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: MinusBuiltIn 
public class MinusBuiltIn{

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    int res = 0;



    while (x > y) {

      y++;
      res++;

    }
  }

}



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 188 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:
Load920(i14, i61, i62) → Cond_Load920(i14 > i61 && i62 + 1 > 0, i14, i61, i62)
Cond_Load920(TRUE, i14, i61, i62) → Load920(i14, i61 + 1, i62 + 1)
The set Q consists of the following terms:
Load920(x0, x1, x2)
Cond_Load920(TRUE, x0, x1, x2)

(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:
Load920(i14, i61, i62) → Cond_Load920(i14 > i61 && i62 + 1 > 0, i14, i61, i62)
Cond_Load920(TRUE, i14, i61, i62) → Load920(i14, i61 + 1, i62 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(i14[0] > i61[0] && i62[0] + 1 > 0, i14[0], i61[0], i62[0])
(1): COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], i61[1] + 1, i62[1] + 1)

(0) -> (1), if ((i14[0] →* i14[1])∧(i14[0] > i61[0] && i62[0] + 1 > 0 →* TRUE)∧(i62[0] →* i62[1])∧(i61[0] →* i61[1]))


(1) -> (0), if ((i62[1] + 1 →* i62[0])∧(i14[1] →* i14[0])∧(i61[1] + 1 →* i61[0]))



The set Q consists of the following terms:
Load920(x0, x1, x2)
Cond_Load920(TRUE, x0, x1, x2)

(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): LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(i14[0] > i61[0] && i62[0] + 1 > 0, i14[0], i61[0], i62[0])
(1): COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], i61[1] + 1, i62[1] + 1)

(0) -> (1), if ((i14[0] →* i14[1])∧(i14[0] > i61[0] && i62[0] + 1 > 0 →* TRUE)∧(i62[0] →* i62[1])∧(i61[0] →* i61[1]))


(1) -> (0), if ((i62[1] + 1 →* i62[0])∧(i14[1] →* i14[0])∧(i61[1] + 1 →* i61[0]))



The set Q consists of the following terms:
Load920(x0, x1, x2)
Cond_Load920(TRUE, x0, x1, x2)

(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 LOAD920(i14, i61, i62) → COND_LOAD920(&&(>(i14, i61), >(+(i62, 1), 0)), i14, i61, i62) the following chains were created:
  • We consider the chain LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0]), COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1)) which results in the following constraint:

    (1)    (i14[0]=i14[1]∧&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0))=TRUE∧i62[0]=i62[1]∧i61[0]=i61[1] ⇒ LOAD920(i14[0], i61[0], i62[0])≥NonInfC∧LOAD920(i14[0], i61[0], i62[0])≥COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])∧(UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥))



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

    (2)    (>(i14[0], i61[0])=TRUE∧>(+(i62[0], 1), 0)=TRUE ⇒ LOAD920(i14[0], i61[0], i62[0])≥NonInfC∧LOAD920(i14[0], i61[0], i62[0])≥COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])∧(UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥))



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

    (3)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i61[0] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (4)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i61[0] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (5)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i61[0] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (6)    (i14[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (7)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)


    (8)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)







For Pair COND_LOAD920(TRUE, i14, i61, i62) → LOAD920(i14, +(i61, 1), +(i62, 1)) the following chains were created:
  • We consider the chain LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0]), COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1)), LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0]) which results in the following constraint:

    (9)    (i14[0]=i14[1]∧&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0))=TRUE∧i62[0]=i62[1]∧i61[0]=i61[1]∧+(i62[1], 1)=i62[0]1∧i14[1]=i14[0]1∧+(i61[1], 1)=i61[0]1 ⇒ COND_LOAD920(TRUE, i14[1], i61[1], i62[1])≥NonInfC∧COND_LOAD920(TRUE, i14[1], i61[1], i62[1])≥LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))∧(UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥))



    We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (10)    (>(i14[0], i61[0])=TRUE∧>(+(i62[0], 1), 0)=TRUE ⇒ COND_LOAD920(TRUE, i14[0], i61[0], i62[0])≥NonInfC∧COND_LOAD920(TRUE, i14[0], i61[0], i62[0])≥LOAD920(i14[0], +(i61[0], 1), +(i62[0], 1))∧(UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥))



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

    (11)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i61[0] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (12)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i61[0] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (13)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i61[0] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



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

    (14)    (i14[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)



    We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (15)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)


    (16)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD920(i14, i61, i62) → COND_LOAD920(&&(>(i14, i61), >(+(i62, 1), 0)), i14, i61, i62)
    • (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)
    • (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

  • COND_LOAD920(TRUE, i14, i61, i62) → LOAD920(i14, +(i61, 1), +(i62, 1))
    • (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)
    • (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 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(LOAD920(x1, x2, x3)) = [2] + [-1]x2 + x1   
POL(COND_LOAD920(x1, x2, x3, x4)) = [2] + [-1]x3 + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))

The following pairs are in Pbound:

LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])
COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))

The following pairs are in P≥:

LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])

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

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

(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): LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(i14[0] > i61[0] && i62[0] + 1 > 0, i14[0], i61[0], i62[0])


The set Q consists of the following terms:
Load920(x0, x1, x2)
Cond_Load920(TRUE, x0, x1, x2)

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load920(x0, x1, x2)
Cond_Load920(TRUE, x0, x1, x2)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE