(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB6
/**
* Example taken from "A Term Rewriting Approach to the Automated Termination
* Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
* and converted to Java.
*/

public class PastaB6 {
public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();

while (x > 0 && y > 0) {
x--;
y--;
}
}
}


public class Random {
static String[] args;
static int index = 0;

public static int random() {
String string = args[index];
index++;
return string.length();
}
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 189 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:
Load548(i36, i45) → Cond_Load548(i45 > 0 && i36 > 0, i36, i45)
Cond_Load548(TRUE, i36, i45) → Load548(i36 + -1, i45 + -1)
The set Q consists of the following terms:
Load548(x0, x1)
Cond_Load548(TRUE, x0, x1)

(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:
Load548(i36, i45) → Cond_Load548(i45 > 0 && i36 > 0, i36, i45)
Cond_Load548(TRUE, i36, i45) → Load548(i36 + -1, i45 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD548(i36[0], i45[0]) → COND_LOAD548(i45[0] > 0 && i36[0] > 0, i36[0], i45[0])
(1): COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(i36[1] + -1, i45[1] + -1)

(0) -> (1), if ((i45[0] > 0 && i36[0] > 0* TRUE)∧(i36[0]* i36[1])∧(i45[0]* i45[1]))


(1) -> (0), if ((i45[1] + -1* i45[0])∧(i36[1] + -1* i36[0]))



The set Q consists of the following terms:
Load548(x0, x1)
Cond_Load548(TRUE, x0, x1)

(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): LOAD548(i36[0], i45[0]) → COND_LOAD548(i45[0] > 0 && i36[0] > 0, i36[0], i45[0])
(1): COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(i36[1] + -1, i45[1] + -1)

(0) -> (1), if ((i45[0] > 0 && i36[0] > 0* TRUE)∧(i36[0]* i36[1])∧(i45[0]* i45[1]))


(1) -> (0), if ((i45[1] + -1* i45[0])∧(i36[1] + -1* i36[0]))



The set Q consists of the following terms:
Load548(x0, x1)
Cond_Load548(TRUE, x0, x1)

(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 LOAD548(i36, i45) → COND_LOAD548(&&(>(i45, 0), >(i36, 0)), i36, i45) the following chains were created:
  • We consider the chain LOAD548(i36[0], i45[0]) → COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0]), COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(+(i36[1], -1), +(i45[1], -1)) which results in the following constraint:

    (1)    (&&(>(i45[0], 0), >(i36[0], 0))=TRUEi36[0]=i36[1]i45[0]=i45[1]LOAD548(i36[0], i45[0])≥NonInfC∧LOAD548(i36[0], i45[0])≥COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])∧(UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥))



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

    (2)    (>(i45[0], 0)=TRUE>(i36[0], 0)=TRUELOAD548(i36[0], i45[0])≥NonInfC∧LOAD548(i36[0], i45[0])≥COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])∧(UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥))



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

    (3)    (i45[0] + [-1] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (4)    (i45[0] + [-1] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (5)    (i45[0] + [-1] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (6)    (i45[0] ≥ 0∧i36[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (7)    (i45[0] ≥ 0∧i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[(5)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)







For Pair COND_LOAD548(TRUE, i36, i45) → LOAD548(+(i36, -1), +(i45, -1)) the following chains were created:
  • We consider the chain COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(+(i36[1], -1), +(i45[1], -1)) which results in the following constraint:

    (8)    (COND_LOAD548(TRUE, i36[1], i45[1])≥NonInfC∧COND_LOAD548(TRUE, i36[1], i45[1])≥LOAD548(+(i36[1], -1), +(i45[1], -1))∧(UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥))



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

    (9)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧[4 + (-1)bso_12] ≥ 0)



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

    (10)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧[4 + (-1)bso_12] ≥ 0)



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

    (11)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧[4 + (-1)bso_12] ≥ 0)



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

    (12)    ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_12] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • LOAD548(i36, i45) → COND_LOAD548(&&(>(i45, 0), >(i36, 0)), i36, i45)
    • (i45[0] ≥ 0∧i36[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])), ≥)∧[(5)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i45[0] + [(2)bni_9]i36[0] ≥ 0∧[(-1)bso_10] ≥ 0)

  • COND_LOAD548(TRUE, i36, i45) → LOAD548(+(i36, -1), +(i45, -1))
    • ((UIncreasing(LOAD548(+(i36[1], -1), +(i45[1], -1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_12] ≥ 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(LOAD548(x1, x2)) = [1] + [2]x2 + [2]x1   
POL(COND_LOAD548(x1, x2, x3)) = [1] + [2]x3 + [2]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(+(i36[1], -1), +(i45[1], -1))

The following pairs are in Pbound:

LOAD548(i36[0], i45[0]) → COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])

The following pairs are in P:

LOAD548(i36[0], i45[0]) → COND_LOAD548(&&(>(i45[0], 0), >(i36[0], 0)), i36[0], i45[0])

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): LOAD548(i36[0], i45[0]) → COND_LOAD548(i45[0] > 0 && i36[0] > 0, i36[0], i45[0])


The set Q consists of the following terms:
Load548(x0, x1)
Cond_Load548(TRUE, x0, x1)

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) 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_LOAD548(TRUE, i36[1], i45[1]) → LOAD548(i36[1] + -1, i45[1] + -1)


The set Q consists of the following terms:
Load548(x0, x1)
Cond_Load548(TRUE, x0, x1)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE