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 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇒)
↳12 IDP
↳13 IDependencyGraphProof (⇔)
↳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_20 (Apple Inc.) Main-Class: AckermannR 
public class AckermannR {

    public static int ack(int m, int n) {
	if (m <= 0) return n + 1;
	else if (n <= 0) return ack(m - 1,1);
	else return ack(m - 1,ack(m,n - 1));
    }

    public static void main(String[] args) {
	Random.args = args;
	ack(Random.random(),Random.random());
    }
}

public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
      if (index >= args.length)
	  return 0;

      String string = args[index];
      index++;
      return string.length();
  }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
AckermannR.main([Ljava/lang/String;)V: Graph of 145 nodes with 0 SCCs.

AckermannR.ack(II)I: Graph of 49 nodes with 0 SCCs.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 33 rules for P and 18 rules for R.


Combined rules. Obtained 4 rules for P and 5 rules for R.


Filtered ground terms:


450_0_ack_GT(x1, x2, x3, x4) → 450_0_ack_GT(x2, x3, x4)
710_0_ack_Return(x1, x2) → 710_0_ack_Return(x2)
629_0_ack_Return(x1, x2, x3, x4) → 629_0_ack_Return(x2, x4)
Cond_450_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_450_0_ack_GT1(x1, x3, x4, x5)
612_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 612_1_ack_InvokeMethod(x1, x2, x4)
Cond_450_0_ack_GT(x1, x2, x3, x4, x5) → Cond_450_0_ack_GT(x1, x3, x4, x5)
502_0_ack_Return(x1, x2, x3, x4) → 502_0_ack_Return(x3, x4)

Filtered duplicate args:


450_0_ack_GT(x1, x2, x3) → 450_0_ack_GT(x2, x3)
Cond_450_0_ack_GT1(x1, x2, x3, x4) → Cond_450_0_ack_GT1(x1, x3, x4)
Cond_450_0_ack_GT(x1, x2, x3, x4) → Cond_450_0_ack_GT(x1, x3, x4)

Filtered unneeded arguments:


612_1_ack_InvokeMethod(x1, x2, x3) → 612_1_ack_InvokeMethod(x1, x3)

Combined rules. Obtained 4 rules for P and 5 rules for R.


Finished conversion. Obtained 4 rules for P and 5 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


The ITRS R consists of the following rules:
450_0_ack_GT(x1, 0) → Cond_450_0_ack_GT(x1 >= 0, x1, 0)
Cond_450_0_ack_GT(TRUE, x1, 0) → 502_0_ack_Return(x1, x1 + 1)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x2), 0) → 629_0_ack_Return(x3, x2)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x3) → 629_0_ack_Return(x1, x0)
701_1_ack_InvokeMethod(502_0_ack_Return(x1, x2), 0, x1) → 710_0_ack_Return(x2)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2) → 710_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(x0[0] > 0, 0, x0[0])
(1): COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, x0[1] - 1)
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x0[3], x1[3] - 1)
(4): COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(x1[4] - 1, x0[4])
(5): 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5])
(6): 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6])

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


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


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


(2) -> (3), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[3])∧(x0[2] →* x0[3]))


(2) -> (4), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[4])∧(x0[2] →* x0[4]))


(3) -> (5), if ((450_0_ack_GT(x1[3] - 1, x0[3]) →* 629_0_ack_Return(x0[5], x2[5]))∧(x0[3] - 1 →* x3[5])∧(x0[3] →* x0[5])∧(x1[3] - 1 →* 0))


(3) -> (6), if ((450_0_ack_GT(x1[3] - 1, x0[3]) →* 710_0_ack_Return(x0[6]))∧(x0[3] - 1 →* x1[6])∧(x0[3] →* x2[6])∧(x1[3] - 1 →* x3[6]))


(4) -> (0), if ((x1[4] - 1 →* 0)∧(x0[4] →* x0[0]))


(4) -> (2), if ((x1[4] - 1 →* x1[2])∧(x0[4] →* x0[2]))


(5) -> (0), if ((x2[5] →* 0)∧(x3[5] →* x0[0]))


(5) -> (2), if ((x2[5] →* x1[2])∧(x3[5] →* x0[2]))


(6) -> (0), if ((x0[6] →* 0)∧(x1[6] →* x0[0]))


(6) -> (2), if ((x0[6] →* x1[2])∧(x1[6] →* x0[2]))



The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(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 450_0_ACK_GT(0, x0) → COND_450_0_ACK_GT(>(x0, 0), 0, x0) the following chains were created:
  • We consider the chain 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 450_0_ACK_GT(0, x0[0])≥NonInfC∧450_0_ACK_GT(0, x0[0])≥COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))



    We simplified constraint (1) using rule (IV) which results in the following new constraint:

    (2)    (>(x0[0], 0)=TRUE ⇒ 450_0_ACK_GT(0, x0[0])≥NonInfC∧450_0_ACK_GT(0, x0[0])≥COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] ≥ 0∧[(-1)bso_40] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] ≥ 0∧[(-1)bso_40] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] ≥ 0∧[(-1)bso_40] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧0 ≥ 0∧[(-1)bni_39 + (-1)Bound*bni_39] ≥ 0∧0 ≥ 0∧[(-1)bso_40] ≥ 0)







For Pair COND_450_0_ACK_GT(TRUE, 0, x0) → 450_0_ACK_GT(1, -(x0, 1)) the following chains were created:
  • We consider the chain 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)), 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧1=0∧-(x0[1], 1)=x0[0]1 ⇒ COND_450_0_ACK_GT(TRUE, 0, x0[1])≥NonInfC∧COND_450_0_ACK_GT(TRUE, 0, x0[1])≥450_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥))



    We solved constraint (7) using rules (I), (II).
  • We consider the chain 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (8)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧1=x1[2]∧-(x0[1], 1)=x0[2] ⇒ COND_450_0_ACK_GT(TRUE, 0, x0[1])≥NonInfC∧COND_450_0_ACK_GT(TRUE, 0, x0[1])≥450_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥))



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

    (9)    (>(x0[0], 0)=TRUE ⇒ COND_450_0_ACK_GT(TRUE, 0, x0[0])≥NonInfC∧COND_450_0_ACK_GT(TRUE, 0, x0[0])≥450_0_ACK_GT(1, -(x0[0], 1))∧(UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥))



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

    (10)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (11)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (12)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (13)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧0 ≥ 0∧[(-1)bni_41 + (-1)Bound*bni_41] ≥ 0∧0 ≥ 0∧[(-1)bso_42] ≥ 0)







For Pair 450_0_ACK_GT(x1, x0) → COND_450_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)) which results in the following constraint:

    (14)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (15)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (16)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (17)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (18)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (19)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_44] ≥ 0)



  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (20)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4] ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (21)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (22)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (23)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (24)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧[(-1)bso_44] ≥ 0)



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

    (25)    (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_44] ≥ 0)







For Pair COND_450_0_ACK_GT1(TRUE, x1, x0) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:

    (26)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3]∧450_0_ack_GT(-(x1[3], 1), x0[3])=629_0_ack_Return(x0[5], x2[5])∧-(x0[3], 1)=x3[5]∧x0[3]=x0[5]∧-(x1[3], 1)=0 ⇒ COND_450_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[3], x0[3])≥680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))∧(UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))



    We simplified constraint (26) using rules (III), (IV) which results in the following new constraint:

    (27)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧450_0_ack_GT(-(x1[2], 1), x0[2])=629_0_ack_Return(x0[2], x2[5])∧-(x1[2], 1)=0 ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[2], 1), x0[2]), -(x0[2], 1), x0[2], -(x1[2], 1))∧(UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))



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

    (28)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧[1 + (-1)bso_46] + [3]x0[2] ≥ 0)



    We simplified constraint (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (29)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧[1 + (-1)bso_46] + [3]x0[2] ≥ 0)



    We simplified constraint (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (30)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧[1 + (-1)bso_46] + [3]x0[2] ≥ 0)



    We simplified constraint (30) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (31)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_46] ≥ 0∧[1] ≥ 0)



  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:

    (32)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3]∧450_0_ack_GT(-(x1[3], 1), x0[3])=710_0_ack_Return(x0[6])∧-(x0[3], 1)=x1[6]∧x0[3]=x2[6]∧-(x1[3], 1)=x3[6] ⇒ COND_450_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[3], x0[3])≥680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))∧(UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))



    We simplified constraint (32) using rules (III), (IV) which results in the following new constraint:

    (33)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧450_0_ack_GT(-(x1[2], 1), x0[2])=710_0_ack_Return(x0[6]) ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[2], 1), x0[2]), -(x0[2], 1), x0[2], -(x1[2], 1))∧(UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))



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

    (34)    (0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧[1 + (-1)bso_46] + [3]x0[2] ≥ 0)



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

    (35)    (0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧[1 + (-1)bso_46] + [3]x0[2] ≥ 0)



    We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (36)    (0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧[1 + (-1)bso_46] + [3]x0[2] ≥ 0)



    We simplified constraint (36) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

    (37)    (0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_46] ≥ 0∧[1] ≥ 0)







For Pair COND_450_0_ACK_GT1(TRUE, x1, x0) → 450_0_ACK_GT(-(x1, 1), x0) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]), 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) which results in the following constraint:

    (38)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4]∧-(x1[4], 1)=0∧x0[4]=x0[0] ⇒ COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥450_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



    We simplified constraint (38) using rules (III), (IV) which results in the following new constraint:

    (39)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧-(x1[2], 1)=0 ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥450_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



    We simplified constraint (39) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (40)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (40) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (41)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (41) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (42)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (42) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (43)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_48] ≥ 0)



  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (44)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4]∧-(x1[4], 1)=x1[2]1∧x0[4]=x0[2]1 ⇒ COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥450_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



    We simplified constraint (44) using rules (III), (IV) which results in the following new constraint:

    (45)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥450_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



    We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (46)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (47)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (48)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧[(-1)bso_48] ≥ 0)



    We simplified constraint (48) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (49)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_48] ≥ 0)







For Pair 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0, x2), x3, x0, 0) → 450_0_ACK_GT(x2, x3) the following chains were created:
  • We consider the chain COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5]), 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) which results in the following constraint:

    (50)    (450_0_ack_GT(-(x1[3], 1), x0[3])=629_0_ack_Return(x0[5], x2[5])∧-(x0[3], 1)=x3[5]∧x0[3]=x0[5]∧-(x1[3], 1)=0∧x2[5]=0∧x3[5]=x0[0] ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0)≥450_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (50) using rules (III), (IV), (VII) which results in the following new constraint:

    (51)    (-(x1[3], 1)=x0∧450_0_ack_GT(x0, x0[3])=629_0_ack_Return(x0[3], 0)∧-(x1[3], 1)=0 ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[3], 0), -(x0[3], 1), x0[3], 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[3], 0), -(x0[3], 1), x0[3], 0)≥450_0_ACK_GT(0, -(x0[3], 1))∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (51) using rule (V) (with possible (I) afterwards) using induction on 450_0_ack_GT(x0, x0[3])=629_0_ack_Return(x0[3], 0) which results in the following new constraint:

    (52)    (Cond_450_0_ack_GT(>=(x1, 0), x1, 0)=629_0_ack_Return(0, 0)∧-(x1[3], 1)=x1∧-(x1[3], 1)=0 ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, 0), -(0, 1), 0, 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, 0), -(0, 1), 0, 0)≥450_0_ACK_GT(0, -(0, 1))∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (52) using rules (III), (VII) which results in the following new constraint:

    (53)    (Cond_450_0_ack_GT(>=(-(x1[3], 1), 0), -(x1[3], 1), 0)=629_0_ack_Return(0, 0)∧-(x1[3], 1)=0 ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, 0), -(0, 1), 0, 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, 0), -(0, 1), 0, 0)≥450_0_ACK_GT(0, -(0, 1))∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (53) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (54)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)



    We simplified constraint (54) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (55)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)



    We simplified constraint (55) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (56)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)



  • We consider the chain COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5]), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (57)    (450_0_ack_GT(-(x1[3], 1), x0[3])=629_0_ack_Return(x0[5], x2[5])∧-(x0[3], 1)=x3[5]∧x0[3]=x0[5]∧-(x1[3], 1)=0∧x2[5]=x1[2]∧x3[5]=x0[2] ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0)≥450_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (57) using rules (III), (IV), (VII) which results in the following new constraint:

    (58)    (-(x1[3], 1)=x4∧450_0_ack_GT(x4, x0[3])=629_0_ack_Return(x0[3], x2[5])∧-(x1[3], 1)=0 ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[3], x2[5]), -(x0[3], 1), x0[3], 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[3], x2[5]), -(x0[3], 1), x0[3], 0)≥450_0_ACK_GT(x2[5], -(x0[3], 1))∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (58) using rule (V) (with possible (I) afterwards) using induction on 450_0_ack_GT(x4, x0[3])=629_0_ack_Return(x0[3], x2[5]) which results in the following new constraint:

    (59)    (Cond_450_0_ack_GT(>=(x5, 0), x5, 0)=629_0_ack_Return(0, x2[5])∧-(x1[3], 1)=x5∧-(x1[3], 1)=0 ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, x2[5]), -(0, 1), 0, 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, x2[5]), -(0, 1), 0, 0)≥450_0_ACK_GT(x2[5], -(0, 1))∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (59) using rules (III), (VII) which results in the following new constraint:

    (60)    (Cond_450_0_ack_GT(>=(-(x1[3], 1), 0), -(x1[3], 1), 0)=629_0_ack_Return(0, x2[5])∧-(x1[3], 1)=0 ⇒ 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, x2[5]), -(0, 1), 0, 0)≥NonInfC∧680_1_ACK_INVOKEMETHOD(629_0_ack_Return(0, x2[5]), -(0, 1), 0, 0)≥450_0_ACK_GT(x2[5], -(0, 1))∧(UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥))



    We simplified constraint (60) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (61)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)



    We simplified constraint (61) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (62)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)



    We simplified constraint (62) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (63)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)



    We simplified constraint (63) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (64)    (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧0 ≥ 0∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧0 ≥ 0∧[(-1)bso_50] ≥ 0)







For Pair 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0), x1, x2, x3) → 450_0_ACK_GT(x0, x1) the following chains were created:
  • We consider the chain COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6]), 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) which results in the following constraint:

    (65)    (450_0_ack_GT(-(x1[3], 1), x0[3])=710_0_ack_Return(x0[6])∧-(x0[3], 1)=x1[6]∧x0[3]=x2[6]∧-(x1[3], 1)=x3[6]∧x0[6]=0∧x1[6]=x0[0] ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥450_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (65) using rules (III), (IV), (VII) which results in the following new constraint:

    (66)    (-(x1[3], 1)=x8∧450_0_ack_GT(x8, x0[3])=710_0_ack_Return(0) ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(0), -(x0[3], 1), x0[3], -(x1[3], 1))≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(0), -(x0[3], 1), x0[3], -(x1[3], 1))≥450_0_ACK_GT(0, -(x0[3], 1))∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (66) using rule (V) (with possible (I) afterwards) using induction on 450_0_ack_GT(x8, x0[3])=710_0_ack_Return(0) which results in the following new constraint:

    (67)    (Cond_450_0_ack_GT(>=(x9, 0), x9, 0)=710_0_ack_Return(0)∧-(x1[3], 1)=x9 ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(0), -(0, 1), 0, -(x1[3], 1))≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(0), -(0, 1), 0, -(x1[3], 1))≥450_0_ACK_GT(0, -(0, 1))∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (67) using rules (III), (VII) which results in the following new constraint:

    (68)    (Cond_450_0_ack_GT(>=(-(x1[3], 1), 0), -(x1[3], 1), 0)=710_0_ack_Return(0) ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(0), -(0, 1), 0, -(x1[3], 1))≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(0), -(0, 1), 0, -(x1[3], 1))≥450_0_ACK_GT(0, -(0, 1))∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (69)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)



    We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (70)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)



    We simplified constraint (70) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (71)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)



  • We consider the chain COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6]), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (72)    (450_0_ack_GT(-(x1[3], 1), x0[3])=710_0_ack_Return(x0[6])∧-(x0[3], 1)=x1[6]∧x0[3]=x2[6]∧-(x1[3], 1)=x3[6]∧x0[6]=x1[2]∧x1[6]=x0[2] ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥450_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (72) using rules (III), (IV), (VII) which results in the following new constraint:

    (73)    (-(x1[3], 1)=x12∧450_0_ack_GT(x12, x0[3])=710_0_ack_Return(x0[6]) ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), -(x0[3], 1), x0[3], -(x1[3], 1))≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), -(x0[3], 1), x0[3], -(x1[3], 1))≥450_0_ACK_GT(x0[6], -(x0[3], 1))∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (73) using rule (V) (with possible (I) afterwards) using induction on 450_0_ack_GT(x12, x0[3])=710_0_ack_Return(x0[6]) which results in the following new constraint:

    (74)    (Cond_450_0_ack_GT(>=(x13, 0), x13, 0)=710_0_ack_Return(x0[6])∧-(x1[3], 1)=x13 ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), -(0, 1), 0, -(x1[3], 1))≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), -(0, 1), 0, -(x1[3], 1))≥450_0_ACK_GT(x0[6], -(0, 1))∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (74) using rules (III), (VII) which results in the following new constraint:

    (75)    (Cond_450_0_ack_GT(>=(-(x1[3], 1), 0), -(x1[3], 1), 0)=710_0_ack_Return(x0[6]) ⇒ 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), -(0, 1), 0, -(x1[3], 1))≥NonInfC∧680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), -(0, 1), 0, -(x1[3], 1))≥450_0_ACK_GT(x0[6], -(0, 1))∧(UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥))



    We simplified constraint (75) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (76)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)



    We simplified constraint (76) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (77)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)



    We simplified constraint (77) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (78)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)



    We simplified constraint (78) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (79)    ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧0 ≥ 0∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧0 ≥ 0∧[(-1)bso_52] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 450_0_ACK_GT(0, x0) → COND_450_0_ACK_GT(>(x0, 0), 0, x0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧0 ≥ 0∧[(-1)bni_39 + (-1)Bound*bni_39] ≥ 0∧0 ≥ 0∧[(-1)bso_40] ≥ 0)

  • COND_450_0_ACK_GT(TRUE, 0, x0) → 450_0_ACK_GT(1, -(x0, 1))
    • (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧0 ≥ 0∧[(-1)bni_41 + (-1)Bound*bni_41] ≥ 0∧0 ≥ 0∧[(-1)bso_42] ≥ 0)

  • 450_0_ACK_GT(x1, x0) → COND_450_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_44] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_43 + (-1)Bound*bni_43] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_44] ≥ 0)

  • COND_450_0_ACK_GT1(TRUE, x1, x0) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1, 1), x0), -(x0, 1), x0, -(x1, 1))
    • (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_46] ≥ 0∧[1] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_45 + (-1)Bound*bni_45] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_46] ≥ 0∧[1] ≥ 0)

  • COND_450_0_ACK_GT1(TRUE, x1, x0) → 450_0_ACK_GT(-(x1, 1), x0)
    • (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_48] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_47 + (-1)Bound*bni_47] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_48] ≥ 0)

  • 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0, x2), x3, x0, 0) → 450_0_ACK_GT(x2, x3)
    • (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧[(-1)bso_50] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(x2[5], x3[5])), ≥)∧0 ≥ 0∧[(-1)bni_49 + (-1)Bound*bni_49] ≥ 0∧0 ≥ 0∧[(-1)bso_50] ≥ 0)

  • 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0), x1, x2, x3) → 450_0_ACK_GT(x0, x1)
    • ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧[(-1)bso_52] ≥ 0)
    • ((UIncreasing(450_0_ACK_GT(x0[6], x1[6])), ≥)∧0 ≥ 0∧[(-1)bni_51 + (-1)Bound*bni_51] ≥ 0∧0 ≥ 0∧[(-1)bso_52] ≥ 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(450_0_ack_GT(x1, x2)) = [1] + [3]x2   
POL(0) = 0   
POL(Cond_450_0_ack_GT(x1, x2, x3)) = [2] + [2]x2   
POL(>=(x1, x2)) = 0   
POL(502_0_ack_Return(x1, x2)) = [3] + [3]x1   
POL(+(x1, x2)) = 0   
POL(1) = 0   
POL(612_1_ack_InvokeMethod(x1, x2)) = 0   
POL(629_0_ack_Return(x1, x2)) = 0   
POL(710_0_ack_Return(x1)) = 0   
POL(701_1_ack_InvokeMethod(x1, x2, x3)) = 0   
POL(450_0_ACK_GT(x1, x2)) = [-1]   
POL(COND_450_0_ACK_GT(x1, x2, x3)) = [-1] + [-1]x2   
POL(>(x1, x2)) = 0   
POL(-(x1, x2)) = 0   
POL(COND_450_0_ACK_GT1(x1, x2, x3)) = [-1]   
POL(&&(x1, x2)) = 0   
POL(680_1_ACK_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x2 + [-1]x1   

The following pairs are in P>:

COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))

The following pairs are in Pbound:

450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])
COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1))
450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_450_0_ACK_GT1(TRUE, x1[3], x0[3]) → 680_1_ACK_INVOKEMETHOD(450_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))
COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])
680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5])
680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6])

The following pairs are in P≥:

450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])
COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1))
450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])
680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5])
680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6])

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

Cond_450_0_ack_GT(>=(x1, 0), x1, 0)1 → 450_0_ack_GT(x1, 0)1
502_0_ack_Return(x1, +(x1, 1))1 → Cond_450_0_ack_GT(TRUE, x1, 0)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:

Integer, Boolean


The ITRS R consists of the following rules:
450_0_ack_GT(x1, 0) → Cond_450_0_ack_GT(x1 >= 0, x1, 0)
Cond_450_0_ack_GT(TRUE, x1, 0) → 502_0_ack_Return(x1, x1 + 1)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x2), 0) → 629_0_ack_Return(x3, x2)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x3) → 629_0_ack_Return(x1, x0)
701_1_ack_InvokeMethod(502_0_ack_Return(x1, x2), 0, x1) → 710_0_ack_Return(x2)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2) → 710_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(x0[0] > 0, 0, x0[0])
(1): COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, x0[1] - 1)
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(x1[4] - 1, x0[4])
(5): 680_1_ACK_INVOKEMETHOD(629_0_ack_Return(x0[5], x2[5]), x3[5], x0[5], 0) → 450_0_ACK_GT(x2[5], x3[5])
(6): 680_1_ACK_INVOKEMETHOD(710_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 450_0_ACK_GT(x0[6], x1[6])

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


(4) -> (0), if ((x1[4] - 1 →* 0)∧(x0[4] →* x0[0]))


(5) -> (0), if ((x2[5] →* 0)∧(x3[5] →* x0[0]))


(6) -> (0), if ((x0[6] →* 0)∧(x1[6] →* x0[0]))


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


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


(4) -> (2), if ((x1[4] - 1 →* x1[2])∧(x0[4] →* x0[2]))


(5) -> (2), if ((x2[5] →* x1[2])∧(x3[5] →* x0[2]))


(6) -> (2), if ((x0[6] →* x1[2])∧(x1[6] →* x0[2]))


(2) -> (4), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[4])∧(x0[2] →* x0[4]))



The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(7) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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


The following domains are used:

Integer, Boolean


The ITRS R consists of the following rules:
450_0_ack_GT(x1, 0) → Cond_450_0_ack_GT(x1 >= 0, x1, 0)
Cond_450_0_ack_GT(TRUE, x1, 0) → 502_0_ack_Return(x1, x1 + 1)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x2), 0) → 629_0_ack_Return(x3, x2)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x3) → 629_0_ack_Return(x1, x0)
701_1_ack_InvokeMethod(502_0_ack_Return(x1, x2), 0, x1) → 710_0_ack_Return(x2)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2) → 710_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(4): COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(x1[4] - 1, x0[4])
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(1): COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, x0[1] - 1)
(0): 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(x0[0] > 0, 0, x0[0])

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


(4) -> (0), if ((x1[4] - 1 →* 0)∧(x0[4] →* x0[0]))


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


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


(4) -> (2), if ((x1[4] - 1 →* x1[2])∧(x0[4] →* x0[2]))


(2) -> (4), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[4])∧(x0[2] →* x0[4]))



The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(9) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(4): COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(x1[4] - 1, x0[4])
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(1): COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, x0[1] - 1)
(0): 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(x0[0] > 0, 0, x0[0])

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


(4) -> (0), if ((x1[4] - 1 →* 0)∧(x0[4] →* x0[0]))


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


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


(4) -> (2), if ((x1[4] - 1 →* x1[2])∧(x0[4] →* x0[2]))


(2) -> (4), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[4])∧(x0[2] →* x0[4]))



The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(11) 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_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]), 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) which results in the following constraint:

    (1)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4]∧-(x1[4], 1)=0∧x0[4]=x0[0] ⇒ COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥450_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (2)    (-(x1[2], 1)=0∧>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥450_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (3)    (x1[2] + [-1] ≥ 0∧x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (4)    (x1[2] + [-1] ≥ 0∧x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (5)    (x1[2] + [-1] ≥ 0∧x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (6)    (x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (7)    (x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (8)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4]∧-(x1[4], 1)=x1[2]1∧x0[4]=x0[2]1 ⇒ COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥450_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (9)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥450_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (13)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)







For Pair 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (15)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4] ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (16)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (17)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (18)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (19)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (20)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (21)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)) the following chains were created:
  • We consider the chain 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)), 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) which results in the following constraint:

    (22)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧1=0∧-(x0[1], 1)=x0[0]1 ⇒ COND_450_0_ACK_GT(TRUE, 0, x0[1])≥NonInfC∧COND_450_0_ACK_GT(TRUE, 0, x0[1])≥450_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥))



    We solved constraint (22) using rules (I), (II).
  • We consider the chain 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (23)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧1=x1[2]∧-(x0[1], 1)=x0[2] ⇒ COND_450_0_ACK_GT(TRUE, 0, x0[1])≥NonInfC∧COND_450_0_ACK_GT(TRUE, 0, x0[1])≥450_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥))



    We simplified constraint (23) using rules (III), (IV) which results in the following new constraint:

    (24)    (>(x0[0], 0)=TRUE ⇒ COND_450_0_ACK_GT(TRUE, 0, x0[0])≥NonInfC∧COND_450_0_ACK_GT(TRUE, 0, x0[0])≥450_0_ACK_GT(1, -(x0[0], 1))∧(UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥))



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

    (25)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (26)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (27)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (28)    (x0[0] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)







For Pair 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]) the following chains were created:
  • We consider the chain 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0]), COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

    (29)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 450_0_ACK_GT(0, x0[0])≥NonInfC∧450_0_ACK_GT(0, x0[0])≥COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))



    We simplified constraint (29) using rule (IV) which results in the following new constraint:

    (30)    (>(x0[0], 0)=TRUE ⇒ 450_0_ACK_GT(0, x0[0])≥NonInfC∧450_0_ACK_GT(0, x0[0])≥COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])∧(UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥))



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

    (31)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (32)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (33)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (34)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])
    • (x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

  • 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1))
    • (x0[0] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

  • 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 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) = [3]   
POL(COND_450_0_ACK_GT1(x1, x2, x3)) = [-1] + x3 + [-1]x1   
POL(450_0_ACK_GT(x1, x2)) = [-1] + x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(COND_450_0_ACK_GT(x1, x2, x3)) = [-1] + x3 + [-1]x2   

The following pairs are in P>:

COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1))

The following pairs are in Pbound:

COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])
450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_450_0_ACK_GT(TRUE, 0, x0[1]) → 450_0_ACK_GT(1, -(x0[1], 1))
450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])

The following pairs are in P≥:

COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])
450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(>(x0[0], 0), 0, x0[0])

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

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

(12) 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:
(4): COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(x1[4] - 1, x0[4])
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(0): 450_0_ACK_GT(0, x0[0]) → COND_450_0_ACK_GT(x0[0] > 0, 0, x0[0])

(4) -> (0), if ((x1[4] - 1 →* 0)∧(x0[4] →* x0[0]))


(4) -> (2), if ((x1[4] - 1 →* x1[2])∧(x0[4] →* x0[2]))


(2) -> (4), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[4])∧(x0[2] →* x0[4]))



The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(x1[4] - 1, x0[4])

(4) -> (2), if ((x1[4] - 1 →* x1[2])∧(x0[4] →* x0[2]))


(2) -> (4), if ((x1[2] > 0 && x0[2] > 0 →* TRUE)∧(x1[2] →* x1[4])∧(x0[2] →* x0[4]))



The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(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 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

    (1)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4] ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (2)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ 450_0_ACK_GT(x1[2], x0[2])≥NonInfC∧450_0_ACK_GT(x1[2], x0[2])≥COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))



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

    (3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [(2)bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [(2)bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [(2)bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (6)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [(2)bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(4)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [(2)bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)







For Pair COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
  • We consider the chain 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4]), 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

    (8)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE∧x1[2]=x1[4]∧x0[2]=x0[4]∧-(x1[4], 1)=x1[2]1∧x0[4]=x0[2]1 ⇒ COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[4], x0[4])≥450_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (9)    (>(x1[2], 0)=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_450_0_ACK_GT1(TRUE, x1[2], x0[2])≥450_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))



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

    (10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[2] + [(2)bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[2] + [(2)bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[2] + [(2)bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (13)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [bni_14]x0[2] + [(2)bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_14 + (3)bni_14] + [bni_14]x0[2] + [(2)bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(4)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[2] + [(2)bni_12]x1[2] ≥ 0∧[(-1)bso_13] ≥ 0)

  • COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(450_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_14 + (3)bni_14] + [bni_14]x0[2] + [(2)bni_14]x1[2] ≥ 0∧[1 + (-1)bso_15] ≥ 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) = [2]   
POL(450_0_ACK_GT(x1, x2)) = [1] + x2 + [2]x1   
POL(COND_450_0_ACK_GT1(x1, x2, x3)) = x3 + [2]x2 + [-1]x1   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(1) = [1]   

The following pairs are in P>:

COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in Pbound:

450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_450_0_ACK_GT1(TRUE, x1[4], x0[4]) → 450_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in P≥:

450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

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

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): 450_0_ACK_GT(x1[2], x0[2]) → COND_450_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])


The set Q consists of the following terms:
450_0_ack_GT(x0, 0)
Cond_450_0_ack_GT(TRUE, x0, 0)
612_1_ack_InvokeMethod(502_0_ack_Return(1, x0), 0)
612_1_ack_InvokeMethod(710_0_ack_Return(x0), x1)
701_1_ack_InvokeMethod(502_0_ack_Return(x0, x1), 0, x0)
701_1_ack_InvokeMethod(710_0_ack_Return(x0), x1, x2)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE