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 IDPNonInfProof (⇒)
↳10 IDP
↳11 IDependencyGraphProof (⇔)
↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Double 
/**
 * A recursive loop.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that the call to <tt>test()</tt> terminates.
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Double {

    private static void test(int n) {
	for (int i = 0; i < n; i++)
	    test(i);
    }

    public static void main(String[] args) {
	test(10);
    }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Double.main([Ljava/lang/String;)V: Graph of 12 nodes with 0 SCCs.

Double.test(I)V: Graph of 20 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 18 rules for P and 2 rules for R.


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


Filtered ground terms:


154_0_test_Store(x1, x2, x3) → 154_0_test_Store(x2)
Cond_264_1_test_InvokeMethod(x1, x2, x3, x4, x5) → Cond_264_1_test_InvokeMethod(x1, x3, x4, x5)
254_0_test_Return(x1) → 254_0_test_Return
Cond_154_0_test_Store(x1, x2, x3, x4) → Cond_154_0_test_Store(x1, x3)

Filtered duplicate args:


264_1_test_InvokeMethod(x1, x2, x3, x4) → 264_1_test_InvokeMethod(x1, x2, x4)
Cond_264_1_test_InvokeMethod(x1, x2, x3, x4) → Cond_264_1_test_InvokeMethod(x1, x2, x4)

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


Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.


(4) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(x0[0] > 0, x0[0])
(1): COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
(3): 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(x1[3] >= 0 && x0[3] > x1[3] + 1, 254_0_test_Return, x0[3], x1[3])
(4): COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(x1[4] + 1), x0[4], x1[4] + 1)
(5): COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(x1[5] + 1)

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


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


(1) -> (3), if ((154_0_test_Store(0) →* 254_0_test_Return)∧(x0[1] →* x0[3])∧(0 →* x1[3]))


(2) -> (0), if ((0 →* x0[0]))


(3) -> (4), if ((x1[3] >= 0 && x0[3] > x1[3] + 1 →* TRUE)∧(x0[3] →* x0[4])∧(x1[3] →* x1[4]))


(3) -> (5), if ((x1[3] >= 0 && x0[3] > x1[3] + 1 →* TRUE)∧(x0[3] →* x0[5])∧(x1[3] →* x1[5]))


(4) -> (3), if ((154_0_test_Store(x1[4] + 1) →* 254_0_test_Return)∧(x0[4] →* x0[3])∧(x1[4] + 1 →* x1[3]))


(5) -> (0), if ((x1[5] + 1 →* x0[0]))



The set Q is empty.

(5) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair 154_0_TEST_STORE(x0) → COND_154_0_TEST_STORE(>(x0, 0), x0) the following chains were created:
  • We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ 154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (3)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (4)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (5)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (6)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)



  • We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (7) using rule (IV) which results in the following new constraint:

    (8)    (>(x0[0], 0)=TRUE ⇒ 154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (9)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (10)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (11)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] + x0[0] ≥ 0)



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

    (12)    (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_154_0_TEST_STORE(TRUE, x0) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0, 0) the following chains were created:
  • We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0), 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]) which results in the following constraint:

    (13)    (>(x0[0], 0)=TRUE∧x0[0]=x0[1]∧154_0_test_Store(0)=254_0_test_Return∧x0[1]=x0[3]∧0=x1[3] ⇒ COND_154_0_TEST_STORE(TRUE, x0[1])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[1])≥264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)∧(UIncreasing(264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)), ≥))



    We solved constraint (13) using rules (I), (II).




For Pair COND_154_0_TEST_STORE(TRUE, x0) → 154_0_TEST_STORE(0) the following chains were created:
  • We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0), 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (14)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2]∧0=x0[0]1 ⇒ COND_154_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[2])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))



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

    (15)    (>(x0[0], 0)=TRUE ⇒ COND_154_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[0])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))



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

    (16)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (17)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (18)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (19)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0, x1) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), 254_0_test_Return, x0, x1) the following chains were created:
  • We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)) which results in the following constraint:

    (20)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥))



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

    (21)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE ⇒ 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥))



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

    (22)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (23)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (24)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (25)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)



  • We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1)) which results in the following constraint:

    (26)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5] ⇒ 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥))



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

    (27)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE ⇒ 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥NonInfC∧264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3])≥COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])∧(UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥))



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

    (28)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (29)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (30)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[3] + [bni_22]x0[3] ≥ 0∧[2 + (-1)bso_23] + x1[3] + x0[3] ≥ 0)



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

    (31)    (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)







For Pair COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1, 1)), x0, +(x1, 1)) the following chains were created:
  • We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1)), 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]) which results in the following constraint:

    (32)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧154_0_test_Store(+(x1[4], 1))=254_0_test_Return∧x0[4]=x0[3]1∧+(x1[4], 1)=x1[3]1 ⇒ COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4])≥NonInfC∧COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4])≥264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))∧(UIncreasing(264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))), ≥))



    We solved constraint (32) using rules (I), (II).




For Pair COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 154_0_TEST_STORE(+(x1, 1)) the following chains were created:
  • We consider the chain 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3]), COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1)), 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (33)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5]∧+(x1[5], 1)=x0[0] ⇒ COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5])≥NonInfC∧COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5])≥154_0_TEST_STORE(+(x1[5], 1))∧(UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥))



    We simplified constraint (33) using rules (III), (IV) which results in the following new constraint:

    (34)    (&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1)))=TRUE ⇒ COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[3], x1[3])≥NonInfC∧COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[3], x1[3])≥154_0_TEST_STORE(+(x1[3], 1))∧(UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥))



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

    (35)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (36)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (37)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (38)    (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 154_0_TEST_STORE(x0) → COND_154_0_TEST_STORE(>(x0, 0), x0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[bni_18] ≥ 0∧[(-1)bni_18 + (-1)Bound*bni_18] ≥ 0∧[1] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_154_0_TEST_STORE(TRUE, x0) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0, 0)

  • COND_154_0_TEST_STORE(TRUE, x0) → 154_0_TEST_STORE(0)
    • (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧0 ≥ 0∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧[(-1)bso_21] ≥ 0)

  • 264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0, x1) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1, 0), >(x0, +(x1, 1))), 254_0_test_Return, x0, x1)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)
    • (0 ≥ 0 ⇒ (UIncreasing(COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])), ≥)∧[bni_22] ≥ 0∧[bni_22] ≥ 0∧[bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_23] ≥ 0)

  • COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1, 1)), x0, +(x1, 1))

  • COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0, x1) → 154_0_TEST_STORE(+(x1, 1))
    • (0 ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(+(x1[5], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 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(154_0_TEST_STORE(x1)) = [-1] + x1   
POL(COND_154_0_TEST_STORE(x1, x2)) = [-1]   
POL(>(x1, x2)) = 0   
POL(0) = 0   
POL(264_1_TEST_INVOKEMETHOD(x1, x2, x3)) = [1] + x3 + x2 + [-1]x1   
POL(154_0_test_Store(x1)) = 0   
POL(254_0_test_Return) = 0   
POL(COND_264_1_TEST_INVOKEMETHOD(x1, x2, x3, x4)) = [-1] + [-1]x2   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = 0   
POL(+(x1, x2)) = 0   
POL(1) = 0   

The following pairs are in P>:

COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)
264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))

The following pairs are in Pbound:

154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_154_0_TEST_STORE(TRUE, x0[1]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(0), x0[1], 0)
COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
264_1_TEST_INVOKEMETHOD(254_0_test_Return, x0[3], x1[3]) → COND_264_1_TEST_INVOKEMETHOD(&&(>=(x1[3], 0), >(x0[3], +(x1[3], 1))), 254_0_test_Return, x0[3], x1[3])
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[4], x1[4]) → 264_1_TEST_INVOKEMETHOD(154_0_test_Store(+(x1[4], 1)), x0[4], +(x1[4], 1))
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1))

The following pairs are in P≥:

154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])
COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(+(x1[5], 1))

There are no usable rules.

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(x0[0] > 0, x0[0])
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
(5): COND_264_1_TEST_INVOKEMETHOD(TRUE, 254_0_test_Return, x0[5], x1[5]) → 154_0_TEST_STORE(x1[5] + 1)

(2) -> (0), if ((0 →* x0[0]))


(5) -> (0), if ((x1[5] + 1 →* x0[0]))


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



The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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

(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


R is empty.

The integer pair graph contains the following rules and edges:
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
(0): 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(x0[0] > 0, x0[0])

(2) -> (0), if ((0 →* x0[0]))


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



The set Q is empty.

(9) 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_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0) the following chains were created:
  • We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0), 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]) which results in the following constraint:

    (1)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2]∧0=x0[0]1 ⇒ COND_154_0_TEST_STORE(TRUE, x0[2])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[2])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))



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

    (2)    (>(x0[0], 0)=TRUE ⇒ COND_154_0_TEST_STORE(TRUE, x0[0])≥NonInfC∧COND_154_0_TEST_STORE(TRUE, x0[0])≥154_0_TEST_STORE(0)∧(UIncreasing(154_0_TEST_STORE(0)), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)







For Pair 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]) the following chains were created:
  • We consider the chain 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0]), COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0) which results in the following constraint:

    (7)    (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



    We simplified constraint (7) using rule (IV) which results in the following new constraint:

    (8)    (>(x0[0], 0)=TRUE ⇒ 154_0_TEST_STORE(x0[0])≥NonInfC∧154_0_TEST_STORE(x0[0])≥COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥))



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

    (9)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] + x0[0] ≥ 0)



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

    (10)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] + x0[0] ≥ 0)



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

    (11)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] + x0[0] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] + x0[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(154_0_TEST_STORE(0)), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] ≥ 0∧[(-1)bso_10] ≥ 0)

  • 154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] + x0[0] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_154_0_TEST_STORE(x1, x2)) = [2]   
POL(154_0_TEST_STORE(x1)) = [2] + x1   
POL(0) = 0   
POL(>(x1, x2)) = 0   

The following pairs are in P>:

154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in Pbound:

COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)
154_0_TEST_STORE(x0[0]) → COND_154_0_TEST_STORE(>(x0[0], 0), x0[0])

The following pairs are in P≥:

COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)

There are no usable rules.

(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:
none


R is empty.

The integer pair graph contains the following rules and edges:
(2): COND_154_0_TEST_STORE(TRUE, x0[2]) → 154_0_TEST_STORE(0)


The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE