Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 DuplicateArgsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 IDP
↳13 IDependencyGraphProof (⇔)
↳14 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: PastaB11

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 180 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:
Load777(i96, i134) → Cond_Load777(i96 < i134 && i96 + i134 > 0, i96, i134)
Cond_Load777(TRUE, i96, i134) → Load777(i96, i134 + -1)
Load777(i134, i134) → Cond_Load7771(i134 + i134 > 0, i134, i134)
Cond_Load7771(TRUE, i134, i134) → Load777(i134 + -1, i134)
Load777(i96, i134) → Cond_Load7772(i96 > i134 && i96 + i134 > 0, i96, i134)
Cond_Load7772(TRUE, i96, i134) → Load777(i96 + -1, i134)
The set Q consists of the following terms:
Load777(x0, x1)
Cond_Load777(TRUE, x0, x1)
Cond_Load7771(TRUE, x0, x0)
Cond_Load7772(TRUE, x0, x1)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Cond_Load7771(x1, x2, x3) → Cond_Load7771(x1, x3)

(6) Obligation:

ITRS problem:

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

The TRS R consists of the following rules:
Load777(i96, i134) → Cond_Load777(i96 < i134 && i96 + i134 > 0, i96, i134)
Cond_Load777(TRUE, i96, i134) → Load777(i96, i134 + -1)
Load777(i134, i134) → Cond_Load7771(i134 + i134 > 0, i134)
Cond_Load7771(TRUE, i134) → Load777(i134 + -1, i134)
Load777(i96, i134) → Cond_Load7772(i96 > i134 && i96 + i134 > 0, i96, i134)
Cond_Load7772(TRUE, i96, i134) → Load777(i96 + -1, i134)
The set Q consists of the following terms:
Load777(x0, x1)
Cond_Load777(TRUE, x0, x1)
Cond_Load7771(TRUE, x0)
Cond_Load7772(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) Obligation:

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


The following domains are used:

Boolean, Integer


The ITRS R consists of the following rules:
Load777(i96, i134) → Cond_Load777(i96 < i134 && i96 + i134 > 0, i96, i134)
Cond_Load777(TRUE, i96, i134) → Load777(i96, i134 + -1)
Load777(i134, i134) → Cond_Load7771(i134 + i134 > 0, i134)
Cond_Load7771(TRUE, i134) → Load777(i134 + -1, i134)
Load777(i96, i134) → Cond_Load7772(i96 > i134 && i96 + i134 > 0, i96, i134)
Cond_Load7772(TRUE, i96, i134) → Load777(i96 + -1, i134)

The integer pair graph contains the following rules and edges:
(0): LOAD777(i96[0], i134[0]) → COND_LOAD777(i96[0] < i134[0] && i96[0] + i134[0] > 0, i96[0], i134[0])
(1): COND_LOAD777(TRUE, i96[1], i134[1]) → LOAD777(i96[1], i134[1] + -1)
(2): LOAD777(i134[2], i134[2]) → COND_LOAD7771(i134[2] + i134[2] > 0, i134[2])
(3): COND_LOAD7771(TRUE, i134[3]) → LOAD777(i134[3] + -1, i134[3])
(4): LOAD777(i96[4], i134[4]) → COND_LOAD7772(i96[4] > i134[4] && i96[4] + i134[4] > 0, i96[4], i134[4])
(5): COND_LOAD7772(TRUE, i96[5], i134[5]) → LOAD777(i96[5] + -1, i134[5])

(0) -> (1), if ((i134[0] →* i134[1])∧(i96[0] < i134[0] && i96[0] + i134[0] > 0 →* TRUE)∧(i96[0] →* i96[1]))


(1) -> (0), if ((i134[1] + -1 →* i134[0])∧(i96[1] →* i96[0]))


(1) -> (2), if ((i134[1] + -1 →* i134[2])∧(i96[1] →* i134[2]))


(1) -> (4), if ((i96[1] →* i96[4])∧(i134[1] + -1 →* i134[4]))


(2) -> (3), if ((i134[2] + i134[2] > 0 →* TRUE)∧(i134[2] →* i134[3]))


(3) -> (0), if ((i134[3] + -1 →* i96[0])∧(i134[3] →* i134[0]))


(3) -> (2), if ((i134[3] + -1 →* i134[2])∧(i134[3] →* i134[2]))


(3) -> (4), if ((i134[3] + -1 →* i96[4])∧(i134[3] →* i134[4]))


(4) -> (5), if ((i96[4] →* i96[5])∧(i96[4] > i134[4] && i96[4] + i134[4] > 0 →* TRUE)∧(i134[4] →* i134[5]))


(5) -> (0), if ((i134[5] →* i134[0])∧(i96[5] + -1 →* i96[0]))


(5) -> (2), if ((i96[5] + -1 →* i134[2])∧(i134[5] →* i134[2]))


(5) -> (4), if ((i134[5] →* i134[4])∧(i96[5] + -1 →* i96[4]))



The set Q consists of the following terms:
Load777(x0, x1)
Cond_Load777(TRUE, x0, x1)
Cond_Load7771(TRUE, x0)
Cond_Load7772(TRUE, x0, x1)

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD777(i96[0], i134[0]) → COND_LOAD777(i96[0] < i134[0] && i96[0] + i134[0] > 0, i96[0], i134[0])
(1): COND_LOAD777(TRUE, i96[1], i134[1]) → LOAD777(i96[1], i134[1] + -1)
(2): LOAD777(i134[2], i134[2]) → COND_LOAD7771(i134[2] + i134[2] > 0, i134[2])
(3): COND_LOAD7771(TRUE, i134[3]) → LOAD777(i134[3] + -1, i134[3])
(4): LOAD777(i96[4], i134[4]) → COND_LOAD7772(i96[4] > i134[4] && i96[4] + i134[4] > 0, i96[4], i134[4])
(5): COND_LOAD7772(TRUE, i96[5], i134[5]) → LOAD777(i96[5] + -1, i134[5])

(0) -> (1), if ((i134[0] →* i134[1])∧(i96[0] < i134[0] && i96[0] + i134[0] > 0 →* TRUE)∧(i96[0] →* i96[1]))


(1) -> (0), if ((i134[1] + -1 →* i134[0])∧(i96[1] →* i96[0]))


(1) -> (2), if ((i134[1] + -1 →* i134[2])∧(i96[1] →* i134[2]))


(1) -> (4), if ((i96[1] →* i96[4])∧(i134[1] + -1 →* i134[4]))


(2) -> (3), if ((i134[2] + i134[2] > 0 →* TRUE)∧(i134[2] →* i134[3]))


(3) -> (0), if ((i134[3] + -1 →* i96[0])∧(i134[3] →* i134[0]))


(3) -> (2), if ((i134[3] + -1 →* i134[2])∧(i134[3] →* i134[2]))


(3) -> (4), if ((i134[3] + -1 →* i96[4])∧(i134[3] →* i134[4]))


(4) -> (5), if ((i96[4] →* i96[5])∧(i96[4] > i134[4] && i96[4] + i134[4] > 0 →* TRUE)∧(i134[4] →* i134[5]))


(5) -> (0), if ((i134[5] →* i134[0])∧(i96[5] + -1 →* i96[0]))


(5) -> (2), if ((i96[5] + -1 →* i134[2])∧(i134[5] →* i134[2]))


(5) -> (4), if ((i134[5] →* i134[4])∧(i96[5] + -1 →* i96[4]))



The set Q consists of the following terms:
Load777(x0, x1)
Cond_Load777(TRUE, x0, x1)
Cond_Load7771(TRUE, x0)
Cond_Load7772(TRUE, x0, x1)

(11) IDPNonInfProof (SOUND transformation)

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


For Pair LOAD777(i96, i134) → COND_LOAD777(&&(<(i96, i134), >(+(i96, i134), 0)), i96, i134) the following chains were created:
  • We consider the chain LOAD777(i96[0], i134[0]) → COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0]), COND_LOAD777(TRUE, i96[1], i134[1]) → LOAD777(i96[1], +(i134[1], -1)) which results in the following constraint:

    (1)    (i134[0]=i134[1]∧&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0))=TRUE∧i96[0]=i96[1] ⇒ LOAD777(i96[0], i134[0])≥NonInfC∧LOAD777(i96[0], i134[0])≥COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])∧(UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥))



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

    (2)    (<(i96[0], i134[0])=TRUE∧>(+(i96[0], i134[0]), 0)=TRUE ⇒ LOAD777(i96[0], i134[0])≥NonInfC∧LOAD777(i96[0], i134[0])≥COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])∧(UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥))



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

    (3)    (i134[0] + [-1] + [-1]i96[0] ≥ 0∧i96[0] + [-1] + i134[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i134[0] + [bni_13]i96[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (4)    (i134[0] + [-1] + [-1]i96[0] ≥ 0∧i96[0] + [-1] + i134[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i134[0] + [bni_13]i96[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (5)    (i134[0] + [-1] + [-1]i96[0] ≥ 0∧i96[0] + [-1] + i134[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i134[0] + [bni_13]i96[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (6)    (i134[0] ≥ 0∧[2]i96[0] + i134[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]i96[0] + [bni_13]i134[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (7)    (i134[0] ≥ 0∧[2]i96[0] + i134[0] ≥ 0∧i96[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]i96[0] + [bni_13]i134[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)


    (8)    (i134[0] ≥ 0∧[-2]i96[0] + i134[0] ≥ 0∧i96[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(-2)bni_13]i96[0] + [bni_13]i134[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (9)    ([2]i96[0] + i134[0] ≥ 0∧i134[0] ≥ 0∧i96[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i134[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)







For Pair COND_LOAD777(TRUE, i96, i134) → LOAD777(i96, +(i134, -1)) the following chains were created:
  • We consider the chain COND_LOAD777(TRUE, i96[1], i134[1]) → LOAD777(i96[1], +(i134[1], -1)) which results in the following constraint:

    (10)    (COND_LOAD777(TRUE, i96[1], i134[1])≥NonInfC∧COND_LOAD777(TRUE, i96[1], i134[1])≥LOAD777(i96[1], +(i134[1], -1))∧(UIncreasing(LOAD777(i96[1], +(i134[1], -1))), ≥))



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

    (11)    ((UIncreasing(LOAD777(i96[1], +(i134[1], -1))), ≥)∧[(-1)bso_16] ≥ 0)



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

    (12)    ((UIncreasing(LOAD777(i96[1], +(i134[1], -1))), ≥)∧[(-1)bso_16] ≥ 0)



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

    (13)    ((UIncreasing(LOAD777(i96[1], +(i134[1], -1))), ≥)∧[(-1)bso_16] ≥ 0)



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

    (14)    ((UIncreasing(LOAD777(i96[1], +(i134[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)







For Pair LOAD777(i134, i134) → COND_LOAD7771(>(+(i134, i134), 0), i134) the following chains were created:
  • We consider the chain LOAD777(i134[2], i134[2]) → COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2]), COND_LOAD7771(TRUE, i134[3]) → LOAD777(+(i134[3], -1), i134[3]) which results in the following constraint:

    (15)    (>(+(i134[2], i134[2]), 0)=TRUE∧i134[2]=i134[3] ⇒ LOAD777(i134[2], i134[2])≥NonInfC∧LOAD777(i134[2], i134[2])≥COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])∧(UIncreasing(COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])), ≥))



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

    (16)    (>(+(i134[2], i134[2]), 0)=TRUE ⇒ LOAD777(i134[2], i134[2])≥NonInfC∧LOAD777(i134[2], i134[2])≥COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])∧(UIncreasing(COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])), ≥))



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

    (17)    ([2]i134[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i134[2] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (18)    ([2]i134[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i134[2] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (19)    ([2]i134[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i134[2] ≥ 0∧[1 + (-1)bso_18] ≥ 0)







For Pair COND_LOAD7771(TRUE, i134) → LOAD777(+(i134, -1), i134) the following chains were created:
  • We consider the chain COND_LOAD7771(TRUE, i134[3]) → LOAD777(+(i134[3], -1), i134[3]) which results in the following constraint:

    (20)    (COND_LOAD7771(TRUE, i134[3])≥NonInfC∧COND_LOAD7771(TRUE, i134[3])≥LOAD777(+(i134[3], -1), i134[3])∧(UIncreasing(LOAD777(+(i134[3], -1), i134[3])), ≥))



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

    (21)    ((UIncreasing(LOAD777(+(i134[3], -1), i134[3])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (22)    ((UIncreasing(LOAD777(+(i134[3], -1), i134[3])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (23)    ((UIncreasing(LOAD777(+(i134[3], -1), i134[3])), ≥)∧[(-1)bso_20] ≥ 0)



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

    (24)    ((UIncreasing(LOAD777(+(i134[3], -1), i134[3])), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)







For Pair LOAD777(i96, i134) → COND_LOAD7772(&&(>(i96, i134), >(+(i96, i134), 0)), i96, i134) the following chains were created:
  • We consider the chain LOAD777(i96[4], i134[4]) → COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4]), COND_LOAD7772(TRUE, i96[5], i134[5]) → LOAD777(+(i96[5], -1), i134[5]) which results in the following constraint:

    (25)    (i96[4]=i96[5]∧&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0))=TRUE∧i134[4]=i134[5] ⇒ LOAD777(i96[4], i134[4])≥NonInfC∧LOAD777(i96[4], i134[4])≥COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])∧(UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥))



    We simplified constraint (25) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (26)    (>(i96[4], i134[4])=TRUE∧>(+(i96[4], i134[4]), 0)=TRUE ⇒ LOAD777(i96[4], i134[4])≥NonInfC∧LOAD777(i96[4], i134[4])≥COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])∧(UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥))



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

    (27)    (i96[4] + [-1] + [-1]i134[4] ≥ 0∧i96[4] + [-1] + i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)



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

    (28)    (i96[4] + [-1] + [-1]i134[4] ≥ 0∧i96[4] + [-1] + i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)



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

    (29)    (i96[4] + [-1] + [-1]i134[4] ≥ 0∧i96[4] + [-1] + i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)



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

    (30)    (i96[4] ≥ 0∧[2]i134[4] + i96[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)



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

    (31)    (i96[4] ≥ 0∧[2]i134[4] + i96[4] ≥ 0∧i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)


    (32)    (i96[4] ≥ 0∧[-2]i134[4] + i96[4] ≥ 0∧i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [(-2)bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)



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

    (33)    ([2]i134[4] + i96[4] ≥ 0∧i96[4] ≥ 0∧i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)







For Pair COND_LOAD7772(TRUE, i96, i134) → LOAD777(+(i96, -1), i134) the following chains were created:
  • We consider the chain COND_LOAD7772(TRUE, i96[5], i134[5]) → LOAD777(+(i96[5], -1), i134[5]) which results in the following constraint:

    (34)    (COND_LOAD7772(TRUE, i96[5], i134[5])≥NonInfC∧COND_LOAD7772(TRUE, i96[5], i134[5])≥LOAD777(+(i96[5], -1), i134[5])∧(UIncreasing(LOAD777(+(i96[5], -1), i134[5])), ≥))



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

    (35)    ((UIncreasing(LOAD777(+(i96[5], -1), i134[5])), ≥)∧[(-1)bso_24] ≥ 0)



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

    (36)    ((UIncreasing(LOAD777(+(i96[5], -1), i134[5])), ≥)∧[(-1)bso_24] ≥ 0)



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

    (37)    ((UIncreasing(LOAD777(+(i96[5], -1), i134[5])), ≥)∧[(-1)bso_24] ≥ 0)



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

    (38)    ((UIncreasing(LOAD777(+(i96[5], -1), i134[5])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_24] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD777(i96, i134) → COND_LOAD777(&&(<(i96, i134), >(+(i96, i134), 0)), i96, i134)
    • (i134[0] ≥ 0∧[2]i96[0] + i134[0] ≥ 0∧i96[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [(2)bni_13]i96[0] + [bni_13]i134[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)
    • ([2]i96[0] + i134[0] ≥ 0∧i134[0] ≥ 0∧i96[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i134[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

  • COND_LOAD777(TRUE, i96, i134) → LOAD777(i96, +(i134, -1))
    • ((UIncreasing(LOAD777(i96[1], +(i134[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  • LOAD777(i134, i134) → COND_LOAD7771(>(+(i134, i134), 0), i134)
    • ([2]i134[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i134[2] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

  • COND_LOAD7771(TRUE, i134) → LOAD777(+(i134, -1), i134)
    • ((UIncreasing(LOAD777(+(i134[3], -1), i134[3])), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)

  • LOAD777(i96, i134) → COND_LOAD7772(&&(>(i96, i134), >(+(i96, i134), 0)), i96, i134)
    • (i96[4] ≥ 0∧[2]i134[4] + i96[4] ≥ 0∧i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]i134[4] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)
    • ([2]i134[4] + i96[4] ≥ 0∧i96[4] ≥ 0∧i134[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])), ≥)∧[(3)bni_21 + (-1)Bound*bni_21] + [bni_21]i96[4] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

  • COND_LOAD7772(TRUE, i96, i134) → LOAD777(+(i96, -1), i134)
    • ((UIncreasing(LOAD777(+(i96[5], -1), i134[5])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_24] ≥ 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(LOAD777(x1, x2)) = [2] + x2 + x1   
POL(COND_LOAD777(x1, x2, x3)) = [1] + x3 + x2   
POL(&&(x1, x2)) = [-1]   
POL(<(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(0) = 0   
POL(-1) = [-1]   
POL(COND_LOAD7771(x1, x2)) = [1] + [2]x2   
POL(COND_LOAD7772(x1, x2, x3)) = [1] + x3 + x2   

The following pairs are in P>:

LOAD777(i96[0], i134[0]) → COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])
LOAD777(i134[2], i134[2]) → COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])
LOAD777(i96[4], i134[4]) → COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])

The following pairs are in Pbound:

LOAD777(i96[0], i134[0]) → COND_LOAD777(&&(<(i96[0], i134[0]), >(+(i96[0], i134[0]), 0)), i96[0], i134[0])
LOAD777(i134[2], i134[2]) → COND_LOAD7771(>(+(i134[2], i134[2]), 0), i134[2])
LOAD777(i96[4], i134[4]) → COND_LOAD7772(&&(>(i96[4], i134[4]), >(+(i96[4], i134[4]), 0)), i96[4], i134[4])

The following pairs are in P≥:

COND_LOAD777(TRUE, i96[1], i134[1]) → LOAD777(i96[1], +(i134[1], -1))
COND_LOAD7771(TRUE, i134[3]) → LOAD777(+(i134[3], -1), i134[3])
COND_LOAD7772(TRUE, i96[5], i134[5]) → LOAD777(+(i96[5], -1), i134[5])

There are no usable rules.

(12) 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_LOAD777(TRUE, i96[1], i134[1]) → LOAD777(i96[1], i134[1] + -1)
(3): COND_LOAD7771(TRUE, i134[3]) → LOAD777(i134[3] + -1, i134[3])
(5): COND_LOAD7772(TRUE, i96[5], i134[5]) → LOAD777(i96[5] + -1, i134[5])


The set Q consists of the following terms:
Load777(x0, x1)
Cond_Load777(TRUE, x0, x1)
Cond_Load7771(TRUE, x0)
Cond_Load7772(TRUE, x0, x1)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE