Termination proof of /tmp/tmpEOwVVI/PastaB15.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 150 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 110 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 170 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB15

/**
 * 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 PastaB15 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        while (x == y && x > z) {
            while (y > z) {
                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:
PastaB15.main([Ljava/lang/String;)V: Graph of 257 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: PastaB15.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 20 rules for P and 0 rules for R.

P rules:
742_0_main_Load(EOS(STATIC_742), i125, i126, i87, i125) → 744_0_main_NE(EOS(STATIC_744), i125, i126, i87, i125, i126)
744_0_main_NE(EOS(STATIC_744), i126, i126, i87, i126, i126) → 748_0_main_NE(EOS(STATIC_748), i126, i126, i87, i126, i126)
748_0_main_NE(EOS(STATIC_748), i126, i126, i87, i126, i126) → 752_0_main_Load(EOS(STATIC_752), i126, i126, i87)
752_0_main_Load(EOS(STATIC_752), i126, i126, i87) → 755_0_main_Load(EOS(STATIC_755), i126, i126, i87, i126)
755_0_main_Load(EOS(STATIC_755), i126, i126, i87, i126) → 757_0_main_LE(EOS(STATIC_757), i126, i126, i87, i126, i87)
757_0_main_LE(EOS(STATIC_757), i126, i126, i87, i126, i87) → 760_0_main_LE(EOS(STATIC_760), i126, i126, i87, i126, i87)
760_0_main_LE(EOS(STATIC_760), i126, i126, i87, i126, i87) → 766_0_main_Load(EOS(STATIC_766), i126, i126, i87) | >(i126, i87)
766_0_main_Load(EOS(STATIC_766), i126, i126, i87) → 793_0_main_Load(EOS(STATIC_793), i126, i126, i87)
793_0_main_Load(EOS(STATIC_793), i134, i135, i87) → 795_0_main_Load(EOS(STATIC_795), i134, i135, i87, i135)
795_0_main_Load(EOS(STATIC_795), i134, i135, i87, i135) → 798_0_main_LE(EOS(STATIC_798), i134, i135, i87, i135, i87)
798_0_main_LE(EOS(STATIC_798), i134, i135, i87, i135, i87) → 800_0_main_LE(EOS(STATIC_800), i134, i135, i87, i135, i87)
798_0_main_LE(EOS(STATIC_798), i134, i135, i87, i135, i87) → 801_0_main_LE(EOS(STATIC_801), i134, i135, i87, i135, i87)
800_0_main_LE(EOS(STATIC_800), i134, i135, i87, i135, i87) → 804_0_main_Load(EOS(STATIC_804), i134, i135, i87) | <=(i135, i87)
804_0_main_Load(EOS(STATIC_804), i134, i135, i87) → 739_0_main_Load(EOS(STATIC_739), i134, i135, i87)
739_0_main_Load(EOS(STATIC_739), i125, i126, i87) → 742_0_main_Load(EOS(STATIC_742), i125, i126, i87, i125)
801_0_main_LE(EOS(STATIC_801), i134, i135, i87, i135, i87) → 806_0_main_Inc(EOS(STATIC_806), i134, i135, i87) | >(i135, i87)
806_0_main_Inc(EOS(STATIC_806), i134, i135, i87) → 808_0_main_Inc(EOS(STATIC_808), +(i134, -1), i135, i87)
808_0_main_Inc(EOS(STATIC_808), i144, i135, i87) → 810_0_main_JMP(EOS(STATIC_810), i144, +(i135, -1), i87)
810_0_main_JMP(EOS(STATIC_810), i144, i145, i87) → 813_0_main_Load(EOS(STATIC_813), i144, i145, i87)
813_0_main_Load(EOS(STATIC_813), i144, i145, i87) → 793_0_main_Load(EOS(STATIC_793), i144, i145, i87)
R rules:

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

P rules:
798_0_main_LE(EOS(STATIC_798), x0, x0, x1, x0, x1) → 798_0_main_LE(EOS(STATIC_798), x0, x0, x1, x0, x1) | FALSE
798_0_main_LE(EOS(STATIC_798), x0, x1, x2, x1, x2) → 798_0_main_LE(EOS(STATIC_798), +(x0, -1), +(x1, -1), x2, +(x1, -1), x2) | <(x2, x1)
R rules:

Filtered ground terms:

798_0_main_LE(x1, x2, x3, x4, x5, x6) → 798_0_main_LE(x2, x3, x4, x5, x6)
EOS(x1) → EOS
Cond_798_0_main_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_798_0_main_LE(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

798_0_main_LE(x1, x2, x3, x4, x5) → 798_0_main_LE(x1, x4, x5)
Cond_798_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_798_0_main_LE(x1, x2, x5, x6)

Filtered unneeded arguments:

Cond_798_0_main_LE(x1, x2, x3, x4) → Cond_798_0_main_LE(x1, x3, x4)
798_0_main_LE(x1, x2, x3) → 798_0_main_LE(x2, x3)

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

P rules:
798_0_main_LE(x1, x2) → 798_0_main_LE(+(x1, -1), x2) | <(x2, x1)
R rules:

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

P rules:
798_0_MAIN_LE(x1, x2) → COND_798_0_MAIN_LE(<(x2, x1), x1, x2)
COND_798_0_MAIN_LE(TRUE, x1, x2) → 798_0_MAIN_LE(+(x1, -1), x2)
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): 798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(x2[0] < x1[0], x1[0], x2[0])
(1): COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(x1[1] + -1, x2[1])

(0) -> (1), if (x2[0] < x1[0] ∧x1[0] →^* x1[1]∧x2[0] →^* x2[1])

(1) -> (0), if (x1[1] + -1 →^* x1[0]∧x2[1] →^* x2[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@bb51061 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 798_0_MAIN_LE(x1, x2) → COND_798_0_MAIN_LE(<(x2, x1), x1, x2) the following chains were created:

We consider the chain 798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0]), COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(+(x1[1], -1), x2[1]) which results in the following constraint:
(1)    (<(x2[0], x1[0])=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1] ⇒ 798_0_MAIN_LE(x1[0], x2[0])≥NonInfC∧798_0_MAIN_LE(x1[0], x2[0])≥COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])∧(U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (<(x2[0], x1[0])=TRUE ⇒ 798_0_MAIN_LE(x1[0], x2[0])≥NonInfC∧798_0_MAIN_LE(x1[0], x2[0])≥COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])∧(U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]x2[0] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]x2[0] + [bni_8]x1[0] ≥ 0∧[1 + (-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] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]x2[0] + [bni_8]x1[0] ≥ 0∧[1 + (-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_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

(8)    (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

For Pair COND_798_0_MAIN_LE(TRUE, x1, x2) → 798_0_MAIN_LE(+(x1, -1), x2) the following chains were created:

We consider the chain COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(+(x1[1], -1), x2[1]) which results in the following constraint:
(9)    (COND_798_0_MAIN_LE(TRUE, x1[1], x2[1])≥NonInfC∧COND_798_0_MAIN_LE(TRUE, x1[1], x2[1])≥798_0_MAIN_LE(+(x1[1], -1), x2[1])∧(U^Increasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

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

798_0_MAIN_LE(x1, x2) → COND_798_0_MAIN_LE(<(x2, x1), x1, x2)
- (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
- (x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x1[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
COND_798_0_MAIN_LE(TRUE, x1, x2) → 798_0_MAIN_LE(+(x1, -1), x2)
- ((U^Increasing(798_0_MAIN_LE(+(x1[1], -1), x2[1])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-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(798_0_MAIN_LE(x₁, x₂)) = [1] + [-1]x₂ + x₁
POL(COND_798_0_MAIN_LE(x₁, x₂, x₃)) = [-1]x₃ + x₂
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])

The following pairs are in P_bound:

798_0_MAIN_LE(x1[0], x2[0]) → COND_798_0_MAIN_LE(<(x2[0], x1[0]), x1[0], x2[0])

The following pairs are in P_≥:

COND_798_0_MAIN_LE(TRUE, x1[1], x2[1]) → 798_0_MAIN_LE(+(x1[1], -1), x2[1])

There are no usable rules.

(8) Obligation: