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 TRUE
    ↳14 IDP
        ↳15 IDependencyGraphProof (⇔)
        ↳16 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB5 
/**
 * 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 PastaB5 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        while (x > 0 && (x % 2) == 0) {
            x--;
        }
    }
}


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 129 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:
Load350(i34) → Cond_Load350(i34 > 0 && 0 = i34 % 2, i34)
Cond_Load350(TRUE, i34) → Load350(i34 + -1)
The set Q consists of the following terms:
Load350(x0)
Cond_Load350(TRUE, x0)

(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:
Load350(i34) → Cond_Load350(i34 > 0 && 0 = i34 % 2, i34)
Cond_Load350(TRUE, i34) → Load350(i34 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD350(i34[0]) → COND_LOAD350(i34[0] > 0 && 0 = i34[0] % 2, i34[0])
(1): COND_LOAD350(TRUE, i34[1]) → LOAD350(i34[1] + -1)

(0) -> (1), if ((i34[0] →* i34[1])∧(i34[0] > 0 && 0 = i34[0] % 2 →* TRUE))


(1) -> (0), if ((i34[1] + -1 →* i34[0]))



The set Q consists of the following terms:
Load350(x0)
Cond_Load350(TRUE, x0)

(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): LOAD350(i34[0]) → COND_LOAD350(i34[0] > 0 && 0 = i34[0] % 2, i34[0])
(1): COND_LOAD350(TRUE, i34[1]) → LOAD350(i34[1] + -1)

(0) -> (1), if ((i34[0] →* i34[1])∧(i34[0] > 0 && 0 = i34[0] % 2 →* TRUE))


(1) -> (0), if ((i34[1] + -1 →* i34[0]))



The set Q consists of the following terms:
Load350(x0)
Cond_Load350(TRUE, x0)

(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 LOAD350(i34) → COND_LOAD350(&&(>(i34, 0), =(0, %(i34, 2))), i34) the following chains were created:
  • We consider the chain LOAD350(i34[0]) → COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0]), COND_LOAD350(TRUE, i34[1]) → LOAD350(+(i34[1], -1)) which results in the following constraint:

    (1)    (i34[0]=i34[1]∧&&(>(i34[0], 0), =(0, %(i34[0], 2)))=TRUE ⇒ LOAD350(i34[0])≥NonInfC∧LOAD350(i34[0])≥COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])∧(UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥))



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

    (2)    (>(i34[0], 0)=TRUE∧>=(0, %(i34[0], 2))=TRUE∧<=(0, %(i34[0], 2))=TRUE ⇒ LOAD350(i34[0])≥NonInfC∧LOAD350(i34[0])≥COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])∧(UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥))



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

    (3)    (i34[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]i34[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (4)    (i34[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]i34[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (5)    (i34[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥)∧[(-1)Bound*bni_10] + [(2)bni_10]i34[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (6)    (i34[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]i34[0] ≥ 0∧[(-1)bso_11] ≥ 0)



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

    (7)    (i34[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]i34[0] ≥ 0∧[(-1)bso_11] ≥ 0)







For Pair COND_LOAD350(TRUE, i34) → LOAD350(+(i34, -1)) the following chains were created:
  • We consider the chain COND_LOAD350(TRUE, i34[1]) → LOAD350(+(i34[1], -1)) which results in the following constraint:

    (8)    (COND_LOAD350(TRUE, i34[1])≥NonInfC∧COND_LOAD350(TRUE, i34[1])≥LOAD350(+(i34[1], -1))∧(UIncreasing(LOAD350(+(i34[1], -1))), ≥))



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

    (9)    ((UIncreasing(LOAD350(+(i34[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)



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

    (10)    ((UIncreasing(LOAD350(+(i34[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)



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

    (11)    ((UIncreasing(LOAD350(+(i34[1], -1))), ≥)∧[2 + (-1)bso_13] ≥ 0)



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

    (12)    ((UIncreasing(LOAD350(+(i34[1], -1))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD350(i34) → COND_LOAD350(&&(>(i34, 0), =(0, %(i34, 2))), i34)
    • (i34[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [(2)bni_10]i34[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  • COND_LOAD350(TRUE, i34) → LOAD350(+(i34, -1))
    • ((UIncreasing(LOAD350(+(i34[1], -1))), ≥)∧0 = 0∧[2 + (-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(LOAD350(x1)) = [2]x1   
POL(COND_LOAD350(x1, x2)) = [2]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(=(x1, x2)) = [-1]   
POL(2) = [2]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

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

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}   
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}   

The following pairs are in P>:

COND_LOAD350(TRUE, i34[1]) → LOAD350(+(i34[1], -1))

The following pairs are in Pbound:

LOAD350(i34[0]) → COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[0])

The following pairs are in P≥:

LOAD350(i34[0]) → COND_LOAD350(&&(>(i34[0], 0), =(0, %(i34[0], 2))), i34[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): LOAD350(i34[0]) → COND_LOAD350(i34[0] > 0 && 0 = i34[0] % 2, i34[0])


The set Q consists of the following terms:
Load350(x0)
Cond_Load350(TRUE, x0)

(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_LOAD350(TRUE, i34[1]) → LOAD350(i34[1] + -1)


The set Q consists of the following terms:
Load350(x0)
Cond_Load350(TRUE, x0)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE