Termination proof of /tmp/tmp2JM5aw/Duplicate.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 IDP

↳7 IDependencyGraphProof (⇔)

↳8 TRUE

(0) Obligation:

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

public class Duplicate{

  public static int round (int x) {

    if (x % 2 == 0) return x;
    else return x+1;
  }


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

    while ((x > y) && (y > 2)) {
      x++;
      y = 2*y;

    }
  }
}


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

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

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Duplicate.main([Ljava/lang/String;)V: Graph of 163 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 15 rules for P and 5 rules for R.

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

Filtered ground terms:

723_0_main_Load(x1, x2, x3, x4) → 723_0_main_Load(x2, x3, x4)
Cond_723_0_main_Load(x1, x2, x3, x4, x5) → Cond_723_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

723_0_main_Load(x1, x2, x3) → 723_0_main_Load(x2, x3)
Cond_723_0_main_Load(x1, x2, x3, x4) → Cond_723_0_main_Load(x1, x3, x4)

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

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

(4) 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): 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(x1[0] > 2 && x1[0] < x0[0] && x0[0] >= 0, x1[0], x0[0])
(1): COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(2 * x1[1], x0[1] + 1)

(0) -> (1), if ((x1[0] > 2 && x1[0] < x0[0] && x0[0] >= 0 →^* TRUE)∧(x1[0] →^* x1[1])∧(x0[0] →^* x0[1]))

(1) -> (0), if ((2 * x1[1] →^* x1[0])∧(x0[1] + 1 →^* x0[0]))

The set Q is empty.

(5) IDPNonInfProof (SOUND transformation)

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 723_0_MAIN_LOAD(x1, x0) → COND_723_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >=(x0, 0)), x1, x0) the following chains were created:

We consider the chain 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0]), COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1)) which results in the following constraint:
(1)    (&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 723_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧723_0_MAIN_LOAD(x1[0], x0[0])≥COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x0[0], 0)=TRUE∧>(x1[0], 2)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ 723_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧723_0_MAIN_LOAD(x1[0], x0[0])≥COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])∧(U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] + [(-1)bni_14]x1[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] + [(-1)bni_14]x1[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] + [(-1)bni_14]x1[0] ≥ 0∧[1 + (-1)bso_15] ≥ 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] + [-3] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_723_0_MAIN_LOAD(TRUE, x1, x0) → 723_0_MAIN_LOAD(*(2, x1), +(x0, 1)) the following chains were created:

We consider the chain 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0]), COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1)), 723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:
(8)    (&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1]∧*(2, x1[1])=x1[0]1∧+(x0[1], 1)=x0[0]1 ⇒ COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))∧(U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>=(x0[0], 0)=TRUE∧>(x1[0], 2)=TRUE∧<(x1[0], x0[0])=TRUE ⇒ COND_723_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥NonInfC∧COND_723_0_MAIN_LOAD(TRUE, x1[0], x0[0])≥723_0_MAIN_LOAD(*(2, x1[0]), +(x0[0], 1))∧(U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] + [(-1)bni_16]x1[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    ([1] + x1[0] + x0[0] ≥ 0∧x1[0] + [-3] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[-3 + (-1)bso_17] + x1[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

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

723_0_MAIN_LOAD(x1, x0) → COND_723_0_MAIN_LOAD(&&(&&(>(x1, 2), <(x1, x0)), >=(x0, 0)), x1, x0)
- ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
COND_723_0_MAIN_LOAD(TRUE, x1, x0) → 723_0_MAIN_LOAD(*(2, x1), +(x0, 1))
- ([4] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))), ≥)∧[(-1)Bound*bni_16 + bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] + x1[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) = [1]
POL(FALSE) = 0
POL(723_0_MAIN_LOAD(x₁, x₂)) = [2] + x₂ + [-1]x₁
POL(COND_723_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₂ + x₁
POL(&&(x₁, x₂)) = [2]
POL(>(x₁, x₂)) = [-1]
POL(2) = [2]
POL(<(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(*(x₁, x₂)) = x₁·x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])

The following pairs are in P_bound:

723_0_MAIN_LOAD(x1[0], x0[0]) → COND_723_0_MAIN_LOAD(&&(&&(>(x1[0], 2), <(x1[0], x0[0])), >=(x0[0], 0)), x1[0], x0[0])
COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))

The following pairs are in P_≥:

COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(*(2, x1[1]), +(x0[1], 1))

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

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

(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:
(1): COND_723_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 723_0_MAIN_LOAD(2 * x1[1], x0[1] + 1)

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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