Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 IDP
↳5 IDPNonInfProof (⇒)
↳6 AND
    ↳7 IDP
        ↳8 IDependencyGraphProof (⇔)
        ↳9 IDP
        ↳10 UsableRulesProof (⇔)
        ↳11 IDP
        ↳12 IDPNonInfProof (⇒)
        ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 AND
            ↳19 IDP
                ↳20 UsableRulesProof (⇔)
                ↳21 IDP
                ↳22 IDPNonInfProof (⇒)
                ↳23 IDP
                ↳24 IDependencyGraphProof (⇔)
                ↳25 TRUE
            ↳26 IDP
                ↳27 UsableRulesProof (⇔)
                ↳28 IDP
                ↳29 IDPNonInfProof (⇒)
                ↳30 IDP
                ↳31 IDependencyGraphProof (⇔)
                ↳32 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB18 
/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB18 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > 0 && y > 0) {
            if (x > y) {
                while (x > 0) {
                    x--;
                }
            } else {
                while (y > 0) {
                    y--;
                }
            }
        }        
    }
}


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

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
PastaB18.main([Ljava/lang/String;)V: Graph of 177 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 32 rules for P and 5 rules for R.


Combined rules. Obtained 6 rules for P and 1 rules for R.


Filtered ground terms:


816_0_main_LE(x1, x2, x3, x4) → 816_0_main_LE(x2, x3, x4)
Cond_816_0_main_LE(x1, x2, x3, x4, x5) → Cond_816_0_main_LE(x1, x3, x4, x5)
529_0_main_LE(x1, x2, x3, x4) → 529_0_main_LE(x2, x3, x4)
Cond_529_0_main_LE1(x1, x2, x3, x4, x5) → Cond_529_0_main_LE1(x1, x3, x4, x5)
814_0_main_LE(x1, x2, x3, x4) → 814_0_main_LE(x2, x3, x4)
Cond_814_0_main_LE(x1, x2, x3, x4, x5) → Cond_814_0_main_LE(x1, x3, x4, x5)
Cond_529_0_main_LE(x1, x2, x3, x4, x5) → Cond_529_0_main_LE(x1, x3, x4, x5)
593_0_main_Return(x1) → 593_0_main_Return

Filtered duplicate args:


816_0_main_LE(x1, x2, x3) → 816_0_main_LE(x2, x3)
Cond_816_0_main_LE(x1, x2, x3, x4) → Cond_816_0_main_LE(x1, x3, x4)
529_0_main_LE(x1, x2, x3) → 529_0_main_LE(x2, x3)
Cond_529_0_main_LE1(x1, x2, x3, x4) → Cond_529_0_main_LE1(x1, x3, x4)
814_0_main_LE(x1, x2, x3) → 814_0_main_LE(x1, x3)
Cond_814_0_main_LE(x1, x2, x3, x4) → Cond_814_0_main_LE(x1, x2, x4)
Cond_529_0_main_LE(x1, x2, x3, x4) → Cond_529_0_main_LE(x1, x3, x4)

Combined rules. Obtained 6 rules for P and 1 rules for R.


Finished conversion. Obtained 6 rules for P and 1 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


The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(x1[0] >= x0[0] && x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
(2): 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(5): 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0, x1[5], x0[5])
(6): COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6])
(7): 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)

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


(1) -> (2), if ((x0[1] →* x0[2])∧(x1[1] →* 0))


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


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


(2) -> (5), if ((0 →* x1[5])∧(x0[2] →* x0[5]))


(3) -> (4), if ((x1[3] > 0 →* TRUE)∧(x0[3] →* x0[4])∧(x1[3] →* x1[4]))


(4) -> (2), if ((x0[4] →* x0[2])∧(x1[4] + -1 →* 0))


(4) -> (3), if ((x0[4] →* x0[3])∧(x1[4] + -1 →* x1[3]))


(5) -> (6), if ((x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0 →* TRUE)∧(x1[5] →* x1[6])∧(x0[5] →* x0[6]))


(6) -> (7), if ((x1[6] →* x1[7])∧(x0[6] →* 0))


(6) -> (8), if ((x1[6] →* x1[8])∧(x0[6] →* x0[8]))


(7) -> (0), if ((x1[7] →* x1[0])∧(0 →* x0[0]))


(7) -> (5), if ((x1[7] →* x1[5])∧(0 →* x0[5]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))


(9) -> (7), if ((x1[9] →* x1[7])∧(x0[9] + -1 →* 0))


(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(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 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]), COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1]) which results in the following constraint:

    (1)    (&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 529_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧529_0_MAIN_LE(x1[0], x0[0])≥COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE∧>=(x1[0], x0[0])=TRUE∧>(x1[0], 0)=TRUE ⇒ 529_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧529_0_MAIN_LE(x1[0], x0[0])≥COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(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] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 0)



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

    (6)    (x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 0)



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

    (7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36 + (2)bni_36] + [(3)bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[1 + (-1)bso_37] + x0[0] + x1[0] ≥ 0)







For Pair COND_529_0_MAIN_LE(TRUE, x1, x0) → 814_0_MAIN_LE(x0, x1) the following chains were created:
  • We consider the chain COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1]), 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]) which results in the following constraint:

    (8)    (x0[1]=x0[2]∧x1[1]=0 ⇒ COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥814_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))



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

    (9)    (COND_529_0_MAIN_LE(TRUE, 0, x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, 0, x0[1])≥814_0_MAIN_LE(x0[1], 0)∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))



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

    (10)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)



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

    (11)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)



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

    (12)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)



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

    (13)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧[(-1)bso_39] ≥ 0)



  • We consider the chain COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1]), 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

    (14)    (x0[1]=x0[3]∧x1[1]=x1[3] ⇒ COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥814_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))



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

    (15)    (COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥814_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))



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

    (16)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)



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

    (17)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)



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

    (18)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)



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

    (19)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_39] ≥ 0)







For Pair 814_0_MAIN_LE(x0, 0) → 529_0_MAIN_LE(0, x0) the following chains were created:
  • We consider the chain 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]), 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

    (20)    (0=x1[0]∧x0[2]=x0[0] ⇒ 814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))



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

    (21)    (814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))



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

    (22)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)



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

    (23)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)



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

    (24)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)



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

    (25)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)



  • We consider the chain 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]), 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:

    (26)    (0=x1[5]∧x0[2]=x0[5] ⇒ 814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))



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

    (27)    (814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))



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

    (28)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)



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

    (29)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)



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

    (30)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)



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

    (31)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)







For Pair 814_0_MAIN_LE(x0, x1) → COND_814_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:
  • We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:

    (32)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))



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

    (33)    (>(x1[3], 0)=TRUE ⇒ 814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))



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

    (34)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] + [bni_42]x0[3] ≥ 0∧[(-1)bso_43] ≥ 0)



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

    (35)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] + [bni_42]x0[3] ≥ 0∧[(-1)bso_43] ≥ 0)



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

    (36)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] + [bni_42]x0[3] ≥ 0∧[(-1)bso_43] ≥ 0)



    We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (37)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)



    We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (38)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_42] = 0∧[(-1)Bound*bni_42] + [bni_42]x1[3] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)







For Pair COND_814_0_MAIN_LE(TRUE, x0, x1) → 814_0_MAIN_LE(x0, +(x1, -1)) the following chains were created:
  • We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)), 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]) which results in the following constraint:

    (39)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x0[2]∧+(x1[4], -1)=0 ⇒ COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥814_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))



    We simplified constraint (39) using rules (III), (IV) which results in the following new constraint:

    (40)    (>(x1[3], 0)=TRUE∧+(x1[3], -1)=0 ⇒ COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥814_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))



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

    (41)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)



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

    (42)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)



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

    (43)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)



    We simplified constraint (43) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (44)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)



    We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (45)    (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)



  • We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)), 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

    (46)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x0[3]1∧+(x1[4], -1)=x1[3]1 ⇒ COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥814_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))



    We simplified constraint (46) using rules (III), (IV) which results in the following new constraint:

    (47)    (>(x1[3], 0)=TRUE ⇒ COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥814_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))



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

    (48)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)



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

    (49)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)



    We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (50)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)



    We simplified constraint (50) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (51)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)



    We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (52)    (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)







For Pair 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0) the following chains were created:
  • We consider the chain 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]), COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6]) which results in the following constraint:

    (53)    (&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0))=TRUE∧x1[5]=x1[6]∧x0[5]=x0[6] ⇒ 529_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧529_0_MAIN_LE(x1[5], x0[5])≥COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))



    We simplified constraint (53) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (54)    (>(x0[5], 0)=TRUE∧>(x1[5], 0)=TRUE∧<(x1[5], x0[5])=TRUE ⇒ 529_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧529_0_MAIN_LE(x1[5], x0[5])≥COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))



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

    (55)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-2 + (-1)bso_47] + x0[5] ≥ 0)



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

    (56)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-2 + (-1)bso_47] + x0[5] ≥ 0)



    We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (57)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-2 + (-1)bso_47] + x0[5] ≥ 0)



    We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (58)    (x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-1 + (-1)bso_47] + x0[5] ≥ 0)



    We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (59)    (x1[5] + x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_46] + [(3)bni_46]x1[5] + [bni_46]x0[5] ≥ 0∧[-1 + (-1)bso_47] + x1[5] + x0[5] ≥ 0)



    We simplified constraint (59) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (60)    ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(3)bni_46 + (-1)Bound*bni_46] + [(3)bni_46]x1[5] + [bni_46]x0[5] ≥ 0∧[(-1)bso_47] + x1[5] + x0[5] ≥ 0)







For Pair COND_529_0_MAIN_LE1(TRUE, x1, x0) → 816_0_MAIN_LE(x1, x0) the following chains were created:
  • We consider the chain COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6]), 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0) which results in the following constraint:

    (61)    (x1[6]=x1[7]∧x0[6]=0 ⇒ COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥816_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))



    We simplified constraint (61) using rules (III), (IV) which results in the following new constraint:

    (62)    (COND_529_0_MAIN_LE1(TRUE, x1[6], 0)≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], 0)≥816_0_MAIN_LE(x1[6], 0)∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))



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

    (63)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)



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

    (64)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)



    We simplified constraint (64) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (65)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)



    We simplified constraint (65) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (66)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧[2 + (-1)bso_49] ≥ 0)



  • We consider the chain COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6]), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

    (67)    (x1[6]=x1[8]∧x0[6]=x0[8] ⇒ COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥816_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))



    We simplified constraint (67) using rule (IV) which results in the following new constraint:

    (68)    (COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥816_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))



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

    (69)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)



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

    (70)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)



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

    (71)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)



    We simplified constraint (71) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (72)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_49] ≥ 0)







For Pair 816_0_MAIN_LE(x1, 0) → 529_0_MAIN_LE(x1, 0) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0), 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

    (73)    (x1[7]=x1[0]∧0=x0[0] ⇒ 816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))



    We simplified constraint (73) using rule (IV) which results in the following new constraint:

    (74)    (816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))



    We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (75)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)



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

    (76)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)



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

    (77)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)



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

    (78)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)



  • We consider the chain 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0), 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:

    (79)    (x1[7]=x1[5]∧0=x0[5] ⇒ 816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))



    We simplified constraint (79) using rule (IV) which results in the following new constraint:

    (80)    (816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))



    We simplified constraint (80) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (81)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)



    We simplified constraint (81) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (82)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)



    We simplified constraint (82) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (83)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)



    We simplified constraint (83) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (84)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)







For Pair 816_0_MAIN_LE(x1, x0) → COND_816_0_MAIN_LE(>(x0, 0), x1, x0) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) which results in the following constraint:

    (85)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9] ⇒ 816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))



    We simplified constraint (85) using rule (IV) which results in the following new constraint:

    (86)    (>(x0[8], 0)=TRUE ⇒ 816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))



    We simplified constraint (86) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (87)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_52 + (-1)Bound*bni_52] + [(2)bni_52]x1[8] ≥ 0∧[(-1)bso_53] ≥ 0)



    We simplified constraint (87) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (88)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_52 + (-1)Bound*bni_52] + [(2)bni_52]x1[8] ≥ 0∧[(-1)bso_53] ≥ 0)



    We simplified constraint (88) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (89)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_52 + (-1)Bound*bni_52] + [(2)bni_52]x1[8] ≥ 0∧[(-1)bso_53] ≥ 0)



    We simplified constraint (89) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (90)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(2)bni_52] = 0∧[(-1)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 = 0∧[(-1)bso_53] ≥ 0)



    We simplified constraint (90) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (91)    (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(2)bni_52] = 0∧[(-1)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 = 0∧[(-1)bso_53] ≥ 0)







For Pair COND_816_0_MAIN_LE(TRUE, x1, x0) → 816_0_MAIN_LE(x1, +(x0, -1)) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0) which results in the following constraint:

    (92)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9]∧x1[9]=x1[7]∧+(x0[9], -1)=0 ⇒ COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



    We simplified constraint (92) using rules (III), (IV) which results in the following new constraint:

    (93)    (>(x0[8], 0)=TRUE∧+(x0[8], -1)=0 ⇒ COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



    We simplified constraint (93) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (94)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (94) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (95)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (95) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (96)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (96) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (97)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (97) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (98)    (x0[8] ≥ 0∧x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)



  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

    (99)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9]∧x1[9]=x1[8]1∧+(x0[9], -1)=x0[8]1 ⇒ COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



    We simplified constraint (99) using rules (III), (IV) which results in the following new constraint:

    (100)    (>(x0[8], 0)=TRUE ⇒ COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



    We simplified constraint (100) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (101)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (101) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (102)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (102) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (103)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (103) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (104)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)



    We simplified constraint (104) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (105)    (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0)
    • (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36 + (2)bni_36] + [(3)bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[1 + (-1)bso_37] + x0[0] + x1[0] ≥ 0)

  • COND_529_0_MAIN_LE(TRUE, x1, x0) → 814_0_MAIN_LE(x0, x1)
    • ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧[(-1)bso_39] ≥ 0)
    • ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_39] ≥ 0)

  • 814_0_MAIN_LE(x0, 0) → 529_0_MAIN_LE(0, x0)
    • ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)
    • ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)

  • 814_0_MAIN_LE(x0, x1) → COND_814_0_MAIN_LE(>(x1, 0), x0, x1)
    • (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_42] = 0∧[(-1)Bound*bni_42] + [bni_42]x1[3] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

  • COND_814_0_MAIN_LE(TRUE, x0, x1) → 814_0_MAIN_LE(x0, +(x1, -1))
    • (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)
    • (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

  • 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0)
    • ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(3)bni_46 + (-1)Bound*bni_46] + [(3)bni_46]x1[5] + [bni_46]x0[5] ≥ 0∧[(-1)bso_47] + x1[5] + x0[5] ≥ 0)

  • COND_529_0_MAIN_LE1(TRUE, x1, x0) → 816_0_MAIN_LE(x1, x0)
    • ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧[2 + (-1)bso_49] ≥ 0)
    • ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_49] ≥ 0)

  • 816_0_MAIN_LE(x1, 0) → 529_0_MAIN_LE(x1, 0)
    • ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)
    • ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)

  • 816_0_MAIN_LE(x1, x0) → COND_816_0_MAIN_LE(>(x0, 0), x1, x0)
    • (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(2)bni_52] = 0∧[(-1)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 = 0∧[(-1)bso_53] ≥ 0)

  • COND_816_0_MAIN_LE(TRUE, x1, x0) → 816_0_MAIN_LE(x1, +(x0, -1))
    • (x0[8] ≥ 0∧x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)
    • (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 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(529_0_main_LE(x1, x2)) = [-1] + [-1]x2 + [-1]x1   
POL(0) = 0   
POL(593_0_main_Return) = [-1]   
POL(529_0_MAIN_LE(x1, x2)) = [-1] + x2 + [2]x1   
POL(COND_529_0_MAIN_LE(x1, x2, x3)) = [-1] + x3 + x2   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(814_0_MAIN_LE(x1, x2)) = [-1] + x2 + x1   
POL(COND_814_0_MAIN_LE(x1, x2, x3)) = [-1] + x3 + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_529_0_MAIN_LE1(x1, x2, x3)) = [1] + [2]x2   
POL(<(x1, x2)) = [-1]   
POL(816_0_MAIN_LE(x1, x2)) = [-1] + [2]x1   
POL(COND_816_0_MAIN_LE(x1, x2, x3)) = [-1] + [2]x2   

The following pairs are in P>:

529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))
COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6])

The following pairs are in Pbound:

529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])

The following pairs are in P≥:

COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])
816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])
COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))

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

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

(6) Complex Obligation (AND)

(7) Obligation:

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


The following domains are used:

Integer, Boolean


The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(1): COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
(2): 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(5): 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0, x1[5], x0[5])
(7): 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)

(1) -> (2), if ((x0[1] →* x0[2])∧(x1[1] →* 0))


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


(2) -> (5), if ((0 →* x1[5])∧(x0[2] →* x0[5]))


(7) -> (5), if ((x1[7] →* x1[5])∧(0 →* x0[5]))


(9) -> (7), if ((x1[9] →* x1[7])∧(x0[9] + -1 →* 0))


(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(8) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

(9) Obligation:

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


The following domains are used:

Integer


The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(10) 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.

(11) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(12) 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_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

    (1)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9]∧x1[9]=x1[8]1∧+(x0[9], -1)=x0[8]1 ⇒ COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



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

    (2)    (>(x0[8], 0)=TRUE ⇒ COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



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

    (3)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) which results in the following constraint:

    (8)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9] ⇒ 816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))



    We simplified constraint (8) using rule (IV) which results in the following new constraint:

    (9)    (>(x0[8], 0)=TRUE ⇒ 816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))



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

    (10)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (13)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)



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

    (14)    (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
    • (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])
    • (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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(COND_816_0_MAIN_LE(x1, x2, x3)) = [-1] + x3   
POL(816_0_MAIN_LE(x1, x2)) = [-1] + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))

The following pairs are in Pbound:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

The following pairs are in P≥:

816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

There are no usable rules.

(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:
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])


The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(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


The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(1): COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
(2): 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(6): COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6])
(7): 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)

(1) -> (2), if ((x0[1] →* x0[2])∧(x1[1] →* 0))


(4) -> (2), if ((x0[4] →* x0[2])∧(x1[4] + -1 →* 0))


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


(4) -> (3), if ((x0[4] →* x0[3])∧(x1[4] + -1 →* x1[3]))


(3) -> (4), if ((x1[3] > 0 →* TRUE)∧(x0[3] →* x0[4])∧(x1[3] →* x1[4]))


(6) -> (7), if ((x1[6] →* x1[7])∧(x0[6] →* 0))


(9) -> (7), if ((x1[9] →* x1[7])∧(x0[9] + -1 →* 0))


(6) -> (8), if ((x1[6] →* x1[8])∧(x0[6] →* x0[8]))


(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

(18) Complex Obligation (AND)

(19) 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


The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

(4) -> (3), if ((x0[4] →* x0[3])∧(x1[4] + -1 →* x1[3]))


(3) -> (4), if ((x1[3] > 0 →* TRUE)∧(x0[3] →* x0[4])∧(x1[3] →* x1[4]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(20) 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.

(21) 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:
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

(4) -> (3), if ((x0[4] →* x0[3])∧(x1[4] + -1 →* x1[3]))


(3) -> (4), if ((x1[3] > 0 →* TRUE)∧(x0[3] →* x0[4])∧(x1[3] →* x1[4]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(22) 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_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)) the following chains were created:
  • We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)), 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

    (1)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧x0[4]=x0[3]1∧+(x1[4], -1)=x1[3]1 ⇒ COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥814_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))



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

    (2)    (>(x1[3], 0)=TRUE ⇒ COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥814_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))



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

    (3)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) the following chains were created:
  • We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:

    (8)    (>(x1[3], 0)=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))



    We simplified constraint (8) using rule (IV) which results in the following new constraint:

    (9)    (>(x1[3], 0)=TRUE ⇒ 814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))



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

    (10)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (13)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)



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

    (14)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))
    • (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
    • (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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(COND_814_0_MAIN_LE(x1, x2, x3)) = [-1] + x3   
POL(814_0_MAIN_LE(x1, x2)) = [-1] + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))

The following pairs are in Pbound:

COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))
814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

The following pairs are in P≥:

814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

There are no usable rules.

(23) 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:
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])


The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(24) IDependencyGraphProof (EQUIVALENT transformation)

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

(25) TRUE

(26) 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


The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(27) 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.

(28) 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:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9] →* x1[8])∧(x0[9] + -1 →* x0[8]))


(8) -> (9), if ((x0[8] > 0 →* TRUE)∧(x1[8] →* x1[9])∧(x0[8] →* x0[9]))



The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(29) 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_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

    (1)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9]∧x1[9]=x1[8]1∧+(x0[9], -1)=x0[8]1 ⇒ COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



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

    (2)    (>(x0[8], 0)=TRUE ⇒ COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))



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

    (3)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) the following chains were created:
  • We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) which results in the following constraint:

    (8)    (>(x0[8], 0)=TRUE∧x1[8]=x1[9]∧x0[8]=x0[9] ⇒ 816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))



    We simplified constraint (8) using rule (IV) which results in the following new constraint:

    (9)    (>(x0[8], 0)=TRUE ⇒ 816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))



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

    (10)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (13)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)



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

    (14)    (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
    • (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])
    • (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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(COND_816_0_MAIN_LE(x1, x2, x3)) = [-1] + x3   
POL(816_0_MAIN_LE(x1, x2)) = [-1] + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))

The following pairs are in Pbound:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

The following pairs are in P≥:

816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

There are no usable rules.

(30) 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:
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])


The set Q consists of the following terms:
529_0_main_LE(x0, 0)

(31) IDependencyGraphProof (EQUIVALENT transformation)

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

(32) TRUE