →* i36[2]))


(1) -> (4), if ((1 →* i37[4])∧(i26[1] →* i36[4]))


(2) -> (3), if ((i37[2] →* i37[3])∧(i36[2] →* i36[3])∧(i37[2] > 0 && i36[2] > i37[2] →* TRUE))


(3) -> (2), if ((2 * i37[3] →* i37[2])∧(i36[3] →* i36[2]))


(3) -> (4), if ((2 * i37[3] →* i37[4])∧(i36[3] →* i36[4]))


(4) -> (5), if ((i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4] →* TRUE)∧(i36[4] →* i36[5])∧(i37[4] →* i37[5]))


(5) -> (0), if ((i36[5] + -1 →* i26[0]))



The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD299(i26[0]) → COND_LOAD299(i26[0] >= 0, i26[0])
(1): COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1)
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])
(3): COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], 2 * i37[3])
(4): LOAD439(i36[4], i37[4]) → COND_LOAD4391(i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4], i36[4], i37[4])
(5): COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(i36[5] + -1)

(0) -> (1), if ((i26[0] →* i26[1])∧(i26[0] >= 0 →* TRUE))


(1) -> (2), if ((1 →* i37[2])∧(i26[1] →* i36[2]))


(1) -> (4), if ((1 →* i37[4])∧(i26[1] →* i36[4]))


(2) -> (3), if ((i37[2] →* i37[3])∧(i36[2] →* i36[3])∧(i37[2] > 0 && i36[2] > i37[2] →* TRUE))


(3) -> (2), if ((2 * i37[3] →* i37[2])∧(i36[3] →* i36[2]))


(3) -> (4), if ((2 * i37[3] →* i37[4])∧(i36[3] →* i36[4]))


(4) -> (5), if ((i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4] →* TRUE)∧(i36[4] →* i36[5])∧(i37[4] →* i37[5]))


(5) -> (0), if ((i36[5] + -1 →* i26[0]))



The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(11) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD299(i26) → COND_LOAD299(>=(i26, 0), i26) the following chains were created:
  • We consider the chain LOAD299(i26[0]) → COND_LOAD299(>=(i26[0], 0), i26[0]), COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1) which results in the following constraint:

    (1)    (i26[0]=i26[1]∧>=(i26[0], 0)=TRUE ⇒ LOAD299(i26[0])≥NonInfC∧LOAD299(i26[0])≥COND_LOAD299(>=(i26[0], 0), i26[0])∧(UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥))



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

    (2)    (>=(i26[0], 0)=TRUE ⇒ LOAD299(i26[0])≥NonInfC∧LOAD299(i26[0])≥COND_LOAD299(>=(i26[0], 0), i26[0])∧(UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥))



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

    (3)    (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (4)    (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (5)    (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)







For Pair COND_LOAD299(TRUE, i26) → LOAD439(i26, 1) the following chains were created:
  • We consider the chain LOAD299(i26[0]) → COND_LOAD299(>=(i26[0], 0), i26[0]), COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1), LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]) which results in the following constraint:

    (6)    (i26[0]=i26[1]∧>=(i26[0], 0)=TRUE∧1=i37[2]∧i26[1]=i36[2] ⇒ COND_LOAD299(TRUE, i26[1])≥NonInfC∧COND_LOAD299(TRUE, i26[1])≥LOAD439(i26[1], 1)∧(UIncreasing(LOAD439(i26[1], 1)), ≥))



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

    (7)    (>=(i26[0], 0)=TRUE ⇒ COND_LOAD299(TRUE, i26[0])≥NonInfC∧COND_LOAD299(TRUE, i26[0])≥LOAD439(i26[0], 1)∧(UIncreasing(LOAD439(i26[1], 1)), ≥))



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

    (8)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (9)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (10)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



  • We consider the chain LOAD299(i26[0]) → COND_LOAD299(>=(i26[0], 0), i26[0]), COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1), LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4]) which results in the following constraint:

    (11)    (i26[0]=i26[1]∧>=(i26[0], 0)=TRUE∧1=i37[4]∧i26[1]=i36[4] ⇒ COND_LOAD299(TRUE, i26[1])≥NonInfC∧COND_LOAD299(TRUE, i26[1])≥LOAD439(i26[1], 1)∧(UIncreasing(LOAD439(i26[1], 1)), ≥))



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

    (12)    (>=(i26[0], 0)=TRUE ⇒ COND_LOAD299(TRUE, i26[0])≥NonInfC∧COND_LOAD299(TRUE, i26[0])≥LOAD439(i26[0], 1)∧(UIncreasing(LOAD439(i26[1], 1)), ≥))



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

    (13)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (14)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (15)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)







For Pair LOAD439(i36, i37) → COND_LOAD439(&&(>(i37, 0), >(i36, i37)), i36, i37) the following chains were created:
  • We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])) which results in the following constraint:

    (16)    (i37[2]=i37[3]∧i36[2]=i36[3]∧&&(>(i37[2], 0), >(i36[2], i37[2]))=TRUE ⇒ LOAD439(i36[2], i37[2])≥NonInfC∧LOAD439(i36[2], i37[2])≥COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])∧(UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥))



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

    (17)    (>(i37[2], 0)=TRUE∧>(i36[2], i37[2])=TRUE ⇒ LOAD439(i36[2], i37[2])≥NonInfC∧LOAD439(i36[2], i37[2])≥COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])∧(UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥))



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

    (18)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (19)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (20)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (21)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (22)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]i37[2] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_LOAD439(TRUE, i36, i37) → LOAD439(i36, *(2, i37)) the following chains were created:
  • We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])), LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]) which results in the following constraint:

    (23)    (i37[2]=i37[3]∧i36[2]=i36[3]∧&&(>(i37[2], 0), >(i36[2], i37[2]))=TRUE∧*(2, i37[3])=i37[2]1∧i36[3]=i36[2]1 ⇒ COND_LOAD439(TRUE, i36[3], i37[3])≥NonInfC∧COND_LOAD439(TRUE, i36[3], i37[3])≥LOAD439(i36[3], *(2, i37[3]))∧(UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥))



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

    (24)    (>(i37[2], 0)=TRUE∧>(i36[2], i37[2])=TRUE ⇒ COND_LOAD439(TRUE, i36[2], i37[2])≥NonInfC∧COND_LOAD439(TRUE, i36[2], i37[2])≥LOAD439(i36[2], *(2, i37[2]))∧(UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥))



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

    (25)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (26)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (27)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (28)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (29)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



  • We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])), LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4]) which results in the following constraint:

    (30)    (i37[2]=i37[3]∧i36[2]=i36[3]∧&&(>(i37[2], 0), >(i36[2], i37[2]))=TRUE∧*(2, i37[3])=i37[4]∧i36[3]=i36[4] ⇒ COND_LOAD439(TRUE, i36[3], i37[3])≥NonInfC∧COND_LOAD439(TRUE, i36[3], i37[3])≥LOAD439(i36[3], *(2, i37[3]))∧(UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥))



    We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (31)    (>(i37[2], 0)=TRUE∧>(i36[2], i37[2])=TRUE ⇒ COND_LOAD439(TRUE, i36[2], i37[2])≥NonInfC∧COND_LOAD439(TRUE, i36[2], i37[2])≥LOAD439(i36[2], *(2, i37[2]))∧(UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥))



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

    (32)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (33)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (34)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (35)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (36)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)







For Pair LOAD439(i36, i37) → COND_LOAD4391(&&(&&(>=(i36, 0), >(i37, 0)), <=(i36, i37)), i36, i37) the following chains were created:
  • We consider the chain LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4]), COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(+(i36[5], -1)) which results in the following constraint:

    (37)    (&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4]))=TRUE∧i36[4]=i36[5]∧i37[4]=i37[5] ⇒ LOAD439(i36[4], i37[4])≥NonInfC∧LOAD439(i36[4], i37[4])≥COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])∧(UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥))



    We simplified constraint (37) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (38)    (<=(i36[4], i37[4])=TRUE∧>=(i36[4], 0)=TRUE∧>(i37[4], 0)=TRUE ⇒ LOAD439(i36[4], i37[4])≥NonInfC∧LOAD439(i36[4], i37[4])≥COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])∧(UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥))



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

    (39)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (40)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (41)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (42)    (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)







For Pair COND_LOAD4391(TRUE, i36, i37) → LOAD299(+(i36, -1)) the following chains were created:
  • We consider the chain LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4]), COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(+(i36[5], -1)), LOAD299(i26[0]) → COND_LOAD299(>=(i26[0], 0), i26[0]) which results in the following constraint:

    (43)    (&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4]))=TRUE∧i36[4]=i36[5]∧i37[4]=i37[5]∧+(i36[5], -1)=i26[0] ⇒ COND_LOAD4391(TRUE, i36[5], i37[5])≥NonInfC∧COND_LOAD4391(TRUE, i36[5], i37[5])≥LOAD299(+(i36[5], -1))∧(UIncreasing(LOAD299(+(i36[5], -1))), ≥))



    We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (44)    (<=(i36[4], i37[4])=TRUE∧>=(i36[4], 0)=TRUE∧>(i37[4], 0)=TRUE ⇒ COND_LOAD4391(TRUE, i36[4], i37[4])≥NonInfC∧COND_LOAD4391(TRUE, i36[4], i37[4])≥LOAD299(+(i36[4], -1))∧(UIncreasing(LOAD299(+(i36[5], -1))), ≥))



    We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (45)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)



    We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (46)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)



    We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (47)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)



    We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (48)    (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD299(i26) → COND_LOAD299(>=(i26, 0), i26)
    • (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)

  • COND_LOAD299(TRUE, i26) → LOAD439(i26, 1)
    • (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)
    • (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

  • LOAD439(i36, i37) → COND_LOAD439(&&(>(i37, 0), >(i36, i37)), i36, i37)
    • (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]i37[2] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_LOAD439(TRUE, i36, i37) → LOAD439(i36, *(2, i37))
    • (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)
    • (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

  • LOAD439(i36, i37) → COND_LOAD4391(&&(&&(>=(i36, 0), >(i37, 0)), <=(i36, i37)), i36, i37)
    • (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)

  • COND_LOAD4391(TRUE, i36, i37) → LOAD299(+(i36, -1))
    • (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 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) = [1]   
POL(LOAD299(x1)) = x1   
POL(COND_LOAD299(x1, x2)) = x2   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(LOAD439(x1, x2)) = [-1] + x1   
POL(1) = [1]   
POL(COND_LOAD439(x1, x2, x3)) = [-1] + x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(*(x1, x2)) = x1·x2   
POL(2) = [2]   
POL(COND_LOAD4391(x1, x2, x3)) = [-1] + x2   
POL(<=(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1)

The following pairs are in Pbound:

LOAD299(i26[0]) → COND_LOAD299(>=(i26[0], 0), i26[0])
COND_LOAD299(TRUE, i26[1]) → LOAD439(i26[1], 1)
LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])
COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3]))
LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])
COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(+(i36[5], -1))

The following pairs are in P≥:

LOAD299(i26[0]) → COND_LOAD299(>=(i26[0], 0), i26[0])
LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])
COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3]))
LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])
COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(+(i36[5], -1))

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

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

(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:
(0): LOAD299(i26[0]) → COND_LOAD299(i26[0] >= 0, i26[0])
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])
(3): COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], 2 * i37[3])
(4): LOAD439(i36[4], i37[4]) → COND_LOAD4391(i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4], i36[4], i37[4])
(5): COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(i36[5] + -1)

(5) -> (0), if ((i36[5] + -1 →* i26[0]))


(3) -> (2), if ((2 * i37[3] →* i37[2])∧(i36[3] →* i36[2]))


(2) -> (3), if ((i37[2] →* i37[3])∧(i36[2] →* i36[3])∧(i37[2] > 0 && i36[2] > i37[2] →* TRUE))


(3) -> (4), if ((2 * i37[3] →* i37[4])∧(i36[3] →* i36[4]))


(4) -> (5), if ((i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4] →* TRUE)∧(i36[4] →* i36[5])∧(i37[4] →* i37[5]))



The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], 2 * i37[3])
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])

(3) -> (2), if ((2 * i37[3] →* i37[2])∧(i36[3] →* i36[2]))


(2) -> (3), if ((i37[2] →* i37[3])∧(i36[2] →* i36[3])∧(i37[2] > 0 && i36[2] > i37[2] →* TRUE))



The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(16) IDPNonInfProof (SOUND transformation)

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


For Pair COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])) the following chains were created:
  • We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])), LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]) which results in the following constraint:

    (1)    (i37[2]=i37[3]∧i36[2]=i36[3]∧&&(>(i37[2], 0), >(i36[2], i37[2]))=TRUE∧*(2, i37[3])=i37[2]1∧i36[3]=i36[2]1 ⇒ COND_LOAD439(TRUE, i36[3], i37[3])≥NonInfC∧COND_LOAD439(TRUE, i36[3], i37[3])≥LOAD439(i36[3], *(2, i37[3]))∧(UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥))



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

    (2)    (>(i37[2], 0)=TRUE∧>(i36[2], i37[2])=TRUE ⇒ COND_LOAD439(TRUE, i36[2], i37[2])≥NonInfC∧COND_LOAD439(TRUE, i36[2], i37[2])≥LOAD439(i36[2], *(2, i37[2]))∧(UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥))



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

    (3)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[(-1)bso_13] + i37[2] ≥ 0)



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

    (4)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[(-1)bso_13] + i37[2] ≥ 0)



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

    (5)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[(-1)bso_13] + i37[2] ≥ 0)



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

    (6)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12 + (-1)bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[1 + (-1)bso_13] + i37[2] ≥ 0)



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

    (7)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]i36[2] ≥ 0∧[1 + (-1)bso_13] + i37[2] ≥ 0)







For Pair LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]) the following chains were created:
  • We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])) which results in the following constraint:

    (8)    (i37[2]=i37[3]∧i36[2]=i36[3]∧&&(>(i37[2], 0), >(i36[2], i37[2]))=TRUE ⇒ LOAD439(i36[2], i37[2])≥NonInfC∧LOAD439(i36[2], i37[2])≥COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])∧(UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥))



    We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(i37[2], 0)=TRUE∧>(i36[2], i37[2])=TRUE ⇒ LOAD439(i36[2], i37[2])≥NonInfC∧LOAD439(i36[2], i37[2])≥COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])∧(UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥))



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

    (10)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (11)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (12)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (13)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14 + (-1)bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (14)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3]))
    • (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]i36[2] ≥ 0∧[1 + (-1)bso_13] + i37[2] ≥ 0)

  • LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])
    • (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [2]   
POL(FALSE) = [2]   
POL(COND_LOAD439(x1, x2, x3)) = [2] + [-1]x3 + x2 + [-1]x1   
POL(LOAD439(x1, x2)) = [-1]x2 + x1   
POL(*(x1, x2)) = x1·x2   
POL(2) = [2]   
POL(&&(x1, x2)) = [2]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3]))

The following pairs are in Pbound:

COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3]))
LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])

The following pairs are in P≥:

LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])

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) Complex Obligation (AND)

(18) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])


The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE

(21) Obligation:

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


The following domains are used:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load299(x0)
Cond_Load299(TRUE, x0)
Load439(x0, x1)
Cond_Load439(TRUE, x0, x1)
Cond_Load4391(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE