Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 IDP
↳5 IDPNonInfProof (⇒)
↳6 AND
    ↳7 IDP
        ↳8 IDependencyGraphProof (⇔)
        ↳9 IDP
        ↳10 IDPNonInfProof (⇒)
        ↳11 IDP
        ↳12 IDependencyGraphProof (⇔)
        ↳13 TRUE
    ↳14 IDP
        ↳15 IDPNonInfProof (⇒)
        ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivMinus2 
public class DivMinus2 {
  public static int div(int x, int y) {
    int res = 0;
    while (x >= y && y > 0) {
      x = minus(x,y);
      res = res + 1;
    }
    return res;
  }

  public static int minus(int x, int y) {
    while (y != 0) {
      if (y > 0)  {
        y--;
        x--;
      } else  {
        y++;
        x++;
      }
    }
    return x;
  }

  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:
DivMinus2.main([Ljava/lang/String;)V: Graph of 189 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 31 rules for P and 8 rules for R.


Combined rules. Obtained 2 rules for P and 0 rules for R.


Filtered ground terms:


1204_0_minus_EQ(x1, x2, x3, x4) → 1204_0_minus_EQ(x2, x3, x4)

Filtered duplicate args:


1204_0_minus_EQ(x1, x2, x3) → 1204_0_minus_EQ(x1, x3)
1204_1_div_InvokeMethod(x1, x2, x3) → 1204_1_div_InvokeMethod(x1, x3)

Combined rules. Obtained 2 rules for P and 0 rules for R.


Finished conversion. Obtained 2 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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(x1[0] > 0, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])
(1): COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1] + -1, x1[1] + -1), x2[1]), x2[1])
(2): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(x3[2] > 0 && x3[2] <= x0[2], 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])
(3): COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])

(0) -> (1), if ((x1[0] > 0 →* TRUE)∧(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]))∧(x2[0] →* x2[1]))


(1) -> (0), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1] + -1, x1[1] + -1), x2[1]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]))∧(x2[1] →* x2[0]))


(1) -> (2), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1] + -1, x1[1] + -1), x2[1]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]))∧(x2[1] →* x3[2]))


(2) -> (3), if ((x3[2] > 0 && x3[2] <= x0[2] →* TRUE)∧(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]))∧(x3[2] →* x3[3]))


(3) -> (0), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]))∧(x3[3] →* x2[0]))


(3) -> (2), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]))∧(x3[3] →* x3[2]))



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 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x1), x2), x2) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1, 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x1), x2), x2) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]), COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1]) which results in the following constraint:

    (1)    (>(x1[0], 0)=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1])∧x2[0]=x2[1] ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥))



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

    (2)    (>(x1[0], 0)=TRUE ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥))



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

    (3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (6)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧0 = 0∧[bni_23] = 0∧[(3)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_24] ≥ 0)



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

    (7)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧0 = 0∧[bni_23] = 0∧[(3)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_24] ≥ 0)







For Pair COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x1), x2), x2) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0, -1), +(x1, -1)), x2), x2) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]), COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1]), 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) which results in the following constraint:

    (8)    (>(x1[0], 0)=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1])∧x2[0]=x2[1]∧1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0]1, x1[0]1), x2[0]1)∧x2[1]=x2[0]1 ⇒ COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥))



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

    (9)    (>(x1[0], 0)=TRUE ⇒ COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[0], -1), +(x1[0], -1)), x2[0]), x2[0])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥))



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

    (10)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (11)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (12)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



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

    (13)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧[bni_25] = 0∧[(3)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)



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

    (14)    (x1[0] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧[bni_25] = 0∧[(3)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)



  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]), COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1]), 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) which results in the following constraint:

    (15)    (>(x1[0], 0)=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1])∧x2[0]=x2[1]∧1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2])∧x2[1]=x3[2] ⇒ COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥))



    We simplified constraint (15) using rules (I), (II), (III), (IV) which results in the following new constraint:

    (16)    (>(x1[0], 0)=TRUE∧+(x1[0], -1)=0 ⇒ COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[0], -1), +(x1[0], -1)), x2[0]), x2[0])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥))



    We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (17)    (x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



    We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (18)    (x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



    We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (19)    (x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[(3)bni_25 + (-1)Bound*bni_25] + [bni_25]x0[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)



    We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (20)    (x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧[bni_25] = 0∧[(3)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)



    We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (21)    (x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧[bni_25] = 0∧[(3)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)







For Pair 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, 0), x3), x3) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3, 0), <=(x3, x0)), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, 0), x3), x3) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]), COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3]) which results in the following constraint:

    (22)    (&&(>(x3[2], 0), <=(x3[2], x0[2]))=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3])∧x3[2]=x3[3] ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥))



    We simplified constraint (22) using rules (I), (II), (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

    (23)    (>(x3[2], 0)=TRUE∧<=(x3[2], x0[2])=TRUE ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥))



    We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (24)    (x3[2] + [-1] ≥ 0∧x0[2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (25)    (x3[2] + [-1] ≥ 0∧x0[2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (26)    (x3[2] + [-1] ≥ 0∧x0[2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (x3[2] ≥ 0∧x0[2] + [-1] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (x3[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(4)bni_27 + (-1)Bound*bni_27] + [bni_27]x3[2] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, 0), x3), x3) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x3), x3), x3) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]), COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3]), 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) which results in the following constraint:

    (29)    (&&(>(x3[2], 0), <=(x3[2], x0[2]))=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3])∧x3[2]=x3[3]∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0])∧x3[3]=x2[0] ⇒ COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥))



    We simplified constraint (29) using rules (I), (II), (III), (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

    (30)    (>(x3[2], 0)=TRUE∧<=(x3[2], x0[2])=TRUE ⇒ COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], x3[2]), x3[2]), x3[2])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥))



    We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (31)    (x3[2] + [-1] ≥ 0∧x0[2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥)∧[(3)bni_29 + (-1)Bound*bni_29] + [bni_29]x0[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (32)    (x3[2] + [-1] ≥ 0∧x0[2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥)∧[(3)bni_29 + (-1)Bound*bni_29] + [bni_29]x0[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (33)    (x3[2] + [-1] ≥ 0∧x0[2] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥)∧[(3)bni_29 + (-1)Bound*bni_29] + [bni_29]x0[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (34)    (x3[2] ≥ 0∧x0[2] + [-1] + [-1]x3[2] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥)∧[(3)bni_29 + (-1)Bound*bni_29] + [bni_29]x0[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (35)    (x3[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥)∧[(4)bni_29 + (-1)Bound*bni_29] + [bni_29]x3[2] + [bni_29]x0[2] ≥ 0∧[(-1)bso_30] ≥ 0)



  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]), COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3]), 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) which results in the following constraint:

    (36)    (&&(>(x3[2], 0), <=(x3[2], x0[2]))=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3])∧x3[2]=x3[3]∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2]1, 0), x3[2]1)∧x3[3]=x3[2]1 ⇒ COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥))



    We solved constraint (36) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO).




To summarize, we get the following constraints P≥ for the following pairs.
  • 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x1), x2), x2) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1, 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x1), x2), x2)
    • (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧0 = 0∧[bni_23] = 0∧[(3)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_24] ≥ 0)

  • COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x1), x2), x2) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0, -1), +(x1, -1)), x2), x2)
    • (x1[0] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧[bni_25] = 0∧[(3)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)
    • (x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧[bni_25] = 0∧[(3)bni_25 + (-1)Bound*bni_25] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

  • 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, 0), x3), x3) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3, 0), <=(x3, x0)), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, 0), x3), x3)
    • (x3[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(4)bni_27 + (-1)Bound*bni_27] + [bni_27]x3[2] + [bni_27]x0[2] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, 0), x3), x3) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0, x3), x3), x3)
    • (x3[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥)∧[(4)bni_29 + (-1)Bound*bni_29] + [bni_29]x3[2] + [bni_29]x0[2] ≥ 0∧[(-1)bso_30] ≥ 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) = [1]   
POL(FALSE) = [1]   
POL(1204_2_MAIN_INVOKEMETHOD(x1, x2)) = [1] + [-1]x2 + [-1]x1   
POL(1204_1_div_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + x1   
POL(1204_0_minus_EQ(x1, x2)) = [-1] + [-1]x1   
POL(COND_1204_2_MAIN_INVOKEMETHOD(x1, x2, x3)) = [1] + [-1]x3 + [-1]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_1204_2_MAIN_INVOKEMETHOD1(x1, x2, x3)) = [2] + [-1]x3 + [-1]x2 + [-1]x1   
POL(&&(x1, x2)) = [1]   
POL(<=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])

The following pairs are in Pbound:

1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])
COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])

The following pairs are in P≥:

1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])
1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])
COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])

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

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

(6) Complex Obligation (AND)

(7) 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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(x1[0] > 0, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])
(2): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(x3[2] > 0 && x3[2] <= x0[2], 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])
(3): COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])

(3) -> (0), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]))∧(x3[3] →* x2[0]))


(3) -> (2), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]))∧(x3[3] →* x3[2]))


(2) -> (3), if ((x3[2] > 0 && x3[2] <= x0[2] →* TRUE)∧(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]))∧(x3[2] →* x3[3]))



The set Q is empty.

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) 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:
(3): COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])
(2): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(x3[2] > 0 && x3[2] <= x0[2], 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])

(3) -> (2), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]))∧(x3[3] →* x3[2]))


(2) -> (3), if ((x3[2] > 0 && x3[2] <= x0[2] →* TRUE)∧(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]))∧(x3[2] →* x3[3]))



The set Q is empty.

(10) 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 COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3]) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]), COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3]), 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) which results in the following constraint:

    (1)    (&&(>(x3[2], 0), <=(x3[2], x0[2]))=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3])∧x3[2]=x3[3]∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2]1, 0), x3[2]1)∧x3[3]=x3[2]1 ⇒ COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])), ≥))



    We solved constraint (1) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO).




For Pair 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]), COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3]) which results in the following constraint:

    (2)    (&&(>(x3[2], 0), <=(x3[2], x0[2]))=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3])∧x3[2]=x3[3] ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥))



    We simplified constraint (2) using rules (I), (II), (IV), (DELETE_TRIVIAL_REDUCESTO) which results in the following new constraint:

    (3)    (&&(>(x3[2], 0), <=(x3[2], x0[2]))=TRUE ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])≥COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥))



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(21)bni_14 + (-1)Bound*bni_14] + [bni_14]x3[2] + [(6)bni_14]x0[2] ≥ 0∧[32 + (-1)bso_15] + [3]x3[2] + [9]x0[2] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(21)bni_14 + (-1)Bound*bni_14] + [bni_14]x3[2] + [(6)bni_14]x0[2] ≥ 0∧[32 + (-1)bso_15] + [3]x3[2] + [9]x0[2] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[(21)bni_14 + (-1)Bound*bni_14] + [bni_14]x3[2] + [(6)bni_14]x0[2] ≥ 0∧[32 + (-1)bso_15] + [3]x3[2] + [9]x0[2] ≥ 0)



    We simplified constraint (6) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (7)    (0 ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[bni_14] ≥ 0∧[(6)bni_14] ≥ 0∧[(21)bni_14 + (-1)Bound*bni_14] ≥ 0∧[32 + (-1)bso_15] ≥ 0∧[1] ≥ 0∧[1] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])

  • 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])
    • (0 ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])), ≥)∧[bni_14] ≥ 0∧[(6)bni_14] ≥ 0∧[(21)bni_14 + (-1)Bound*bni_14] ≥ 0∧[32 + (-1)bso_15] ≥ 0∧[1] ≥ 0∧[1] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_1204_2_MAIN_INVOKEMETHOD1(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(1204_1_div_InvokeMethod(x1, x2)) = [1] + x2 + [3]x1   
POL(1204_0_minus_EQ(x1, x2)) = [3] + x1   
POL(0) = 0   
POL(1204_2_MAIN_INVOKEMETHOD(x1, x2)) = [1] + [-1]x2 + [2]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = 0   
POL(<=(x1, x2)) = 0   

The following pairs are in P>:

COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])
1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])

The following pairs are in Pbound:

COND_1204_2_MAIN_INVOKEMETHOD1(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], 0), x3[3]), x3[3]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[3], x3[3]), x3[3]), x3[3])
1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2]) → COND_1204_2_MAIN_INVOKEMETHOD1(&&(>(x3[2], 0), <=(x3[2], x0[2])), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[2], 0), x3[2]), x3[2])

The following pairs are in P≥:
none

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

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

(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:
none


R is empty.

The integer pair graph is empty.

The set Q is empty.

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(0): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(x1[0] > 0, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])
(1): COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1] + -1, x1[1] + -1), x2[1]), x2[1])

(1) -> (0), if ((1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1] + -1, x1[1] + -1), x2[1]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]))∧(x2[1] →* x2[0]))


(0) -> (1), if ((x1[0] > 0 →* TRUE)∧(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]) →* 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]))∧(x2[0] →* x2[1]))



The set Q is empty.

(15) 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 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]), COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1]) which results in the following constraint:

    (1)    (>(x1[0], 0)=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1])∧x2[0]=x2[1] ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥))



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

    (2)    (>(x1[0], 0)=TRUE ⇒ 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])∧(UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥))



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

    (3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (6)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧0 = 0∧0 = 0∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)



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

    (7)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧0 = 0∧0 = 0∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)







For Pair COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1]) the following chains were created:
  • We consider the chain 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]), COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1]), 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) which results in the following constraint:

    (8)    (>(x1[0], 0)=TRUE∧1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1])∧x2[0]=x2[1]∧1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1])=1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0]1, x1[0]1), x2[0]1)∧x2[1]=x2[0]1 ⇒ COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥))



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

    (9)    (>(x1[0], 0)=TRUE ⇒ COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥NonInfC∧COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])≥1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[0], -1), +(x1[0], -1)), x2[0]), x2[0])∧(UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥))



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

    (10)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (11)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (12)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (13)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧0 = 0∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)



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

    (14)    (x1[0] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧0 = 0∧[(2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])
    • (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])), ≥)∧0 = 0∧0 = 0∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

  • COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])
    • (x1[0] ≥ 0 ⇒ (UIncreasing(1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])), ≥)∧0 = 0∧0 = 0∧[(2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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(1204_2_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(1204_1_div_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + x1   
POL(1204_0_minus_EQ(x1, x2)) = [-1] + [-1]x2   
POL(COND_1204_2_MAIN_INVOKEMETHOD(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])

The following pairs are in Pbound:

1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])
COND_1204_2_MAIN_INVOKEMETHOD(TRUE, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[1], x1[1]), x2[1]), x2[1]) → 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(+(x0[1], -1), +(x1[1], -1)), x2[1]), x2[1])

The following pairs are in P≥:

1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(>(x1[0], 0), 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])

There are no usable rules.

(16) 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:
(0): 1204_2_MAIN_INVOKEMETHOD(1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0]) → COND_1204_2_MAIN_INVOKEMETHOD(x1[0] > 0, 1204_1_div_InvokeMethod(1204_0_minus_EQ(x0[0], x1[0]), x2[0]), x2[0])


The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE