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 IDPNonInfProof (⇒)
        ↳9 AND
            ↳10 IDP
                ↳11 IDependencyGraphProof (⇔)
                ↳12 TRUE
            ↳13 IDP
                ↳14 IDependencyGraphProof (⇔)
                ↳15 TRUE
    ↳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: MinusMin 
public class MinusMin{

  public static int min (int x, int y) {

    if (x < y) return x;
    else return y;
  }


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



    while (min(x-1,y) == 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:
MinusMin.main([Ljava/lang/String;)V: Graph of 174 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 24 rules for P and 2 rules for R.


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


Filtered ground terms:


968_0_main_ConstantStackPush(x1, x2, x3, x4) → 968_0_main_ConstantStackPush(x2, x3, x4)
Cond_968_0_main_ConstantStackPush1(x1, x2, x3, x4, x5) → Cond_968_0_main_ConstantStackPush1(x1, x3, x4, x5)
Cond_968_0_main_ConstantStackPush(x1, x2, x3, x4, x5) → Cond_968_0_main_ConstantStackPush(x1, x3, x4, x5)

Filtered duplicate args:


968_0_main_ConstantStackPush(x1, x2, x3) → 968_0_main_ConstantStackPush(x2, x3)
Cond_968_0_main_ConstantStackPush1(x1, x2, x3, x4) → Cond_968_0_main_ConstantStackPush1(x1, x3, x4)
Cond_968_0_main_ConstantStackPush(x1, x2, x3, x4) → Cond_968_0_main_ConstantStackPush(x1, x3, x4)

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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(x1[0] <= x0[0] - 1 && x0[0] >= 0, x1[0], x0[0])
(1): COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])
(2): 968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_968_0_MAIN_CONSTANTSTACKPUSH1(x0[2] >= 0 && x0[2] - 1 < x0[2] - 1 && x0_-1[2] = x0[2] - 1, x0_-1[2], x0[2])
(3): COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 968_0_MAIN_CONSTANTSTACKPUSH(x0[3] - 1 + 1, x0[3])

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


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


(1) -> (2), if ((x1[1] + 1 →* x0_-1[2])∧(x0[1] →* x0[2]))


(2) -> (3), if ((x0[2] >= 0 && x0[2] - 1 < x0[2] - 1 && x0_-1[2] = x0[2] - 1 →* TRUE)∧(x0_-1[2] →* x0_-1[3])∧(x0[2] →* x0[3]))


(3) -> (0), if ((x0[3] - 1 + 1 →* x1[0])∧(x0[3] →* x0[0]))


(3) -> (2), if ((x0[3] - 1 + 1 →* x0_-1[2])∧(x0[3] →* x0[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 968_0_MAIN_CONSTANTSTACKPUSH(x1, x0) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1, -(x0, 1)), >=(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0]), COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

    (1)    (&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (2)    (<=(x1[0], -(x0[0], 1))=TRUE∧>=(x0[0], 0)=TRUE ⇒ 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (3)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)


    (7)    (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (8)    (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x0) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), x0) the following chains were created:
  • We consider the chain COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

    (9)    (COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥NonInfC∧COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])∧(UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥))



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

    (10)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[(-1)bso_16] ≥ 0)



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

    (11)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[(-1)bso_16] ≥ 0)



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

    (12)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[(-1)bso_16] ≥ 0)



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

    (13)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)







For Pair 968_0_MAIN_CONSTANTSTACKPUSH(x0_-1, x0) → COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0, 0), <(-(x0, 1), -(x0, 1))), =(x0_-1, -(x0, 1))), x0_-1, x0) the following chains were created:
  • We consider the chain 968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2]), COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3]) which results in the following constraint:

    (14)    (&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1)))=TRUE∧x0_-1[2]=x0_-1[3]∧x0[2]=x0[3] ⇒ 968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥NonInfC∧968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])∧(UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥))



    We simplified constraint (14) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (15)    (>=(x0[2], 0)=TRUE∧<(-(x0[2], 1), -(x0[2], 1))=TRUE∧>=(x0_-1[2], -(x0[2], 1))=TRUE∧<=(x0_-1[2], -(x0[2], 1))=TRUE ⇒ 968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥NonInfC∧968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])∧(UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥))



    We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (16)    (x0[2] ≥ 0∧[-1] ≥ 0∧x0-1[2] + [1] + [-1]x0[2] ≥ 0∧x0[2] + [-1] + [-1]x0-1[2] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[-2 + (-1)bso_18] ≥ 0)



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

    (17)    (x0[2] ≥ 0∧[-1] ≥ 0∧x0-1[2] + [1] + [-1]x0[2] ≥ 0∧x0[2] + [-1] + [-1]x0-1[2] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[-2 + (-1)bso_18] ≥ 0)



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

    (18)    (x0[2] ≥ 0∧[-1] ≥ 0∧x0-1[2] + [1] + [-1]x0[2] ≥ 0∧x0[2] + [-1] + [-1]x0-1[2] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[-2 + (-1)bso_18] ≥ 0)



    We solved constraint (18) using rule (IDP_SMT_SPLIT).




For Pair COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1, x0) → 968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0, 1), 1), x0) the following chains were created:
  • We consider the chain COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3]) which results in the following constraint:

    (19)    (COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3])≥NonInfC∧COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3])≥968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])∧(UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥))



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

    (20)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[2 + (-1)bso_20] ≥ 0)



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

    (21)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[2 + (-1)bso_20] ≥ 0)



    We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (22)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[2 + (-1)bso_20] ≥ 0)



    We simplified constraint (22) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (23)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_20] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 968_0_MAIN_CONSTANTSTACKPUSH(x1, x0) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1, -(x0, 1)), >=(x0, 0)), x1, x0)
    • (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)
    • (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x0) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), x0)
    • ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

  • 968_0_MAIN_CONSTANTSTACKPUSH(x0_-1, x0) → COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0, 0), <(-(x0, 1), -(x0, 1))), =(x0_-1, -(x0, 1))), x0_-1, x0)

  • COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1, x0) → 968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0, 1), 1), x0)
    • ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[2 + (-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(968_0_MAIN_CONSTANTSTACKPUSH(x1, x2)) = [-1] + x2   
POL(COND_968_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x3)) = [-1] + x3   
POL(&&(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(COND_968_0_MAIN_CONSTANTSTACKPUSH1(x1, x2, x3)) = [1] + x3   
POL(<(x1, x2)) = [-1]   
POL(=(x1, x2)) = [-1]   

The following pairs are in P>:

968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])
COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])

The following pairs are in Pbound:

968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])
968_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_968_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>=(x0[2], 0), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])

The following pairs are in P≥:

968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])
COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])

There are no usable rules.

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(x1[0] <= x0[0] - 1 && x0[0] >= 0, x1[0], x0[0])
(1): COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])

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


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



The set Q is empty.

(8) 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 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0]) the following chains were created:
  • We consider the chain 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0]), COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

    (1)    (&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (2)    (<=(x1[0], -(x0[0], 1))=TRUE∧>=(x0[0], 0)=TRUE ⇒ 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (3)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (4)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (5)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (6)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)


    (7)    (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (8)    (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)







For Pair COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) the following chains were created:
  • We consider the chain COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

    (9)    (COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥NonInfC∧COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])∧(UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥))



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

    (10)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)



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

    (11)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)



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

    (12)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)



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

    (13)    ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])
    • (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)
    • (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

  • COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])
    • ((UIncreasing(968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 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(968_0_MAIN_CONSTANTSTACKPUSH(x1, x2)) = [1] + x2 + [-1]x1   
POL(COND_968_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x3)) = [1] + x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   

The following pairs are in P>:

COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])

The following pairs are in Pbound:

968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])

The following pairs are in P≥:

968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >=(x0[0], 0)), x1[0], x0[0])

There are no usable rules.

(9) Complex Obligation (AND)

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 968_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_968_0_MAIN_CONSTANTSTACKPUSH(x1[0] <= x0[0] - 1 && x0[0] >= 0, x1[0], x0[0])


The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE

(13) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])


The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(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:
(1): COND_968_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 968_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])
(3): COND_968_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 968_0_MAIN_CONSTANTSTACKPUSH(x0[3] - 1 + 1, x0[3])


The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(18) TRUE