Termination proof of /tmp/tmpWq4lYJ/LogIterative.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 359 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 0 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 187 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:

public class LogIterative {
  public static int log(int x, int y) {
    int res = 0;
    while (x >= y && y > 1) {
      res++;
      x = x/y;
    }
    return res;
  } 

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    log(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:
LogIterative.main([Ljava/lang/String;)V: Graph of 198 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: LogIterative.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses:

Used field analysis yielded the following read fields:
Marker field analysis yielded the following relations that could be markers:

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 15 rules for P and 0 rules for R.

P rules:
f815_0_log_Load(EOS(STATIC_815), i113, i112, i113, i112) → f817_0_log_LT(EOS(STATIC_817), i113, i112, i113, i112, i113)
f817_0_log_LT(EOS(STATIC_817), i113, i112, i113, i112, i113) → f820_0_log_LT(EOS(STATIC_820), i113, i112, i113, i112, i113)
f820_0_log_LT(EOS(STATIC_820), i113, i112, i113, i112, i113) → f825_0_log_Load(EOS(STATIC_825), i113, i112, i113) | >=(i112, i113)
f825_0_log_Load(EOS(STATIC_825), i113, i112, i113) → f830_0_log_ConstantStackPush(EOS(STATIC_830), i113, i112, i113, i113)
f830_0_log_ConstantStackPush(EOS(STATIC_830), i113, i112, i113, i113) → f835_0_log_LE(EOS(STATIC_835), i113, i112, i113, i113, 1)
f835_0_log_LE(EOS(STATIC_835), i125, i112, i125, i125, matching1) → f841_0_log_LE(EOS(STATIC_841), i125, i112, i125, i125, 1) | =(matching1, 1)
f841_0_log_LE(EOS(STATIC_841), i125, i112, i125, i125, matching1) → f849_0_log_Inc(EOS(STATIC_849), i125, i112, i125) | &&(>(i125, 1), =(matching1, 1))
f849_0_log_Inc(EOS(STATIC_849), i125, i112, i125) → f855_0_log_Load(EOS(STATIC_855), i125, i112, i125)
f855_0_log_Load(EOS(STATIC_855), i125, i112, i125) → f867_0_log_Load(EOS(STATIC_867), i125, i125, i112)
f867_0_log_Load(EOS(STATIC_867), i125, i125, i112) → f872_0_log_IntArithmetic(EOS(STATIC_872), i125, i125, i112, i125)
f872_0_log_IntArithmetic(EOS(STATIC_872), i125, i125, i112, i125) → f876_0_log_Store(EOS(STATIC_876), i125, i125, /(i112, i125)) | >(i125, 1)
f876_0_log_Store(EOS(STATIC_876), i125, i125, i129) → f878_0_log_JMP(EOS(STATIC_878), i125, i129, i125)
f878_0_log_JMP(EOS(STATIC_878), i125, i129, i125) → f907_0_log_Load(EOS(STATIC_907), i125, i129, i125)
f907_0_log_Load(EOS(STATIC_907), i125, i129, i125) → f811_0_log_Load(EOS(STATIC_811), i125, i129, i125)
f811_0_log_Load(EOS(STATIC_811), i113, i112, i113) → f815_0_log_Load(EOS(STATIC_815), i113, i112, i113, i112)
R rules:

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

P rules:
f815_0_log_Load(EOS(STATIC_815), x0, x1, x0, x1) → f815_0_log_Load(EOS(STATIC_815), x0, /(x1, x0), x0, /(x1, x0)) | &&(>=(x1, x0), >(x0, 1))
R rules:

Filtered ground terms:

f815_0_log_Load(x1, x2, x3, x4, x5) → f815_0_log_Load(x2, x3, x4, x5)
Cond_f815_0_log_Load(x1, x2, x3, x4, x5, x6) → Cond_f815_0_log_Load(x1, x3, x4, x5, x6)
EOS(x1) → EOS

Filtered duplicate args:

f815_0_log_Load(x1, x2, x3, x4) → f815_0_log_Load(x3, x4)
Cond_f815_0_log_Load(x1, x2, x3, x4, x5) → Cond_f815_0_log_Load(x1, x4, x5)

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

P rules:
F815_0_LOG_LOAD(x0, x1) → F815_0_LOG_LOAD(x0, /(x1, x0)) | &&(>(x0, 1), >=(x1, x0))
R rules:

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

P rules:
F815_0_LOG_LOAD'(x0, x1) → COND_F815_0_LOG_LOAD(&&(>(x0, 1), >=(x1, x0)), x0, x1)
COND_F815_0_LOG_LOAD(TRUE, x0, x1) → F815_0_LOG_LOAD'(x0, /(x1, 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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(x0[0] > 1 && x1[0] >= x0[0], x0[0], x1[0])
(1): COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], x1[1] / x0[1])

(0) -> (1), if (x0[0] > 1 && x1[0] >= x0[0] ∧x0[0] →^* x0[1]∧x1[0] →^* x1[1])

(1) -> (0), if (x0[1] →^* x0[0]∧x1[1] / x0[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.IdpDefaultShapeHeuristic@4c7bd82c Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 F815_0_LOG_LOAD'(x0, x1) → COND_F815_0_LOG_LOAD(&&(>(x0, 1), >=(x1, x0)), x0, x1) the following chains were created:

We consider the chain F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0]), COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1])) which results in the following constraint:
(1)    (&&(>(x0[0], 1), >=(x1[0], x0[0]))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ F815_0_LOG_LOAD'(x0[0], x1[0])≥NonInfC∧F815_0_LOG_LOAD'(x0[0], x1[0])≥COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])∧(U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 1)=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ F815_0_LOG_LOAD'(x0[0], x1[0])≥NonInfC∧F815_0_LOG_LOAD'(x0[0], x1[0])≥COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])∧(U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [(-1)bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_F815_0_LOG_LOAD(TRUE, x0, x1) → F815_0_LOG_LOAD'(x0, /(x1, x0)) the following chains were created:

We consider the chain F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0]), COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1])), F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0]) which results in the following constraint:
(8)    (&&(>(x0[0], 1), >=(x1[0], x0[0]))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x0[0]1∧/(x1[1], x0[1])=x1[0]1 ⇒ COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1])≥NonInfC∧COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1])≥F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))∧(U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x0[0], 1)=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ COND_F815_0_LOG_LOAD(TRUE, x0[0], x1[0])≥NonInfC∧COND_F815_0_LOG_LOAD(TRUE, x0[0], x1[0])≥F815_0_LOG_LOAD'(x0[0], /(x1[0], x0[0]))∧(U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] + [-1]max{x1[0], [-1]x1[0]} + min{max{x0[0], [-1]x0[0]} + [-1], max{x1[0], [-1]x1[0]}} ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[(-1)bso_19] + x1[0] + [-1]max{x1[0], [-1]x1[0]} + min{max{x0[0], [-1]x0[0]} + [-1], max{x1[0], [-1]x1[0]}} ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧[2]x1[0] ≥ 0∧[2]x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[-1 + (-1)bso_19] + x0[0] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0∧[2]x1[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(-1)Bound*bni_15] + [bni_15]x1[0] + [(-1)bni_15]x0[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[4] + [2]x0[0] + [2]x1[0] ≥ 0∧[4] + [2]x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:
(15)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 0)

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

F815_0_LOG_LOAD'(x0, x1) → COND_F815_0_LOG_LOAD(&&(>(x0, 1), >=(x1, x0)), x0, x1)
- (x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)
COND_F815_0_LOG_LOAD(TRUE, x0, x1) → F815_0_LOG_LOAD'(x0, /(x1, x0))
- (x0[0] ≥ 0∧x1[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (U^Increasing(F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[1 + (-1)bso_19] + x0[0] ≥ 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) = [3]
POL(FALSE) = 0
POL(F815_0_LOG_LOAD'(x₁, x₂)) = [2] + x₂ + [-1]x₁
POL(COND_F815_0_LOG_LOAD(x₁, x₂, x₃)) = [2] + x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(>=(x₁, x₂)) = [-1]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(/(x₁, x0[0])¹ @ {F815_0_LOG_LOAD'_2/1}) = max{x₁, [-1]x₁} + [-1]min{max{x₂, [-1]x₂} + [-1], max{x₁, [-1]x₁}}

The following pairs are in P_>:

COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))

The following pairs are in P_bound:

F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])
COND_F815_0_LOG_LOAD(TRUE, x0[1], x1[1]) → F815_0_LOG_LOAD'(x0[1], /(x1[1], x0[1]))

The following pairs are in P_≥:

F815_0_LOG_LOAD'(x0[0], x1[0]) → COND_F815_0_LOG_LOAD(&&(>(x0[0], 1), >=(x1[0], x0[0])), x0[0], x1[0])

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE¹ → &&(TRUE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹
/¹ →

(8) Obligation: