Termination proof of /tmp/tmplJ0JFB/Velroyen08-simple-whileDecr.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 110 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 40 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 50 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_26 (Sun Microsystems Inc.) Main-Class: simple/whileDecr/Main

package simple.whileDecr;

public class Main {

	/**
	 * @param args
	 */
	public static void main(String[] args) {
		WhileDecr.decrease(args.length);
	}

}


package simple.whileDecr;

public class WhileDecr {

	// this example does acutally terminate.
	
	public static void decrease(int i) {
		while (i > 5) {
			i--;
		}
	}
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
simple.whileDecr.Main.main([Ljava/lang/String;)V: Graph of 39 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: simple.whileDecr.Main.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 7 rules for P and 0 rules for R.

P rules:
113_0_decrease_ConstantStackPush(EOS(STATIC_113), i14, i14) → 116_0_decrease_LE(EOS(STATIC_116), i14, i14, 5)
116_0_decrease_LE(EOS(STATIC_116), i27, i27, matching1) → 120_0_decrease_LE(EOS(STATIC_120), i27, i27, 5) | =(matching1, 5)
120_0_decrease_LE(EOS(STATIC_120), i27, i27, matching1) → 124_0_decrease_Inc(EOS(STATIC_124), i27) | &&(>(i27, 5), =(matching1, 5))
124_0_decrease_Inc(EOS(STATIC_124), i27) → 129_0_decrease_JMP(EOS(STATIC_129), +(i27, -1)) | >(i27, 0)
129_0_decrease_JMP(EOS(STATIC_129), i29) → 147_0_decrease_Load(EOS(STATIC_147), i29)
147_0_decrease_Load(EOS(STATIC_147), i29) → 109_0_decrease_Load(EOS(STATIC_109), i29)
109_0_decrease_Load(EOS(STATIC_109), i14) → 113_0_decrease_ConstantStackPush(EOS(STATIC_113), i14, i14)
R rules:

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

P rules:
113_0_decrease_ConstantStackPush(EOS(STATIC_113), x0, x0) → 113_0_decrease_ConstantStackPush(EOS(STATIC_113), +(x0, -1), +(x0, -1)) | >(x0, 5)
R rules:

Filtered ground terms:

113_0_decrease_ConstantStackPush(x1, x2, x3) → 113_0_decrease_ConstantStackPush(x2, x3)
EOS(x1) → EOS
Cond_113_0_decrease_ConstantStackPush(x1, x2, x3, x4) → Cond_113_0_decrease_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

113_0_decrease_ConstantStackPush(x1, x2) → 113_0_decrease_ConstantStackPush(x2)
Cond_113_0_decrease_ConstantStackPush(x1, x2, x3) → Cond_113_0_decrease_ConstantStackPush(x1, x3)

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

P rules:
113_0_decrease_ConstantStackPush(x0) → 113_0_decrease_ConstantStackPush(+(x0, -1)) | >(x0, 5)
R rules:

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

P rules:
113_0_DECREASE_CONSTANTSTACKPUSH(x0) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0, 5), x0)
COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0) → 113_0_DECREASE_CONSTANTSTACKPUSH(+(x0, -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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 113_0_DECREASE_CONSTANTSTACKPUSH(x0[0]) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(x0[0] > 5, x0[0])
(1): COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 113_0_DECREASE_CONSTANTSTACKPUSH(x0[1] + -1)

(0) -> (1), if (x0[0] > 5 ∧x0[0] →^* x0[1])

(1) -> (0), if (x0[1] + -1 →^* 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@4da9ec16 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 113_0_DECREASE_CONSTANTSTACKPUSH(x0) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0, 5), x0) the following chains were created:

We consider the chain 113_0_DECREASE_CONSTANTSTACKPUSH(x0[0]) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0]), COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(1)    (>(x0[0], 5)=TRUE∧x0[0]=x0[1] ⇒ 113_0_DECREASE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧113_0_DECREASE_CONSTANTSTACKPUSH(x0[0])≥COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])∧(U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x0[0], 5)=TRUE ⇒ 113_0_DECREASE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧113_0_DECREASE_CONSTANTSTACKPUSH(x0[0])≥COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])∧(U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-6] ≥ 0 ⇒ (U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-6] ≥ 0 ⇒ (U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-6] ≥ 0 ⇒ (U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥)∧[(-1)Bound*bni_8 + (12)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0) → 113_0_DECREASE_CONSTANTSTACKPUSH(+(x0, -1)) the following chains were created:

We consider the chain COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1)) which results in the following constraint:
(7)    (COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1])≥113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))∧(U^Increasing(113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

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

113_0_DECREASE_CONSTANTSTACKPUSH(x0) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0, 5), x0)
- (x0[0] ≥ 0 ⇒ (U^Increasing(COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])), ≥)∧[(-1)Bound*bni_8 + (12)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0) → 113_0_DECREASE_CONSTANTSTACKPUSH(+(x0, -1))
- ((U^Increasing(113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 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(113_0_DECREASE_CONSTANTSTACKPUSH(x₁)) = [2]x₁
POL(COND_113_0_DECREASE_CONSTANTSTACKPUSH(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(5) = [5]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 113_0_DECREASE_CONSTANTSTACKPUSH(+(x0[1], -1))

The following pairs are in P_bound:

113_0_DECREASE_CONSTANTSTACKPUSH(x0[0]) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[0])

The following pairs are in P_≥:

113_0_DECREASE_CONSTANTSTACKPUSH(x0[0]) → COND_113_0_DECREASE_CONSTANTSTACKPUSH(>(x0[0], 5), x0[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:

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_113_0_DECREASE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 113_0_DECREASE_CONSTANTSTACKPUSH(x0[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