Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 50 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 60 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 150 ms)
↳8 AND
    ↳9 IDP
        ↳10 IDependencyGraphProof (⇔, 0 ms)
        ↳11 TRUE
    ↳12 IDP
        ↳13 IDependencyGraphProof (⇔, 0 ms)
        ↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Double3 
/**
 * A recursive loop.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that all calls terminate.
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Double3 {

    private static void test(int n) {
	while (--n > 0) test(n);
    }

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

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Double3.main([Ljava/lang/String;)V: Graph of 31 nodes with 0 SCCs.

Double3.test(I)V: Graph of 15 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: Double3.test(I)V
SCC calls the following helper methods: Double3.test(I)V
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 13 rules for P and 2 rules for R.


P rules:
248_0_test_Inc(EOS(STATIC_248), i33) → 250_0_test_Load(EOS(STATIC_250), +(i33, -1))
250_0_test_Load(EOS(STATIC_250), i36) → 252_0_test_LE(EOS(STATIC_252), i36, i36)
252_0_test_LE(EOS(STATIC_252), i40, i40) → 255_0_test_LE(EOS(STATIC_255), i40, i40)
255_0_test_LE(EOS(STATIC_255), i40, i40) → 258_0_test_Load(EOS(STATIC_258), i40) | >(i40, 0)
258_0_test_Load(EOS(STATIC_258), i40) → 261_0_test_InvokeMethod(EOS(STATIC_261), i40, i40)
261_0_test_InvokeMethod(EOS(STATIC_261), i40, i40) → 265_1_test_InvokeMethod(265_0_test_Inc(EOS(STATIC_265), i40), i40, i40)
265_0_test_Inc(EOS(STATIC_265), i40) → 269_0_test_Inc(EOS(STATIC_269), i40)
265_1_test_InvokeMethod(256_0_test_Return(EOS(STATIC_256)), i43, i43) → 276_0_test_Return(EOS(STATIC_276), i43, i43)
269_0_test_Inc(EOS(STATIC_269), i40) → 210_0_test_Inc(EOS(STATIC_210), i40)
210_0_test_Inc(EOS(STATIC_210), i24) → 248_0_test_Inc(EOS(STATIC_248), i24)
276_0_test_Return(EOS(STATIC_276), i43, i43) → 278_0_test_JMP(EOS(STATIC_278), i43)
278_0_test_JMP(EOS(STATIC_278), i43) → 280_0_test_Inc(EOS(STATIC_280), i43)
280_0_test_Inc(EOS(STATIC_280), i43) → 248_0_test_Inc(EOS(STATIC_248), i43)
R rules:
252_0_test_LE(EOS(STATIC_252), i39, i39) → 254_0_test_LE(EOS(STATIC_254), i39, i39)
254_0_test_LE(EOS(STATIC_254), i39, i39) → 256_0_test_Return(EOS(STATIC_256)) | <=(i39, 0)

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


P rules:
248_0_test_Inc(EOS(STATIC_248), x0) → 265_1_test_InvokeMethod(248_0_test_Inc(EOS(STATIC_248), +(x0, -1)), +(x0, -1), +(x0, -1)) | >(x0, 1)
265_1_test_InvokeMethod(256_0_test_Return(EOS(STATIC_256)), x0, x0) → 248_0_test_Inc(EOS(STATIC_248), x0)
R rules:

Filtered ground terms:



248_0_test_Inc(x1, x2) → 248_0_test_Inc(x2)
256_0_test_Return(x1) → 256_0_test_Return
Cond_248_0_test_Inc(x1, x2, x3) → Cond_248_0_test_Inc(x1, x3)

Filtered duplicate args:



265_1_test_InvokeMethod(x1, x2, x3) → 265_1_test_InvokeMethod(x1, x3)

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


P rules:
248_0_test_Inc(x0) → 265_1_test_InvokeMethod(248_0_test_Inc(+(x0, -1)), +(x0, -1)) | >(x0, 1)
265_1_test_InvokeMethod(256_0_test_Return, x0) → 248_0_test_Inc(x0)
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.


P rules:
248_0_TEST_INC(x0) → COND_248_0_TEST_INC(>(x0, 1), x0)
COND_248_0_TEST_INC(TRUE, x0) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0, -1)), +(x0, -1))
COND_248_0_TEST_INC(TRUE, x0) → 248_0_TEST_INC(+(x0, -1))
265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0) → 248_0_TEST_INC(x0)
R 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): 248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(x0[0] > 1, x0[0])
(1): COND_248_0_TEST_INC(TRUE, x0[1]) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(x0[1] + -1), x0[1] + -1)
(2): COND_248_0_TEST_INC(TRUE, x0[2]) → 248_0_TEST_INC(x0[2] + -1)
(3): 265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3]) → 248_0_TEST_INC(x0[3])

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


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


(1) -> (3), if (248_0_test_Inc(x0[1] + -1) →* 256_0_test_Return∧x0[1] + -1 →* x0[3])


(2) -> (0), if (x0[2] + -1 →* x0[0])


(3) -> (0), if (x0[3] →* x0[0])



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@142b452b Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 248_0_TEST_INC(x0) → COND_248_0_TEST_INC(>(x0, 1), x0) the following chains were created:
  • We consider the chain 248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(>(x0[0], 1), x0[0]), COND_248_0_TEST_INC(TRUE, x0[1]) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1)) which results in the following constraint:

    (1)    (>(x0[0], 1)=TRUE∧x0[0]=x0[1] ⇒ 248_0_TEST_INC(x0[0])≥NonInfC∧248_0_TEST_INC(x0[0])≥COND_248_0_TEST_INC(>(x0[0], 1), x0[0])∧(UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥))



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

    (2)    (>(x0[0], 1)=TRUE ⇒ 248_0_TEST_INC(x0[0])≥NonInfC∧248_0_TEST_INC(x0[0])≥COND_248_0_TEST_INC(>(x0[0], 1), x0[0])∧(UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥))



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

    (3)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (4)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (5)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



  • We consider the chain 248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(>(x0[0], 1), x0[0]), COND_248_0_TEST_INC(TRUE, x0[2]) → 248_0_TEST_INC(+(x0[2], -1)) which results in the following constraint:

    (7)    (>(x0[0], 1)=TRUE∧x0[0]=x0[2] ⇒ 248_0_TEST_INC(x0[0])≥NonInfC∧248_0_TEST_INC(x0[0])≥COND_248_0_TEST_INC(>(x0[0], 1), x0[0])∧(UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥))



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

    (8)    (>(x0[0], 1)=TRUE ⇒ 248_0_TEST_INC(x0[0])≥NonInfC∧248_0_TEST_INC(x0[0])≥COND_248_0_TEST_INC(>(x0[0], 1), x0[0])∧(UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥))



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

    (9)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (10)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (11)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)



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

    (12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)







For Pair COND_248_0_TEST_INC(TRUE, x0) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0, -1)), +(x0, -1)) the following chains were created:
  • We consider the chain COND_248_0_TEST_INC(TRUE, x0[1]) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1)) which results in the following constraint:

    (13)    (COND_248_0_TEST_INC(TRUE, x0[1])≥NonInfC∧COND_248_0_TEST_INC(TRUE, x0[1])≥265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))∧(UIncreasing(265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥))



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

    (14)    ((UIncreasing(265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)



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

    (15)    ((UIncreasing(265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)



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

    (16)    ((UIncreasing(265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)



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

    (17)    ((UIncreasing(265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[(-1)bso_15] ≥ 0)







For Pair COND_248_0_TEST_INC(TRUE, x0) → 248_0_TEST_INC(+(x0, -1)) the following chains were created:
  • We consider the chain COND_248_0_TEST_INC(TRUE, x0[2]) → 248_0_TEST_INC(+(x0[2], -1)) which results in the following constraint:

    (18)    (COND_248_0_TEST_INC(TRUE, x0[2])≥NonInfC∧COND_248_0_TEST_INC(TRUE, x0[2])≥248_0_TEST_INC(+(x0[2], -1))∧(UIncreasing(248_0_TEST_INC(+(x0[2], -1))), ≥))



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

    (19)    ((UIncreasing(248_0_TEST_INC(+(x0[2], -1))), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)



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

    (20)    ((UIncreasing(248_0_TEST_INC(+(x0[2], -1))), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)



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

    (21)    ((UIncreasing(248_0_TEST_INC(+(x0[2], -1))), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)



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

    (22)    ((UIncreasing(248_0_TEST_INC(+(x0[2], -1))), ≥)∧[bni_16] = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)







For Pair 265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0) → 248_0_TEST_INC(x0) the following chains were created:
  • We consider the chain 265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3]) → 248_0_TEST_INC(x0[3]), 248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(>(x0[0], 1), x0[0]) which results in the following constraint:

    (23)    (x0[3]=x0[0] ⇒ 265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3])≥NonInfC∧265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3])≥248_0_TEST_INC(x0[3])∧(UIncreasing(248_0_TEST_INC(x0[3])), ≥))



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

    (24)    (265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3])≥NonInfC∧265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3])≥248_0_TEST_INC(x0[3])∧(UIncreasing(248_0_TEST_INC(x0[3])), ≥))



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

    (25)    ((UIncreasing(248_0_TEST_INC(x0[3])), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)



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

    (26)    ((UIncreasing(248_0_TEST_INC(x0[3])), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)



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

    (27)    ((UIncreasing(248_0_TEST_INC(x0[3])), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)



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

    (28)    ((UIncreasing(248_0_TEST_INC(x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 248_0_TEST_INC(x0) → COND_248_0_TEST_INC(>(x0, 1), x0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)
    • (x0[0] ≥ 0 ⇒ (UIncreasing(COND_248_0_TEST_INC(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

  • COND_248_0_TEST_INC(TRUE, x0) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0, -1)), +(x0, -1))
    • ((UIncreasing(265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

  • COND_248_0_TEST_INC(TRUE, x0) → 248_0_TEST_INC(+(x0, -1))
    • ((UIncreasing(248_0_TEST_INC(+(x0[2], -1))), ≥)∧[bni_16] = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

  • 265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0) → 248_0_TEST_INC(x0)
    • ((UIncreasing(248_0_TEST_INC(x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 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(248_0_TEST_INC(x1)) = [2]x1   
POL(COND_248_0_TEST_INC(x1, x2)) = [2]x2   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(265_1_TEST_INVOKEMETHOD(x1, x2)) = [2] + [2]x2   
POL(248_0_test_Inc(x1)) = x1   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(256_0_test_Return) = [-1]   

The following pairs are in P>:

COND_248_0_TEST_INC(TRUE, x0[2]) → 248_0_TEST_INC(+(x0[2], -1))
265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3]) → 248_0_TEST_INC(x0[3])

The following pairs are in Pbound:

248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(>(x0[0], 1), x0[0])

The following pairs are in P≥:

248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(>(x0[0], 1), x0[0])
COND_248_0_TEST_INC(TRUE, x0[1]) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))

There are no usable rules.

(8) Complex Obligation (AND)

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 248_0_TEST_INC(x0[0]) → COND_248_0_TEST_INC(x0[0] > 1, x0[0])
(1): COND_248_0_TEST_INC(TRUE, x0[1]) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(x0[1] + -1), x0[1] + -1)

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



The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_248_0_TEST_INC(TRUE, x0[1]) → 265_1_TEST_INVOKEMETHOD(248_0_test_Inc(x0[1] + -1), x0[1] + -1)
(2): COND_248_0_TEST_INC(TRUE, x0[2]) → 248_0_TEST_INC(x0[2] + -1)
(3): 265_1_TEST_INVOKEMETHOD(256_0_test_Return, x0[3]) → 248_0_TEST_INC(x0[3])

(1) -> (3), if (248_0_test_Inc(x0[1] + -1) →* 256_0_test_Return∧x0[1] + -1 →* x0[3])



The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(14) TRUE