Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 GroundTermsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 IDP
        ↳16 IDPNonInfProof (⇐)
        ↳17 IDP
        ↳18 IDependencyGraphProof (⇔)
        ↳19 TRUE
    ↳20 IDP
        ↳21 IDependencyGraphProof (⇔)
        ↳22 IDP
        ↳23 IDPNonInfProof (⇐)
        ↳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: PastaB16

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 167 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:
Load492(i69, i80) → Cond_Load492(i80 > 0, i69, i80)
Cond_Load492(TRUE, i69, i80) → Load492(i69, i80 + -1)
Load430(i69, i80) → Cond_Load430(i80 > 0 && i69 > 0, i69, i80)
Cond_Load430(TRUE, i69, i80) → Load492(i69, i80 + -1)
Load492(i69, 0) → Load430(i69 + -1, 0)
Load430(i69, 0) → Cond_Load4301(i69 > 0, i69, 0)
Cond_Load4301(TRUE, i69, 0) → Load430(i69 + -1, 0)
The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0, 0)

(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_Load4301(x1, x2, x3) → Cond_Load4301(x1, x2)

(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:
Load492(i69, i80) → Cond_Load492(i80 > 0, i69, i80)
Cond_Load492(TRUE, i69, i80) → Load492(i69, i80 + -1)
Load430(i69, i80) → Cond_Load430(i80 > 0 && i69 > 0, i69, i80)
Cond_Load430(TRUE, i69, i80) → Load492(i69, i80 + -1)
Load492(i69, 0) → Load430(i69 + -1, 0)
Load430(i69, 0) → Cond_Load4301(i69 > 0, i69)
Cond_Load4301(TRUE, i69) → Load430(i69 + -1, 0)
The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(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:
Load492(i69, i80) → Cond_Load492(i80 > 0, i69, i80)
Cond_Load492(TRUE, i69, i80) → Load492(i69, i80 + -1)
Load430(i69, i80) → Cond_Load430(i80 > 0 && i69 > 0, i69, i80)
Cond_Load430(TRUE, i69, i80) → Load492(i69, i80 + -1)
Load492(i69, 0) → Load430(i69 + -1, 0)
Load430(i69, 0) → Cond_Load4301(i69 > 0, i69)
Cond_Load4301(TRUE, i69) → Load430(i69 + -1, 0)

The integer pair graph contains the following rules and edges:
(0): LOAD492(i69[0], i80[0]) → COND_LOAD492(i80[0] > 0, i69[0], i80[0])
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)
(2): LOAD430(i69[2], i80[2]) → COND_LOAD430(i80[2] > 0 && i69[2] > 0, i69[2], i80[2])
(3): COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], i80[3] + -1)
(4): LOAD492(i69[4], 0) → LOAD430(i69[4] + -1, 0)
(5): LOAD430(i69[5], 0) → COND_LOAD4301(i69[5] > 0, i69[5])
(6): COND_LOAD4301(TRUE, i69[6]) → LOAD430(i69[6] + -1, 0)

(0) -> (1), if ((i80[0] →* i80[1])∧(i69[0] →* i69[1])∧(i80[0] > 0 →* TRUE))


(1) -> (0), if ((i69[1] →* i69[0])∧(i80[1] + -1 →* i80[0]))


(1) -> (4), if ((i80[1] + -1 →* 0)∧(i69[1] →* i69[4]))


(2) -> (3), if ((i69[2] →* i69[3])∧(i80[2] > 0 && i69[2] > 0 →* TRUE)∧(i80[2] →* i80[3]))


(3) -> (0), if ((i69[3] →* i69[0])∧(i80[3] + -1 →* i80[0]))


(3) -> (4), if ((i80[3] + -1 →* 0)∧(i69[3] →* i69[4]))


(4) -> (2), if ((0 →* i80[2])∧(i69[4] + -1 →* i69[2]))


(4) -> (5), if (i69[4] + -1 →* i69[5])


(5) -> (6), if ((i69[5] →* i69[6])∧(i69[5] > 0 →* TRUE))


(6) -> (2), if ((i69[6] + -1 →* i69[2])∧(0 →* i80[2]))


(6) -> (5), if (i69[6] + -1 →* i69[5])



The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(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): LOAD492(i69[0], i80[0]) → COND_LOAD492(i80[0] > 0, i69[0], i80[0])
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)
(2): LOAD430(i69[2], i80[2]) → COND_LOAD430(i80[2] > 0 && i69[2] > 0, i69[2], i80[2])
(3): COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], i80[3] + -1)
(4): LOAD492(i69[4], 0) → LOAD430(i69[4] + -1, 0)
(5): LOAD430(i69[5], 0) → COND_LOAD4301(i69[5] > 0, i69[5])
(6): COND_LOAD4301(TRUE, i69[6]) → LOAD430(i69[6] + -1, 0)

(0) -> (1), if ((i80[0] →* i80[1])∧(i69[0] →* i69[1])∧(i80[0] > 0 →* TRUE))


(1) -> (0), if ((i69[1] →* i69[0])∧(i80[1] + -1 →* i80[0]))


(1) -> (4), if ((i80[1] + -1 →* 0)∧(i69[1] →* i69[4]))


(2) -> (3), if ((i69[2] →* i69[3])∧(i80[2] > 0 && i69[2] > 0 →* TRUE)∧(i80[2] →* i80[3]))


(3) -> (0), if ((i69[3] →* i69[0])∧(i80[3] + -1 →* i80[0]))


(3) -> (4), if ((i80[3] + -1 →* 0)∧(i69[3] →* i69[4]))


(4) -> (2), if ((0 →* i80[2])∧(i69[4] + -1 →* i69[2]))


(4) -> (5), if (i69[4] + -1 →* i69[5])


(5) -> (6), if ((i69[5] →* i69[6])∧(i69[5] > 0 →* TRUE))


(6) -> (2), if ((i69[6] + -1 →* i69[2])∧(0 →* i80[2]))


(6) -> (5), if (i69[6] + -1 →* i69[5])



The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(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 LOAD492(i69, i80) → COND_LOAD492(>(i80, 0), i69, i80) the following chains were created:
  • We consider the chain LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]), COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

    (1)    (i80[0]=i80[1]∧i69[0]=i69[1]∧>(i80[0], 0)=TRUE ⇒ LOAD492(i69[0], i80[0])≥NonInfC∧LOAD492(i69[0], i80[0])≥COND_LOAD492(>(i80[0], 0), i69[0], i80[0])∧(UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥))



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

    (2)    (>(i80[0], 0)=TRUE ⇒ LOAD492(i69[0], i80[0])≥NonInfC∧LOAD492(i69[0], i80[0])≥COND_LOAD492(>(i80[0], 0), i69[0], i80[0])∧(UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥))



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

    (3)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i69[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (4)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i69[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (5)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i69[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (6)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)



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

    (7)    (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)







For Pair COND_LOAD492(TRUE, i69, i80) → LOAD492(i69, +(i80, -1)) the following chains were created:
  • We consider the chain COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

    (8)    (COND_LOAD492(TRUE, i69[1], i80[1])≥NonInfC∧COND_LOAD492(TRUE, i69[1], i80[1])≥LOAD492(i69[1], +(i80[1], -1))∧(UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥))



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

    (9)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (10)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (11)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (12)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair LOAD430(i69, i80) → COND_LOAD430(&&(>(i80, 0), >(i69, 0)), i69, i80) the following chains were created:
  • We consider the chain LOAD430(i69[2], i80[2]) → COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2]), COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], +(i80[3], -1)) which results in the following constraint:

    (13)    (i69[2]=i69[3]∧&&(>(i80[2], 0), >(i69[2], 0))=TRUE∧i80[2]=i80[3] ⇒ LOAD430(i69[2], i80[2])≥NonInfC∧LOAD430(i69[2], i80[2])≥COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])∧(UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥))



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

    (14)    (>(i80[2], 0)=TRUE∧>(i69[2], 0)=TRUE ⇒ LOAD430(i69[2], i80[2])≥NonInfC∧LOAD430(i69[2], i80[2])≥COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])∧(UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥))



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

    (15)    (i80[2] + [-1] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (16)    (i80[2] + [-1] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (17)    (i80[2] + [-1] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (18)    (i80[2] ≥ 0∧i69[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (19)    (i80[2] ≥ 0∧i69[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_LOAD430(TRUE, i69, i80) → LOAD492(i69, +(i80, -1)) the following chains were created:
  • We consider the chain COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], +(i80[3], -1)) which results in the following constraint:

    (20)    (COND_LOAD430(TRUE, i69[3], i80[3])≥NonInfC∧COND_LOAD430(TRUE, i69[3], i80[3])≥LOAD492(i69[3], +(i80[3], -1))∧(UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥))



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

    (21)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧[(-1)bso_21] ≥ 0)



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

    (22)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧[(-1)bso_21] ≥ 0)



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

    (23)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧[(-1)bso_21] ≥ 0)



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

    (24)    ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)







For Pair LOAD492(i69, 0) → LOAD430(+(i69, -1), 0) the following chains were created:
  • We consider the chain LOAD492(i69[4], 0) → LOAD430(+(i69[4], -1), 0) which results in the following constraint:

    (25)    (LOAD492(i69[4], 0)≥NonInfC∧LOAD492(i69[4], 0)≥LOAD430(+(i69[4], -1), 0)∧(UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥))



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

    (26)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧[1 + (-1)bso_23] ≥ 0)



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

    (27)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧[1 + (-1)bso_23] ≥ 0)



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

    (28)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧[1 + (-1)bso_23] ≥ 0)



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

    (29)    ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 0)







For Pair LOAD430(i69, 0) → COND_LOAD4301(>(i69, 0), i69) the following chains were created:
  • We consider the chain LOAD430(i69[5], 0) → COND_LOAD4301(>(i69[5], 0), i69[5]), COND_LOAD4301(TRUE, i69[6]) → LOAD430(+(i69[6], -1), 0) which results in the following constraint:

    (30)    (i69[5]=i69[6]∧>(i69[5], 0)=TRUE ⇒ LOAD430(i69[5], 0)≥NonInfC∧LOAD430(i69[5], 0)≥COND_LOAD4301(>(i69[5], 0), i69[5])∧(UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥))



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

    (31)    (>(i69[5], 0)=TRUE ⇒ LOAD430(i69[5], 0)≥NonInfC∧LOAD430(i69[5], 0)≥COND_LOAD4301(>(i69[5], 0), i69[5])∧(UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥))



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

    (32)    (i69[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (33)    (i69[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (34)    (i69[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (35)    (i69[5] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)







For Pair COND_LOAD4301(TRUE, i69) → LOAD430(+(i69, -1), 0) the following chains were created:
  • We consider the chain COND_LOAD4301(TRUE, i69[6]) → LOAD430(+(i69[6], -1), 0) which results in the following constraint:

    (36)    (COND_LOAD4301(TRUE, i69[6])≥NonInfC∧COND_LOAD4301(TRUE, i69[6])≥LOAD430(+(i69[6], -1), 0)∧(UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥))



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

    (37)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧[1 + (-1)bso_27] ≥ 0)



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

    (38)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧[1 + (-1)bso_27] ≥ 0)



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

    (39)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧[1 + (-1)bso_27] ≥ 0)



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

    (40)    ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_27] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD492(i69, i80) → COND_LOAD492(>(i80, 0), i69, i80)
    • (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

  • COND_LOAD492(TRUE, i69, i80) → LOAD492(i69, +(i80, -1))
    • ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  • LOAD430(i69, i80) → COND_LOAD430(&&(>(i80, 0), >(i69, 0)), i69, i80)
    • (i80[2] ≥ 0∧i69[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i69[2] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_LOAD430(TRUE, i69, i80) → LOAD492(i69, +(i80, -1))
    • ((UIncreasing(LOAD492(i69[3], +(i80[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  • LOAD492(i69, 0) → LOAD430(+(i69, -1), 0)
    • ((UIncreasing(LOAD430(+(i69[4], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_23] ≥ 0)

  • LOAD430(i69, 0) → COND_LOAD4301(>(i69, 0), i69)
    • (i69[5] ≥ 0 ⇒ (UIncreasing(COND_LOAD4301(>(i69[5], 0), i69[5])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i69[5] ≥ 0∧[(-1)bso_25] ≥ 0)

  • COND_LOAD4301(TRUE, i69) → LOAD430(+(i69, -1), 0)
    • ((UIncreasing(LOAD430(+(i69[6], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_27] ≥ 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(LOAD492(x1, x2)) = [-1] + x1   
POL(COND_LOAD492(x1, x2, x3)) = [-1] + x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(LOAD430(x1, x2)) = [-1] + x1   
POL(COND_LOAD430(x1, x2, x3)) = [-1] + x2   
POL(&&(x1, x2)) = [-1]   
POL(COND_LOAD4301(x1, x2)) = [-1] + x2   

The following pairs are in P>:

LOAD492(i69[4], 0) → LOAD430(+(i69[4], -1), 0)
COND_LOAD4301(TRUE, i69[6]) → LOAD430(+(i69[6], -1), 0)

The following pairs are in Pbound:

LOAD430(i69[2], i80[2]) → COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])
LOAD430(i69[5], 0) → COND_LOAD4301(>(i69[5], 0), i69[5])

The following pairs are in P≥:

LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])
COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1))
LOAD430(i69[2], i80[2]) → COND_LOAD430(&&(>(i80[2], 0), >(i69[2], 0)), i69[2], i80[2])
COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], +(i80[3], -1))
LOAD430(i69[5], 0) → COND_LOAD4301(>(i69[5], 0), i69[5])

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:
(0): LOAD492(i69[0], i80[0]) → COND_LOAD492(i80[0] > 0, i69[0], i80[0])
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)
(2): LOAD430(i69[2], i80[2]) → COND_LOAD430(i80[2] > 0 && i69[2] > 0, i69[2], i80[2])
(3): COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], i80[3] + -1)
(5): LOAD430(i69[5], 0) → COND_LOAD4301(i69[5] > 0, i69[5])

(1) -> (0), if ((i69[1] →* i69[0])∧(i80[1] + -1 →* i80[0]))


(3) -> (0), if ((i69[3] →* i69[0])∧(i80[3] + -1 →* i80[0]))


(0) -> (1), if ((i80[0] →* i80[1])∧(i69[0] →* i69[1])∧(i80[0] > 0 →* TRUE))


(2) -> (3), if ((i69[2] →* i69[3])∧(i80[2] > 0 && i69[2] > 0 →* TRUE)∧(i80[2] →* i80[3]))



The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(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


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)
(0): LOAD492(i69[0], i80[0]) → COND_LOAD492(i80[0] > 0, i69[0], i80[0])

(1) -> (0), if ((i69[1] →* i69[0])∧(i80[1] + -1 →* i80[0]))


(0) -> (1), if ((i80[0] →* i80[1])∧(i69[0] →* i69[1])∧(i80[0] > 0 →* TRUE))



The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(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_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) the following chains were created:
  • We consider the chain COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

    (1)    (COND_LOAD492(TRUE, i69[1], i80[1])≥NonInfC∧COND_LOAD492(TRUE, i69[1], i80[1])≥LOAD492(i69[1], +(i80[1], -1))∧(UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥))



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

    (2)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_7] ≥ 0)



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

    (3)    ((UIncreasing(LOAD492(i69[1], +(i80[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(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_7] ≥ 0)



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

    (5)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_7] ≥ 0)







For Pair LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]) the following chains were created:
  • We consider the chain LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]), COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

    (6)    (i80[0]=i80[1]∧i69[0]=i69[1]∧>(i80[0], 0)=TRUE ⇒ LOAD492(i69[0], i80[0])≥NonInfC∧LOAD492(i69[0], i80[0])≥COND_LOAD492(>(i80[0], 0), i69[0], i80[0])∧(UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥))



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

    (7)    (>(i80[0], 0)=TRUE ⇒ LOAD492(i69[0], i80[0])≥NonInfC∧LOAD492(i69[0], i80[0])≥COND_LOAD492(>(i80[0], 0), i69[0], i80[0])∧(UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥))



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

    (8)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)



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

    (9)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)



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

    (10)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)



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

    (11)    (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(4)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-1)bso_9] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1))
    • ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_7] ≥ 0)

  • LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])
    • (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(4)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]i80[0] ≥ 0∧[2 + (-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_LOAD492(x1, x2, x3)) = [2]x3   
POL(LOAD492(x1, x2)) = [2] + [2]x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])

The following pairs are in Pbound:

LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])

The following pairs are in P≥:

COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1))

There are no usable rules.

(17) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)


The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(20) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD492(i69[0], i80[0]) → COND_LOAD492(i80[0] > 0, i69[0], i80[0])
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)
(3): COND_LOAD430(TRUE, i69[3], i80[3]) → LOAD492(i69[3], i80[3] + -1)
(4): LOAD492(i69[4], 0) → LOAD430(i69[4] + -1, 0)
(6): COND_LOAD4301(TRUE, i69[6]) → LOAD430(i69[6] + -1, 0)

(1) -> (0), if ((i69[1] →* i69[0])∧(i80[1] + -1 →* i80[0]))


(3) -> (0), if ((i69[3] →* i69[0])∧(i80[3] + -1 →* i80[0]))


(0) -> (1), if ((i80[0] →* i80[1])∧(i69[0] →* i69[1])∧(i80[0] > 0 →* TRUE))


(1) -> (4), if ((i80[1] + -1 →* 0)∧(i69[1] →* i69[4]))


(3) -> (4), if ((i80[3] + -1 →* 0)∧(i69[3] →* i69[4]))



The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(22) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)
(0): LOAD492(i69[0], i80[0]) → COND_LOAD492(i80[0] > 0, i69[0], i80[0])

(1) -> (0), if ((i69[1] →* i69[0])∧(i80[1] + -1 →* i80[0]))


(0) -> (1), if ((i80[0] →* i80[1])∧(i69[0] →* i69[1])∧(i80[0] > 0 →* TRUE))



The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(23) 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_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) the following chains were created:
  • We consider the chain COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

    (1)    (COND_LOAD492(TRUE, i69[1], i80[1])≥NonInfC∧COND_LOAD492(TRUE, i69[1], i80[1])≥LOAD492(i69[1], +(i80[1], -1))∧(UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥))



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

    (2)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)



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

    (3)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)



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

    (4)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)



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

    (5)    ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_8] ≥ 0)







For Pair LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]) the following chains were created:
  • We consider the chain LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0]), COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1)) which results in the following constraint:

    (6)    (i80[0]=i80[1]∧i69[0]=i69[1]∧>(i80[0], 0)=TRUE ⇒ LOAD492(i69[0], i80[0])≥NonInfC∧LOAD492(i69[0], i80[0])≥COND_LOAD492(>(i80[0], 0), i69[0], i80[0])∧(UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥))



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

    (7)    (>(i80[0], 0)=TRUE ⇒ LOAD492(i69[0], i80[0])≥NonInfC∧LOAD492(i69[0], i80[0])≥COND_LOAD492(>(i80[0], 0), i69[0], i80[0])∧(UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥))



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

    (8)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)



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

    (9)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)



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

    (10)    (i80[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)



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

    (11)    (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1))
    • ((UIncreasing(LOAD492(i69[1], +(i80[1], -1))), ≥)∧0 = 0∧[(-1)bso_8] ≥ 0)

  • LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])
    • (i80[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD492(>(i80[0], 0), i69[0], i80[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i80[0] ≥ 0∧[2 + (-1)bso_10] ≥ 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_LOAD492(x1, x2, x3)) = [-1] + [2]x3   
POL(LOAD492(x1, x2)) = [1] + [2]x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])

The following pairs are in Pbound:

LOAD492(i69[0], i80[0]) → COND_LOAD492(>(i80[0], 0), i69[0], i80[0])

The following pairs are in P≥:

COND_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], +(i80[1], -1))

There are no usable rules.

(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_LOAD492(TRUE, i69[1], i80[1]) → LOAD492(i69[1], i80[1] + -1)


The set Q consists of the following terms:
Load492(x0, x1)
Cond_Load492(TRUE, x0, x1)
Load430(x0, x1)
Cond_Load430(TRUE, x0, x1)
Cond_Load4301(TRUE, x0)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE