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 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 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: LogBuiltIn

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 115 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:
Load352(1, i32, i29) → Cond_Load352(i32 > 1 && i29 + 1 > 0, 1, i32, i29)
Cond_Load352(TRUE, 1, i32, i29) → Load352(1, i32 / 2, i29 + 1)
The set Q consists of the following terms:
Load352(1, x0, x1)
Cond_Load352(TRUE, 1, x0, x1)

(5) GroundTermsRemoverProof (EQUIVALENT transformation)

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

We removed arguments according to the following replacements:

Load352(x1, x2, x3) → Load352(x2, x3)
Cond_Load352(x1, x2, x3, x4) → Cond_Load352(x1, x3, x4)

(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:
Load352(i32, i29) → Cond_Load352(i32 > 1 && i29 + 1 > 0, i32, i29)
Cond_Load352(TRUE, i32, i29) → Load352(i32 / 2, i29 + 1)
The set Q consists of the following terms:
Load352(x0, x1)
Cond_Load352(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:
Load352(i32, i29) → Cond_Load352(i32 > 1 && i29 + 1 > 0, i32, i29)
Cond_Load352(TRUE, i32, i29) → Load352(i32 / 2, i29 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD352(i32[0], i29[0]) → COND_LOAD352(i32[0] > 1 && i29[0] + 1 > 0, i32[0], i29[0])
(1): COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(i32[1] / 2, i29[1] + 1)

(0) -> (1), if ((i29[0] →* i29[1])∧(i32[0] →* i32[1])∧(i32[0] > 1 && i29[0] + 1 > 0 →* TRUE))


(1) -> (0), if ((i32[1] / 2 →* i32[0])∧(i29[1] + 1 →* i29[0]))



The set Q consists of the following terms:
Load352(x0, x1)
Cond_Load352(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): LOAD352(i32[0], i29[0]) → COND_LOAD352(i32[0] > 1 && i29[0] + 1 > 0, i32[0], i29[0])
(1): COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(i32[1] / 2, i29[1] + 1)

(0) -> (1), if ((i29[0] →* i29[1])∧(i32[0] →* i32[1])∧(i32[0] > 1 && i29[0] + 1 > 0 →* TRUE))


(1) -> (0), if ((i32[1] / 2 →* i32[0])∧(i29[1] + 1 →* i29[0]))



The set Q consists of the following terms:
Load352(x0, x1)
Cond_Load352(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 LOAD352(i32, i29) → COND_LOAD352(&&(>(i32, 1), >(+(i29, 1), 0)), i32, i29) the following chains were created:
  • We consider the chain LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0]), COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(/(i32[1], 2), +(i29[1], 1)) which results in the following constraint:

    (1)    (i29[0]=i29[1]∧i32[0]=i32[1]∧&&(>(i32[0], 1), >(+(i29[0], 1), 0))=TRUE ⇒ LOAD352(i32[0], i29[0])≥NonInfC∧LOAD352(i32[0], i29[0])≥COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])∧(UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥))



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

    (2)    (>(i32[0], 1)=TRUE∧>(+(i29[0], 1), 0)=TRUE ⇒ LOAD352(i32[0], i29[0])≥NonInfC∧LOAD352(i32[0], i29[0])≥COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])∧(UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥))



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

    (3)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (4)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (5)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (6)    (i32[0] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)







For Pair COND_LOAD352(TRUE, i32, i29) → LOAD352(/(i32, 2), +(i29, 1)) the following chains were created:
  • We consider the chain LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0]), COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(/(i32[1], 2), +(i29[1], 1)), LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0]) which results in the following constraint:

    (7)    (i29[0]=i29[1]∧i32[0]=i32[1]∧&&(>(i32[0], 1), >(+(i29[0], 1), 0))=TRUE∧/(i32[1], 2)=i32[0]1∧+(i29[1], 1)=i29[0]1 ⇒ COND_LOAD352(TRUE, i32[1], i29[1])≥NonInfC∧COND_LOAD352(TRUE, i32[1], i29[1])≥LOAD352(/(i32[1], 2), +(i29[1], 1))∧(UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥))



    We simplified constraint (7) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (8)    (>(i32[0], 1)=TRUE∧>(+(i29[0], 1), 0)=TRUE ⇒ COND_LOAD352(TRUE, i32[0], i29[0])≥NonInfC∧COND_LOAD352(TRUE, i32[0], i29[0])≥LOAD352(/(i32[0], 2), +(i29[0], 1))∧(UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥))



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

    (9)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] + i32[0] + [-1]max{i32[0], [-1]i32[0]} ≥ 0)



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

    (10)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] + i32[0] + [-1]max{i32[0], [-1]i32[0]} ≥ 0)



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

    (11)    (i32[0] + [-2] ≥ 0∧i29[0] ≥ 0∧[2]i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (12)    (i32[0] ≥ 0∧i29[0] ≥ 0∧[4] + [2]i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (13)    (i32[0] ≥ 0∧i29[0] ≥ 0∧[2] + i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD352(i32, i29) → COND_LOAD352(&&(>(i32, 1), >(+(i29, 1), 0)), i32, i29)
    • (i32[0] ≥ 0∧i29[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]i32[0] ≥ 0∧[(-1)bso_13] ≥ 0)

  • COND_LOAD352(TRUE, i32, i29) → LOAD352(/(i32, 2), +(i29, 1))
    • (i32[0] ≥ 0∧i29[0] ≥ 0∧[2] + i32[0] ≥ 0 ⇒ (UIncreasing(LOAD352(/(i32[1], 2), +(i29[1], 1))), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i32[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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) = [3]   
POL(LOAD352(x1, x2)) = [-1] + x1   
POL(COND_LOAD352(x1, x2, x3)) = [-1] + x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(+(x1, x2)) = x1 + x2   
POL(0) = 0   
POL(2) = [2]   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {LOAD352_2/0}) = max{x1, [-1]x1} + [-1]   

The following pairs are in P>:

COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(/(i32[1], 2), +(i29[1], 1))

The following pairs are in Pbound:

LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])
COND_LOAD352(TRUE, i32[1], i29[1]) → LOAD352(/(i32[1], 2), +(i29[1], 1))

The following pairs are in P≥:

LOAD352(i32[0], i29[0]) → COND_LOAD352(&&(>(i32[0], 1), >(+(i29[0], 1), 0)), i32[0], i29[0])

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

TRUE1 → &&(TRUE, TRUE)1
FALSE1 → &&(FALSE, FALSE)1
/1 →

(12) Complex Obligation (AND)

(13) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD352(i32[0], i29[0]) → COND_LOAD352(i32[0] > 1 && i29[0] + 1 > 0, i32[0], i29[0])


The set Q consists of the following terms:
Load352(x0, x1)
Cond_Load352(TRUE, x0, x1)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load352(x0, x1)
Cond_Load352(TRUE, x0, x1)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE