Termination proof of /tmp/tmp71f1Z

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 60 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 90 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 160 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: Loop1

/**
 * A very simple loop over an array.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that all calls terminate
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Loop1 {
    public static void main(String[] args) {
	for (int i = 0; i < args.length; i++) {}
    }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
Loop1.main([Ljava/lang/String;)V: Graph of 40 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: Loop1.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 8 rules for P and 0 rules for R.

P rules:
203_0_main_Load(EOS(STATIC_203), java.lang.Object(ARRAY(i6)), i26, i26) → 205_0_main_ArrayLength(EOS(STATIC_205), java.lang.Object(ARRAY(i6)), i26, i26, java.lang.Object(ARRAY(i6)))
205_0_main_ArrayLength(EOS(STATIC_205), java.lang.Object(ARRAY(i6)), i26, i26, java.lang.Object(ARRAY(i6))) → 208_0_main_GE(EOS(STATIC_208), java.lang.Object(ARRAY(i6)), i26, i26, i6) | >=(i6, 0)
208_0_main_GE(EOS(STATIC_208), java.lang.Object(ARRAY(i6)), i26, i26, i6) → 211_0_main_GE(EOS(STATIC_211), java.lang.Object(ARRAY(i6)), i26, i26, i6)
211_0_main_GE(EOS(STATIC_211), java.lang.Object(ARRAY(i6)), i26, i26, i6) → 215_0_main_Inc(EOS(STATIC_215), java.lang.Object(ARRAY(i6)), i26) | <(i26, i6)
215_0_main_Inc(EOS(STATIC_215), java.lang.Object(ARRAY(i6)), i26) → 218_0_main_JMP(EOS(STATIC_218), java.lang.Object(ARRAY(i6)), +(i26, 1)) | >=(i26, 0)
218_0_main_JMP(EOS(STATIC_218), java.lang.Object(ARRAY(i6)), i30) → 223_0_main_Load(EOS(STATIC_223), java.lang.Object(ARRAY(i6)), i30)
223_0_main_Load(EOS(STATIC_223), java.lang.Object(ARRAY(i6)), i30) → 199_0_main_Load(EOS(STATIC_199), java.lang.Object(ARRAY(i6)), i30)
199_0_main_Load(EOS(STATIC_199), java.lang.Object(ARRAY(i6)), i26) → 203_0_main_Load(EOS(STATIC_203), java.lang.Object(ARRAY(i6)), i26, i26)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
203_0_main_Load(EOS(STATIC_203), java.lang.Object(ARRAY(x0)), x1, x1) → 203_0_main_Load(EOS(STATIC_203), java.lang.Object(ARRAY(x0)), +(x1, 1), +(x1, 1)) | &&(&&(>(+(x1, 1), 0), <(x1, x0)), >(+(x0, 1), 0))
R rules:

Filtered ground terms:

203_0_main_Load(x1, x2, x3, x4) → 203_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_203_0_main_Load(x1, x2, x3, x4, x5) → Cond_203_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

203_0_main_Load(x1, x2, x3) → 203_0_main_Load(x1, x3)
Cond_203_0_main_Load(x1, x2, x3, x4) → Cond_203_0_main_Load(x1, x2, x4)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.

P rules:
203_0_main_Load(java.lang.Object(ARRAY(x0)), x1) → 203_0_main_Load(java.lang.Object(ARRAY(x0)), +(x1, 1)) | &&(&&(>(x1, -1), <(x1, x0)), >(x0, -1))
R rules:

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

P rules:
203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_203_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1)
COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1))
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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_203_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0] && x0[0] > -1, java.lang.Object(ARRAY(x0[0])), x1[0])
(1): COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), x1[1] + 1)

(0) -> (1), if (x1[0] > -1 && x1[0] < x0[0] && x0[0] > -1 ∧java.lang.Object(ARRAY(x0[0])) →^* java.lang.Object(ARRAY(x0[1]))∧x1[0] →^* x1[1])

(1) -> (0), if (java.lang.Object(ARRAY(x0[1])) →^* java.lang.Object(ARRAY(x0[0]))∧x1[1] + 1 →^* x1[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@4ab27bb5 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 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_203_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1) the following chains were created:

We consider the chain 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0]), COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:
(1)    (&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1))=TRUE∧java.lang.Object(ARRAY(x0[0]))=java.lang.Object(ARRAY(x0[1]))∧x1[0]=x1[1] ⇒ 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥NonInfC∧203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])∧(U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], -1)=TRUE∧>(x1[0], -1)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥NonInfC∧203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])∧(U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    ([1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1)) the following chains were created:

We consider the chain COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:
(7)    (COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1])≥NonInfC∧COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1])≥203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))∧(U^Increasing(203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_203_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1)
- ([1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1))
- ((U^Increasing(203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(203_0_MAIN_LOAD(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁)) = x₁
POL(COND_203_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))

The following pairs are in P_bound:

203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])

The following pairs are in P_≥:

203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_203_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])

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:

Boolean, Integer

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(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_203_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 203_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), x1[1] + 1)

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE