Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 ITRStoIDPProof (⇔)
↳6 IDP
↳7 UsableRulesProof (⇔)
↳8 IDP
↳9 IDPNonInfProof (⇐)
↳10 AND
    ↳11 IDP
        ↳12 IDependencyGraphProof (⇔)
        ↳13 IDP
        ↳14 IDPNonInfProof (⇐)
        ↳15 AND
            ↳16 IDP
                ↳17 IDependencyGraphProof (⇔)
                ↳18 TRUE
            ↳19 IDP
                ↳20 IDependencyGraphProof (⇔)
                ↳21 TRUE
    ↳22 IDP
        ↳23 IDependencyGraphProof (⇔)
        ↳24 IDP
        ↳25 IDPNonInfProof (⇐)
        ↳26 AND
            ↳27 IDP
                ↳28 IDependencyGraphProof (⇔)
                ↳29 TRUE
            ↳30 IDP
                ↳31 IDependencyGraphProof (⇔)
                ↳32 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: PastaB17

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 232 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:
Load1129(i128, i129, i88) → Cond_Load1129(i88 >= 0 && i129 > i88, i128, i129, i88)
Cond_Load1129(TRUE, i128, i129, i88) → Load1129(i128, i129 + -1, i88)
Load1115(i128, i129, i88) → Cond_Load1115(i88 >= 0 && i129 > i88 && i128 > i88, i128, i129, i88)
Cond_Load1115(TRUE, i128, i129, i88) → Load1129(i128, i129 + -1, i88)
Load1129(i128, i129, i88) → Cond_Load11291(i129 <= i88, i128, i129, i88)
Cond_Load11291(TRUE, i128, i129, i88) → Load1115(i128 + -1, i129, i88)
Load1115(i128, i129, i88) → Cond_Load11151(i129 <= i88 && i88 >= 0 && i128 > i88, i128, i129, i88)
Cond_Load11151(TRUE, i128, i129, i88) → Load1115(i128 + -1, i129, i88)
The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load1129(i128, i129, i88) → Cond_Load1129(i88 >= 0 && i129 > i88, i128, i129, i88)
Cond_Load1129(TRUE, i128, i129, i88) → Load1129(i128, i129 + -1, i88)
Load1115(i128, i129, i88) → Cond_Load1115(i88 >= 0 && i129 > i88 && i128 > i88, i128, i129, i88)
Cond_Load1115(TRUE, i128, i129, i88) → Load1129(i128, i129 + -1, i88)
Load1129(i128, i129, i88) → Cond_Load11291(i129 <= i88, i128, i129, i88)
Cond_Load11291(TRUE, i128, i129, i88) → Load1115(i128 + -1, i129, i88)
Load1115(i128, i129, i88) → Cond_Load11151(i129 <= i88 && i88 >= 0 && i128 > i88, i128, i129, i88)
Cond_Load11151(TRUE, i128, i129, i88) → Load1115(i128 + -1, i129, i88)

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(1): COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])
(2): LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2], i128[2], i129[2], i88[2])
(3): COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], i129[3] + -1, i88[3])
(4): LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(i129[4] <= i88[4], i128[4], i129[4], i88[4])
(5): COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(i128[5] + -1, i129[5], i88[5])
(6): LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6], i128[6], i129[6], i88[6])
(7): COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(i128[7] + -1, i129[7], i88[7])

(0) -> (1), if ((i88[0] →* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0] →* TRUE)∧(i129[0] →* i129[1])∧(i128[0] →* i128[1]))


(1) -> (0), if ((i128[1] →* i128[0])∧(i129[1] + -1 →* i129[0])∧(i88[1] →* i88[0]))


(1) -> (4), if ((i128[1] →* i128[4])∧(i129[1] + -1 →* i129[4])∧(i88[1] →* i88[4]))


(2) -> (3), if ((i128[2] →* i128[3])∧(i88[2] →* i88[3])∧(i129[2] →* i129[3])∧(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2] →* TRUE))


(3) -> (0), if ((i88[3] →* i88[0])∧(i129[3] + -1 →* i129[0])∧(i128[3] →* i128[0]))


(3) -> (4), if ((i129[3] + -1 →* i129[4])∧(i128[3] →* i128[4])∧(i88[3] →* i88[4]))


(4) -> (5), if ((i88[4] →* i88[5])∧(i129[4] →* i129[5])∧(i129[4] <= i88[4] →* TRUE)∧(i128[4] →* i128[5]))


(5) -> (2), if ((i88[5] →* i88[2])∧(i128[5] + -1 →* i128[2])∧(i129[5] →* i129[2]))


(5) -> (6), if ((i88[5] →* i88[6])∧(i128[5] + -1 →* i128[6])∧(i129[5] →* i129[6]))


(6) -> (7), if ((i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6] →* TRUE)∧(i128[6] →* i128[7])∧(i88[6] →* i88[7])∧(i129[6] →* i129[7]))


(7) -> (2), if ((i128[7] + -1 →* i128[2])∧(i129[7] →* i129[2])∧(i88[7] →* i88[2]))


(7) -> (6), if ((i129[7] →* i129[6])∧(i128[7] + -1 →* i128[6])∧(i88[7] →* i88[6]))



The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(7) 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.

(8) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(1): COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])
(2): LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2], i128[2], i129[2], i88[2])
(3): COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], i129[3] + -1, i88[3])
(4): LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(i129[4] <= i88[4], i128[4], i129[4], i88[4])
(5): COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(i128[5] + -1, i129[5], i88[5])
(6): LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6], i128[6], i129[6], i88[6])
(7): COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(i128[7] + -1, i129[7], i88[7])

(0) -> (1), if ((i88[0] →* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0] →* TRUE)∧(i129[0] →* i129[1])∧(i128[0] →* i128[1]))


(1) -> (0), if ((i128[1] →* i128[0])∧(i129[1] + -1 →* i129[0])∧(i88[1] →* i88[0]))


(1) -> (4), if ((i128[1] →* i128[4])∧(i129[1] + -1 →* i129[4])∧(i88[1] →* i88[4]))


(2) -> (3), if ((i128[2] →* i128[3])∧(i88[2] →* i88[3])∧(i129[2] →* i129[3])∧(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2] →* TRUE))


(3) -> (0), if ((i88[3] →* i88[0])∧(i129[3] + -1 →* i129[0])∧(i128[3] →* i128[0]))


(3) -> (4), if ((i129[3] + -1 →* i129[4])∧(i128[3] →* i128[4])∧(i88[3] →* i88[4]))


(4) -> (5), if ((i88[4] →* i88[5])∧(i129[4] →* i129[5])∧(i129[4] <= i88[4] →* TRUE)∧(i128[4] →* i128[5]))


(5) -> (2), if ((i88[5] →* i88[2])∧(i128[5] + -1 →* i128[2])∧(i129[5] →* i129[2]))


(5) -> (6), if ((i88[5] →* i88[6])∧(i128[5] + -1 →* i128[6])∧(i129[5] →* i129[6]))


(6) -> (7), if ((i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6] →* TRUE)∧(i128[6] →* i128[7])∧(i88[6] →* i88[7])∧(i129[6] →* i129[7]))


(7) -> (2), if ((i128[7] + -1 →* i128[2])∧(i129[7] →* i129[2])∧(i88[7] →* i88[2]))


(7) -> (6), if ((i129[7] →* i129[6])∧(i128[7] + -1 →* i128[6])∧(i88[7] →* i88[6]))



The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(9) 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 LOAD1129(i128, i129, i88) → COND_LOAD1129(&&(>=(i88, 0), >(i129, i88)), i128, i129, i88) the following chains were created:
  • We consider the chain LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]), COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

    (1)    (i88[0]=i88[1]∧&&(>=(i88[0], 0), >(i129[0], i88[0]))=TRUE∧i129[0]=i129[1]∧i128[0]=i128[1] ⇒ LOAD1129(i128[0], i129[0], i88[0])≥NonInfC∧LOAD1129(i128[0], i129[0], i88[0])≥COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])∧(UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥))



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

    (2)    (>=(i88[0], 0)=TRUE∧>(i129[0], i88[0])=TRUE ⇒ LOAD1129(i128[0], i129[0], i88[0])≥NonInfC∧LOAD1129(i128[0], i129[0], i88[0])≥COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])∧(UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥))



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

    (3)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] + [bni_16]i128[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (4)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] + [bni_16]i128[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (5)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] + [bni_16]i128[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (6)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_16] = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)



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

    (7)    (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_16] = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair COND_LOAD1129(TRUE, i128, i129, i88) → LOAD1129(i128, +(i129, -1), i88) the following chains were created:
  • We consider the chain COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

    (8)    (COND_LOAD1129(TRUE, i128[1], i129[1], i88[1])≥NonInfC∧COND_LOAD1129(TRUE, i128[1], i129[1], i88[1])≥LOAD1129(i128[1], +(i129[1], -1), i88[1])∧(UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥))



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

    (9)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[(-1)bso_19] ≥ 0)



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

    (10)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[(-1)bso_19] ≥ 0)



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

    (11)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[(-1)bso_19] ≥ 0)



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

    (12)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)







For Pair LOAD1115(i128, i129, i88) → COND_LOAD1115(&&(&&(>=(i88, 0), >(i129, i88)), >(i128, i88)), i128, i129, i88) the following chains were created:
  • We consider the chain LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2]), COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], +(i129[3], -1), i88[3]) which results in the following constraint:

    (13)    (i128[2]=i128[3]∧i88[2]=i88[3]∧i129[2]=i129[3]∧&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2]))=TRUE ⇒ LOAD1115(i128[2], i129[2], i88[2])≥NonInfC∧LOAD1115(i128[2], i129[2], i88[2])≥COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])∧(UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥))



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

    (14)    (>(i128[2], i88[2])=TRUE∧>=(i88[2], 0)=TRUE∧>(i129[2], i88[2])=TRUE ⇒ LOAD1115(i128[2], i129[2], i88[2])≥NonInfC∧LOAD1115(i128[2], i129[2], i88[2])≥COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])∧(UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥))



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

    (15)    (i128[2] + [-1] + [-1]i88[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i88[2] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (16)    (i128[2] + [-1] + [-1]i88[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i88[2] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (17)    (i128[2] + [-1] + [-1]i88[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i88[2] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (18)    (i128[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (19)    (i128[2] ≥ 0∧i88[2] ≥ 0∧i129[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair COND_LOAD1115(TRUE, i128, i129, i88) → LOAD1129(i128, +(i129, -1), i88) the following chains were created:
  • We consider the chain COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], +(i129[3], -1), i88[3]) which results in the following constraint:

    (20)    (COND_LOAD1115(TRUE, i128[3], i129[3], i88[3])≥NonInfC∧COND_LOAD1115(TRUE, i128[3], i129[3], i88[3])≥LOAD1129(i128[3], +(i129[3], -1), i88[3])∧(UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥))



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

    (21)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧[(-1)bso_23] ≥ 0)



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

    (22)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧[(-1)bso_23] ≥ 0)



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

    (23)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧[(-1)bso_23] ≥ 0)



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

    (24)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)







For Pair LOAD1129(i128, i129, i88) → COND_LOAD11291(<=(i129, i88), i128, i129, i88) the following chains were created:
  • We consider the chain LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4]), COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(+(i128[5], -1), i129[5], i88[5]) which results in the following constraint:

    (25)    (i88[4]=i88[5]∧i129[4]=i129[5]∧<=(i129[4], i88[4])=TRUE∧i128[4]=i128[5] ⇒ LOAD1129(i128[4], i129[4], i88[4])≥NonInfC∧LOAD1129(i128[4], i129[4], i88[4])≥COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])∧(UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥))



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

    (26)    (<=(i129[4], i88[4])=TRUE ⇒ LOAD1129(i128[4], i129[4], i88[4])≥NonInfC∧LOAD1129(i128[4], i129[4], i88[4])≥COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])∧(UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥))



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

    (27)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] + [bni_24]i128[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (28)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] + [bni_24]i128[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (29)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] + [bni_24]i128[4] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (30)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)



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

    (31)    (i88[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)



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

    (32)    (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)


    (33)    (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)







For Pair COND_LOAD11291(TRUE, i128, i129, i88) → LOAD1115(+(i128, -1), i129, i88) the following chains were created:
  • We consider the chain COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(+(i128[5], -1), i129[5], i88[5]) which results in the following constraint:

    (34)    (COND_LOAD11291(TRUE, i128[5], i129[5], i88[5])≥NonInfC∧COND_LOAD11291(TRUE, i128[5], i129[5], i88[5])≥LOAD1115(+(i128[5], -1), i129[5], i88[5])∧(UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥))



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

    (35)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧[1 + (-1)bso_27] ≥ 0)



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

    (36)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧[1 + (-1)bso_27] ≥ 0)



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

    (37)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧[1 + (-1)bso_27] ≥ 0)



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

    (38)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)







For Pair LOAD1115(i128, i129, i88) → COND_LOAD11151(&&(&&(<=(i129, i88), >=(i88, 0)), >(i128, i88)), i128, i129, i88) the following chains were created:
  • We consider the chain LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6]), COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(+(i128[7], -1), i129[7], i88[7]) which results in the following constraint:

    (39)    (&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6]))=TRUE∧i128[6]=i128[7]∧i88[6]=i88[7]∧i129[6]=i129[7] ⇒ LOAD1115(i128[6], i129[6], i88[6])≥NonInfC∧LOAD1115(i128[6], i129[6], i88[6])≥COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])∧(UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥))



    We simplified constraint (39) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (40)    (>(i128[6], i88[6])=TRUE∧<=(i129[6], i88[6])=TRUE∧>=(i88[6], 0)=TRUE ⇒ LOAD1115(i128[6], i129[6], i88[6])≥NonInfC∧LOAD1115(i128[6], i129[6], i88[6])≥COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])∧(UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥))



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

    (41)    (i128[6] + [-1] + [-1]i88[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i88[6] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (42)    (i128[6] + [-1] + [-1]i88[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i88[6] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)



    We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (43)    (i128[6] + [-1] + [-1]i88[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i88[6] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)



    We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (44)    (i128[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (45)    (i128[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)


    (46)    (i128[6] ≥ 0∧i88[6] + i129[6] ≥ 0∧i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)



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

    (47)    (i128[6] ≥ 0∧i88[6] ≥ 0∧i129[6] + i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)







For Pair COND_LOAD11151(TRUE, i128, i129, i88) → LOAD1115(+(i128, -1), i129, i88) the following chains were created:
  • We consider the chain COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(+(i128[7], -1), i129[7], i88[7]) which results in the following constraint:

    (48)    (COND_LOAD11151(TRUE, i128[7], i129[7], i88[7])≥NonInfC∧COND_LOAD11151(TRUE, i128[7], i129[7], i88[7])≥LOAD1115(+(i128[7], -1), i129[7], i88[7])∧(UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥))



    We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (49)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧[1 + (-1)bso_31] ≥ 0)



    We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (50)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧[1 + (-1)bso_31] ≥ 0)



    We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (51)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧[1 + (-1)bso_31] ≥ 0)



    We simplified constraint (51) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (52)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1129(i128, i129, i88) → COND_LOAD1129(&&(>=(i88, 0), >(i129, i88)), i128, i129, i88)
    • (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_16] = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  • COND_LOAD1129(TRUE, i128, i129, i88) → LOAD1129(i128, +(i129, -1), i88)
    • ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

  • LOAD1115(i128, i129, i88) → COND_LOAD1115(&&(&&(>=(i88, 0), >(i129, i88)), >(i128, i88)), i128, i129, i88)
    • (i128[2] ≥ 0∧i88[2] ≥ 0∧i129[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

  • COND_LOAD1115(TRUE, i128, i129, i88) → LOAD1129(i128, +(i129, -1), i88)
    • ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

  • LOAD1129(i128, i129, i88) → COND_LOAD11291(<=(i129, i88), i128, i129, i88)
    • (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)
    • (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)

  • COND_LOAD11291(TRUE, i128, i129, i88) → LOAD1115(+(i128, -1), i129, i88)
    • ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

  • LOAD1115(i128, i129, i88) → COND_LOAD11151(&&(&&(<=(i129, i88), >=(i88, 0)), >(i128, i88)), i128, i129, i88)
    • (i128[6] ≥ 0∧i88[6] + i129[6] ≥ 0∧i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)
    • (i128[6] ≥ 0∧i88[6] ≥ 0∧i129[6] + i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

  • COND_LOAD11151(TRUE, i128, i129, i88) → LOAD1115(+(i128, -1), i129, i88)
    • ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 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(LOAD1129(x1, x2, x3)) = [-1] + [-1]x3 + x1   
POL(COND_LOAD1129(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(LOAD1115(x1, x2, x3)) = [-1] + [-1]x3 + x1   
POL(COND_LOAD1115(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2   
POL(COND_LOAD11291(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2   
POL(<=(x1, x2)) = [-1]   
POL(COND_LOAD11151(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2   

The following pairs are in P>:

COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(+(i128[5], -1), i129[5], i88[5])
COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(+(i128[7], -1), i129[7], i88[7])

The following pairs are in Pbound:

LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])
LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])

The following pairs are in P≥:

LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])
COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1])
LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])
COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], +(i129[3], -1), i88[3])
LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])
LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])

There are no usable rules.

(10) Complex Obligation (AND)

(11) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(1): COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])
(2): LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2], i128[2], i129[2], i88[2])
(3): COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], i129[3] + -1, i88[3])
(4): LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(i129[4] <= i88[4], i128[4], i129[4], i88[4])
(6): LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6], i128[6], i129[6], i88[6])

(1) -> (0), if ((i128[1] →* i128[0])∧(i129[1] + -1 →* i129[0])∧(i88[1] →* i88[0]))


(3) -> (0), if ((i88[3] →* i88[0])∧(i129[3] + -1 →* i129[0])∧(i128[3] →* i128[0]))


(0) -> (1), if ((i88[0] →* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0] →* TRUE)∧(i129[0] →* i129[1])∧(i128[0] →* i128[1]))


(2) -> (3), if ((i128[2] →* i128[3])∧(i88[2] →* i88[3])∧(i129[2] →* i129[3])∧(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2] →* TRUE))


(1) -> (4), if ((i128[1] →* i128[4])∧(i129[1] + -1 →* i129[4])∧(i88[1] →* i88[4]))


(3) -> (4), if ((i129[3] + -1 →* i129[4])∧(i128[3] →* i128[4])∧(i88[3] →* i88[4]))



The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

(1) -> (0), if ((i128[1] →* i128[0])∧(i129[1] + -1 →* i129[0])∧(i88[1] →* i88[0]))


(0) -> (1), if ((i88[0] →* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0] →* TRUE)∧(i129[0] →* i129[1])∧(i128[0] →* i128[1]))



The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(14) 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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) the following chains were created:
  • We consider the chain COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

    (1)    (COND_LOAD1129(TRUE, i128[1], i129[1], i88[1])≥NonInfC∧COND_LOAD1129(TRUE, i128[1], i129[1], i88[1])≥LOAD1129(i128[1], +(i129[1], -1), i88[1])∧(UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥))



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

    (2)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)



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

    (3)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)



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

    (4)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)



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

    (5)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)







For Pair LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]) the following chains were created:
  • We consider the chain LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]), COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

    (6)    (i88[0]=i88[1]∧&&(>=(i88[0], 0), >(i129[0], i88[0]))=TRUE∧i129[0]=i129[1]∧i128[0]=i128[1] ⇒ LOAD1129(i128[0], i129[0], i88[0])≥NonInfC∧LOAD1129(i128[0], i129[0], i88[0])≥COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])∧(UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥))



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

    (7)    (>=(i88[0], 0)=TRUE∧>(i129[0], i88[0])=TRUE ⇒ LOAD1129(i128[0], i129[0], i88[0])≥NonInfC∧LOAD1129(i128[0], i129[0], i88[0])≥COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])∧(UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥))



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

    (8)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i129[0] + [bni_10]i88[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (9)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i129[0] + [bni_10]i88[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (10)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i129[0] + [bni_10]i88[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (11)    (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i88[0] + [bni_10]i129[0] ≥ 0∧[(-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1])
    • ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

  • LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])
    • (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i88[0] + [bni_10]i129[0] ≥ 0∧[(-1)bso_11] ≥ 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_LOAD1129(x1, x2, x3, x4)) = [1] + x4 + x3   
POL(LOAD1129(x1, x2, x3)) = [1] + x2 + x3   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(&&(x1, x2)) = [1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1])

The following pairs are in Pbound:

LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])

The following pairs are in P≥:

LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])

There are no usable rules.

(15) Complex Obligation (AND)

(16) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])


The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE

(19) 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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])


The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(1): COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])
(3): COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], i129[3] + -1, i88[3])
(4): LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(i129[4] <= i88[4], i128[4], i129[4], i88[4])
(5): COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(i128[5] + -1, i129[5], i88[5])
(7): COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(i128[7] + -1, i129[7], i88[7])

(1) -> (0), if ((i128[1] →* i128[0])∧(i129[1] + -1 →* i129[0])∧(i88[1] →* i88[0]))


(3) -> (0), if ((i88[3] →* i88[0])∧(i129[3] + -1 →* i129[0])∧(i128[3] →* i128[0]))


(0) -> (1), if ((i88[0] →* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0] →* TRUE)∧(i129[0] →* i129[1])∧(i128[0] →* i128[1]))


(1) -> (4), if ((i128[1] →* i128[4])∧(i129[1] + -1 →* i129[4])∧(i88[1] →* i88[4]))


(3) -> (4), if ((i129[3] + -1 →* i129[4])∧(i128[3] →* i128[4])∧(i88[3] →* i88[4]))


(4) -> (5), if ((i88[4] →* i88[5])∧(i129[4] →* i129[5])∧(i129[4] <= i88[4] →* TRUE)∧(i128[4] →* i128[5]))



The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

(1) -> (0), if ((i128[1] →* i128[0])∧(i129[1] + -1 →* i129[0])∧(i88[1] →* i88[0]))


(0) -> (1), if ((i88[0] →* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0] →* TRUE)∧(i129[0] →* i129[1])∧(i128[0] →* i128[1]))



The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(25) 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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) the following chains were created:
  • We consider the chain COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

    (1)    (COND_LOAD1129(TRUE, i128[1], i129[1], i88[1])≥NonInfC∧COND_LOAD1129(TRUE, i128[1], i129[1], i88[1])≥LOAD1129(i128[1], +(i129[1], -1), i88[1])∧(UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥))



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

    (2)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)



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

    (3)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)



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

    (4)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)



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

    (5)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)







For Pair LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]) the following chains were created:
  • We consider the chain LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]), COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

    (6)    (i88[0]=i88[1]∧&&(>=(i88[0], 0), >(i129[0], i88[0]))=TRUE∧i129[0]=i129[1]∧i128[0]=i128[1] ⇒ LOAD1129(i128[0], i129[0], i88[0])≥NonInfC∧LOAD1129(i128[0], i129[0], i88[0])≥COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])∧(UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥))



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

    (7)    (>=(i88[0], 0)=TRUE∧>(i129[0], i88[0])=TRUE ⇒ LOAD1129(i128[0], i129[0], i88[0])≥NonInfC∧LOAD1129(i128[0], i129[0], i88[0])≥COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])∧(UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥))



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

    (8)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i129[0] + [bni_12]i88[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (9)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i129[0] + [bni_12]i88[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (10)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i129[0] + [bni_12]i88[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (11)    (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]i88[0] + [bni_12]i129[0] ≥ 0∧[(-1)bso_13] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1])
    • ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

  • LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])
    • (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]i88[0] + [bni_12]i129[0] ≥ 0∧[(-1)bso_13] ≥ 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_LOAD1129(x1, x2, x3, x4)) = [-1] + x4 + x3   
POL(LOAD1129(x1, x2, x3)) = [-1] + x2 + x3   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1])

The following pairs are in Pbound:

LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])

The following pairs are in P≥:

LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])

There are no usable rules.

(26) Complex Obligation (AND)

(27) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])


The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE

(30) 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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], i129[1] + -1, i88[1])


The set Q consists of the following terms:
Load1129(x0, x1, x2)
Cond_Load1129(TRUE, x0, x1, x2)
Load1115(x0, x1, x2)
Cond_Load1115(TRUE, x0, x1, x2)
Cond_Load11291(TRUE, x0, x1, x2)
Cond_Load11151(TRUE, x0, x1, x2)

(31) IDependencyGraphProof (EQUIVALENT transformation)

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

(32) TRUE