Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 IDP
↳5 IDPNonInfProof (⇒)
↳6 IDP
↳7 IDependencyGraphProof (⇔)
↳8 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivMinus 
public class DivMinus {
  public static int div(int x, int y) {
    int res = 0;
    while (x >= y && y > 0) {
      x = x-y;
      res = res + 1;
    }
    return res;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    div(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:
DivMinus.main([Ljava/lang/String;)V: Graph of 176 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 17 rules for P and 8 rules for R.


Combined rules. Obtained 1 rules for P and 0 rules for R.


Filtered ground terms:


958_0_div_Load(x1, x2, x3, x4) → 958_0_div_Load(x2, x3, x4)

Filtered duplicate args:


958_0_div_Load(x1, x2, x3) → 958_0_div_Load(x2, x3)

Combined rules. Obtained 1 rules for P and 0 rules for R.


Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.


(4) 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): 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(x1[0] > 0 && x1[0] <= x0[0], 958_0_div_Load(x1[0], x0[0]), x1[0])
(1): COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1]) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], x0[1] - x1[1]), x1[1])

(0) -> (1), if ((x1[0] > 0 && x1[0] <= x0[0] →* TRUE)∧(958_0_div_Load(x1[0], x0[0]) →* 958_0_div_Load(x1[1], x0[1]))∧(x1[0] →* x1[1]))


(1) -> (0), if ((958_0_div_Load(x1[1], x0[1] - x1[1]) →* 958_0_div_Load(x1[0], x0[0]))∧(x1[1] →* x1[0]))



The set Q is empty.

(5) 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 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1, x0), x1) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1, 0), <=(x1, x0)), 958_0_div_Load(x1, x0), x1) the following chains were created:
  • We consider the chain 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0]), COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1]) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1]) which results in the following constraint:

    (1)    (&&(>(x1[0], 0), <=(x1[0], x0[0]))=TRUE∧958_0_div_Load(x1[0], x0[0])=958_0_div_Load(x1[1], x0[1])∧x1[0]=x1[1] ⇒ 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0])≥NonInfC∧958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0])≥COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])∧(UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥))



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

    (2)    (>(x1[0], 0)=TRUE∧<=(x1[0], x0[0])=TRUE ⇒ 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0])≥NonInfC∧958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0])≥COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])∧(UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥))



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

    (3)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (4)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (5)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)







For Pair COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1, x0), x1) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1, -(x0, x1)), x1) the following chains were created:
  • We consider the chain 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0]), COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1]) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1]), 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0]) which results in the following constraint:

    (8)    (&&(>(x1[0], 0), <=(x1[0], x0[0]))=TRUE∧958_0_div_Load(x1[0], x0[0])=958_0_div_Load(x1[1], x0[1])∧x1[0]=x1[1]∧958_0_div_Load(x1[1], -(x0[1], x1[1]))=958_0_div_Load(x1[0]1, x0[0]1)∧x1[1]=x1[0]1 ⇒ COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1])≥NonInfC∧COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1])≥958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])∧(UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥))



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

    (9)    (>(x1[0], 0)=TRUE∧<=(x1[0], x0[0])=TRUE ⇒ COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[0], x0[0]), x1[0])≥NonInfC∧COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[0], x0[0]), x1[0])≥958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], -(x0[0], x1[0])), x1[0])∧(UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥))



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

    (10)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] ≥ 0)



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

    (11)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] ≥ 0)



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

    (12)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] ≥ 0)



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

    (13)    (x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] + x1[0] ≥ 0)



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

    (14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x1[0] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] + x1[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1, x0), x1) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1, 0), <=(x1, x0)), 958_0_div_Load(x1, x0), x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

  • COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1, x0), x1) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1, -(x0, x1)), x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x1[0] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] + x1[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) = 0   
POL(FALSE) = 0   
POL(958_1_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(958_0_div_Load(x1, x2)) = [-1]x2 + [-1]x1   
POL(COND_958_1_MAIN_INVOKEMETHOD(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<=(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   

The following pairs are in P>:

COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1]) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])

The following pairs are in Pbound:

958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[0])
COND_958_1_MAIN_INVOKEMETHOD(TRUE, 958_0_div_Load(x1[1], x0[1]), x1[1]) → 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[1], -(x0[1], x1[1])), x1[1])

The following pairs are in P≥:

958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(&&(>(x1[0], 0), <=(x1[0], x0[0])), 958_0_div_Load(x1[0], x0[0]), x1[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
FALSE1 → &&(FALSE, FALSE)1

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 958_1_MAIN_INVOKEMETHOD(958_0_div_Load(x1[0], x0[0]), x1[0]) → COND_958_1_MAIN_INVOKEMETHOD(x1[0] > 0 && x1[0] <= x0[0], 958_0_div_Load(x1[0], x0[0]), x1[0])


The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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

(8) TRUE