Termination proof of /tmp/tmpE7Bvcd/PastaB14.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 830 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 40 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 120 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: PastaB14

/**
 * 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 PastaB14 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

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


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

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

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
PastaB14.main([Ljava/lang/String;)V: Graph of 189 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: PastaB14.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 19 rules for P and 0 rules for R.

P rules:
1160_0_main_Load(EOS(STATIC_1160), i441, i442, i441) → 1162_0_main_NE(EOS(STATIC_1162), i441, i442, i441, i442)
1162_0_main_NE(EOS(STATIC_1162), i442, i442, i442, i442) → 1165_0_main_NE(EOS(STATIC_1165), i442, i442, i442, i442)
1165_0_main_NE(EOS(STATIC_1165), i442, i442, i442, i442) → 1168_0_main_Load(EOS(STATIC_1168), i442, i442)
1168_0_main_Load(EOS(STATIC_1168), i442, i442) → 1170_0_main_LE(EOS(STATIC_1170), i442, i442, i442)
1170_0_main_LE(EOS(STATIC_1170), i447, i447, i447) → 1173_0_main_LE(EOS(STATIC_1173), i447, i447, i447)
1173_0_main_LE(EOS(STATIC_1173), i447, i447, i447) → 1177_0_main_Load(EOS(STATIC_1177), i447, i447) | >(i447, 0)
1177_0_main_Load(EOS(STATIC_1177), i447, i447) → 1189_0_main_Load(EOS(STATIC_1189), i447, i447)
1189_0_main_Load(EOS(STATIC_1189), i451, i452) → 1296_0_main_Load(EOS(STATIC_1296), i451, i452)
1296_0_main_Load(EOS(STATIC_1296), i566, i567) → 1298_0_main_LE(EOS(STATIC_1298), i566, i567, i567)
1298_0_main_LE(EOS(STATIC_1298), i566, matching1, matching2) → 1300_0_main_LE(EOS(STATIC_1300), i566, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
1298_0_main_LE(EOS(STATIC_1298), i566, i572, i572) → 1301_0_main_LE(EOS(STATIC_1301), i566, i572, i572)
1300_0_main_LE(EOS(STATIC_1300), i566, matching1, matching2) → 1303_0_main_Load(EOS(STATIC_1303), i566, 0) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
1303_0_main_Load(EOS(STATIC_1303), i566, matching1) → 1158_0_main_Load(EOS(STATIC_1158), i566, 0) | =(matching1, 0)
1158_0_main_Load(EOS(STATIC_1158), i441, i442) → 1160_0_main_Load(EOS(STATIC_1160), i441, i442, i441)
1301_0_main_LE(EOS(STATIC_1301), i566, i572, i572) → 1305_0_main_Inc(EOS(STATIC_1305), i566, i572) | >(i572, 0)
1305_0_main_Inc(EOS(STATIC_1305), i566, i572) → 1453_0_main_Inc(EOS(STATIC_1453), +(i566, -1), i572)
1453_0_main_Inc(EOS(STATIC_1453), i690, i572) → 1454_0_main_JMP(EOS(STATIC_1454), i690, +(i572, -1)) | >(i572, 0)
1454_0_main_JMP(EOS(STATIC_1454), i690, i691) → 1457_0_main_Load(EOS(STATIC_1457), i690, i691)
1457_0_main_Load(EOS(STATIC_1457), i690, i691) → 1296_0_main_Load(EOS(STATIC_1296), i690, i691)
R rules:

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

P rules:
1298_0_main_LE(EOS(STATIC_1298), 0, x0, x1) → 1298_0_main_LE(EOS(STATIC_1298), 0, 0, 0) | FALSE
1298_0_main_LE(EOS(STATIC_1298), x0, x1, x1) → 1298_0_main_LE(EOS(STATIC_1298), +(x0, -1), +(x1, -1), +(x1, -1)) | >(x1, 0)
R rules:

Filtered ground terms:

1298_0_main_LE(x1, x2, x3, x4) → 1298_0_main_LE(x2, x3, x4)
EOS(x1) → EOS
Cond_1298_0_main_LE(x1, x2, x3, x4, x5) → Cond_1298_0_main_LE(x1, x3, x4, x5)

Filtered duplicate args:

1298_0_main_LE(x1, x2, x3) → 1298_0_main_LE(x1, x3)
Cond_1298_0_main_LE(x1, x2, x3, x4) → Cond_1298_0_main_LE(x1, x2, x4)

Filtered unneeded arguments:

Cond_1298_0_main_LE(x1, x2, x3) → Cond_1298_0_main_LE(x1, x3)
1298_0_main_LE(x1, x2) → 1298_0_main_LE(x2)

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

P rules:
1298_0_main_LE(x1) → 1298_0_main_LE(+(x1, -1)) | >(x1, 0)
R rules:

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

P rules:
1298_0_MAIN_LE(x1) → COND_1298_0_MAIN_LE(>(x1, 0), x1)
COND_1298_0_MAIN_LE(TRUE, x1) → 1298_0_MAIN_LE(+(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(x1[0] > 0, x1[0])
(1): COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(x1[1] + -1)

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

(1) -> (0), if (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@3661eeb 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 1298_0_MAIN_LE(x1) → COND_1298_0_MAIN_LE(>(x1, 0), x1) the following chains were created:

We consider the chain 1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0]), COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(+(x1[1], -1)) which results in the following constraint:
(1)    (>(x1[0], 0)=TRUE∧x1[0]=x1[1] ⇒ 1298_0_MAIN_LE(x1[0])≥NonInfC∧1298_0_MAIN_LE(x1[0])≥COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])∧(U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE ⇒ 1298_0_MAIN_LE(x1[0])≥NonInfC∧1298_0_MAIN_LE(x1[0])≥COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])∧(U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_1298_0_MAIN_LE(TRUE, x1) → 1298_0_MAIN_LE(+(x1, -1)) the following chains were created:

We consider the chain COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(+(x1[1], -1)) which results in the following constraint:
(7)    (COND_1298_0_MAIN_LE(TRUE, x1[1])≥NonInfC∧COND_1298_0_MAIN_LE(TRUE, x1[1])≥1298_0_MAIN_LE(+(x1[1], -1))∧(U^Increasing(1298_0_MAIN_LE(+(x1[1], -1))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(1298_0_MAIN_LE(+(x1[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(1298_0_MAIN_LE(+(x1[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(1298_0_MAIN_LE(+(x1[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(1298_0_MAIN_LE(+(x1[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

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

1298_0_MAIN_LE(x1) → COND_1298_0_MAIN_LE(>(x1, 0), x1)
- (x1[0] ≥ 0 ⇒ (U^Increasing(COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_1298_0_MAIN_LE(TRUE, x1) → 1298_0_MAIN_LE(+(x1, -1))
- ((U^Increasing(1298_0_MAIN_LE(+(x1[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(1298_0_MAIN_LE(x₁)) = [2]x₁
POL(COND_1298_0_MAIN_LE(x₁, x₂)) = [2]x₂
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

COND_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(+(x1[1], -1))

The following pairs are in P_bound:

1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(>(x1[0], 0), x1[0])

The following pairs are in P_≥:

1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(>(x1[0], 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1298_0_MAIN_LE(x1[0]) → COND_1298_0_MAIN_LE(x1[0] > 0, x1[0])

The set Q is empty.

(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_1298_0_MAIN_LE(TRUE, x1[1]) → 1298_0_MAIN_LE(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