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 IDP
↳11 IDependencyGraphProof (⇔)
↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Duplicate 
public class Duplicate{

  public static int round (int x) {

    if (x % 2 == 0) return x;
    else return x+1;
  }


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

    while ((x > y) && (y > 2)) {
      x++;
      y = 2*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 194 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:
Load551(2, i11, i21) → Cond_Load551(i21 > 2 && i11 > i21, 2, i11, i21)
Cond_Load551(TRUE, 2, i11, i21) → Load551(2, i11 + 1, 2 * i21)
The set Q consists of the following terms:
Load551(2, x0, x1)
Cond_Load551(TRUE, 2, 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:
Load551(2, i11, i21) → Cond_Load551(i21 > 2 && i11 > i21, 2, i11, i21)
Cond_Load551(TRUE, 2, i11, i21) → Load551(2, i11 + 1, 2 * i21)

The integer pair graph contains the following rules and edges:
(0): LOAD551(2, i11[0], i21[0]) → COND_LOAD551(i21[0] > 2 && i11[0] > i21[0], 2, i11[0], i21[0])
(1): COND_LOAD551(TRUE, 2, i11[1], i21[1]) → LOAD551(2, i11[1] + 1, 2 * i21[1])

(0) -> (1), if ((i11[0] →* i11[1])∧(i21[0] > 2 && i11[0] > i21[0] →* TRUE)∧(i21[0] →* i21[1]))


(1) -> (0), if ((2 * i21[1] →* i21[0])∧(i11[1] + 1 →* i11[0]))



The set Q consists of the following terms:
Load551(2, x0, x1)
Cond_Load551(TRUE, 2, 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): LOAD551(2, i11[0], i21[0]) → COND_LOAD551(i21[0] > 2 && i11[0] > i21[0], 2, i11[0], i21[0])
(1): COND_LOAD551(TRUE, 2, i11[1], i21[1]) → LOAD551(2, i11[1] + 1, 2 * i21[1])

(0) -> (1), if ((i11[0] →* i11[1])∧(i21[0] > 2 && i11[0] > i21[0] →* TRUE)∧(i21[0] →* i21[1]))


(1) -> (0), if ((2 * i21[1] →* i21[0])∧(i11[1] + 1 →* i11[0]))



The set Q consists of the following terms:
Load551(2, x0, x1)
Cond_Load551(TRUE, 2, 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 LOAD551(2, i11, i21) → COND_LOAD551(&&(>(i21, 2), >(i11, i21)), 2, i11, i21) the following chains were created:
  • We consider the chain LOAD551(2, i11[0], i21[0]) → COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0]), COND_LOAD551(TRUE, 2, i11[1], i21[1]) → LOAD551(2, +(i11[1], 1), *(2, i21[1])) which results in the following constraint:

    (1)    (i11[0]=i11[1]∧&&(>(i21[0], 2), >(i11[0], i21[0]))=TRUE∧i21[0]=i21[1] ⇒ LOAD551(2, i11[0], i21[0])≥NonInfC∧LOAD551(2, i11[0], i21[0])≥COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])∧(UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥))



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

    (2)    (>(i21[0], 2)=TRUE∧>(i11[0], i21[0])=TRUE ⇒ LOAD551(2, i11[0], i21[0])≥NonInfC∧LOAD551(2, i11[0], i21[0])≥COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])∧(UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥))



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

    (3)    (i21[0] + [-3] ≥ 0∧i11[0] + [-1] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i21[0] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (4)    (i21[0] + [-3] ≥ 0∧i11[0] + [-1] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i21[0] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (5)    (i21[0] + [-3] ≥ 0∧i11[0] + [-1] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i21[0] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (6)    (i21[0] ≥ 0∧i11[0] + [-4] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥)∧[(-2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i21[0] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (7)    (i21[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







For Pair COND_LOAD551(TRUE, 2, i11, i21) → LOAD551(2, +(i11, 1), *(2, i21)) the following chains were created:
  • We consider the chain LOAD551(2, i11[0], i21[0]) → COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0]), COND_LOAD551(TRUE, 2, i11[1], i21[1]) → LOAD551(2, +(i11[1], 1), *(2, i21[1])), LOAD551(2, i11[0], i21[0]) → COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0]) which results in the following constraint:

    (8)    (i11[0]=i11[1]∧&&(>(i21[0], 2), >(i11[0], i21[0]))=TRUE∧i21[0]=i21[1]∧*(2, i21[1])=i21[0]1∧+(i11[1], 1)=i11[0]1 ⇒ COND_LOAD551(TRUE, 2, i11[1], i21[1])≥NonInfC∧COND_LOAD551(TRUE, 2, i11[1], i21[1])≥LOAD551(2, +(i11[1], 1), *(2, i21[1]))∧(UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥))



    We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(i21[0], 2)=TRUE∧>(i11[0], i21[0])=TRUE ⇒ COND_LOAD551(TRUE, 2, i11[0], i21[0])≥NonInfC∧COND_LOAD551(TRUE, 2, i11[0], i21[0])≥LOAD551(2, +(i11[0], 1), *(2, i21[0]))∧(UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥))



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

    (10)    (i21[0] + [-3] ≥ 0∧i11[0] + [-1] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i21[0] + [bni_16]i11[0] ≥ 0∧[-2 + (-1)bso_17] + i21[0] ≥ 0)



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

    (11)    (i21[0] + [-3] ≥ 0∧i11[0] + [-1] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i21[0] + [bni_16]i11[0] ≥ 0∧[-2 + (-1)bso_17] + i21[0] ≥ 0)



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

    (12)    (i21[0] + [-3] ≥ 0∧i11[0] + [-1] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥)∧[(-1)Bound*bni_16] + [(-1)bni_16]i21[0] + [bni_16]i11[0] ≥ 0∧[-2 + (-1)bso_17] + i21[0] ≥ 0)



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

    (13)    (i21[0] ≥ 0∧i11[0] + [-4] + [-1]i21[0] ≥ 0 ⇒ (UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥)∧[(-1)Bound*bni_16 + (-3)bni_16] + [(-1)bni_16]i21[0] + [bni_16]i11[0] ≥ 0∧[1 + (-1)bso_17] + i21[0] ≥ 0)



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

    (14)    (i21[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]i11[0] ≥ 0∧[1 + (-1)bso_17] + i21[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD551(2, i11, i21) → COND_LOAD551(&&(>(i21, 2), >(i11, i21)), 2, i11, i21)
    • (i21[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

  • COND_LOAD551(TRUE, 2, i11, i21) → LOAD551(2, +(i11, 1), *(2, i21))
    • (i21[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(LOAD551(2, +(i11[1], 1), *(2, i21[1]))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]i11[0] ≥ 0∧[1 + (-1)bso_17] + i21[0] ≥ 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) = [2]   
POL(FALSE) = 0   
POL(LOAD551(x1, x2, x3)) = [-1] + [-1]x3 + x2 + x1   
POL(2) = [2]   
POL(COND_LOAD551(x1, x2, x3, x4)) = [-1]x4 + x3   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(*(x1, x2)) = x1·x2   

The following pairs are in P>:

LOAD551(2, i11[0], i21[0]) → COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])
COND_LOAD551(TRUE, 2, i11[1], i21[1]) → LOAD551(2, +(i11[1], 1), *(2, i21[1]))

The following pairs are in Pbound:

LOAD551(2, i11[0], i21[0]) → COND_LOAD551(&&(>(i21[0], 2), >(i11[0], i21[0])), 2, i11[0], i21[0])
COND_LOAD551(TRUE, 2, i11[1], i21[1]) → LOAD551(2, +(i11[1], 1), *(2, i21[1]))

The following pairs are in P≥:
none

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

FALSE1 → &&(FALSE, TRUE)1

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load551(2, x0, x1)
Cond_Load551(TRUE, 2, x0, x1)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE