Termination proof of /tmp/tmpInBv6A/IntRTA.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 AND

    ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDependencyGraphProof (⇔)

        ↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: IntRTA

public class IntRTA {
  // only wrap a primitive int
  private int val;

  // count up to the value
  // in "limit"
  public static void count(
      IntRTA orig, IntRTA limit) {

    if (orig == null
        || limit == null) {
      return;
    }

    // introduce sharing
    IntRTA copy = orig;

    while (orig.val < limit.val) {
      copy.val++;
    }
  }

  public static void main(String[] args) {
    Random.args = args;
    IntRTA x = new IntRTA();
    x.val = Random.random();
    IntRTA y = new IntRTA();
    y.val = Random.random();
    count(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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
IntRTA.main([Ljava/lang/String;)V: Graph of 168 nodes with 0 SCCs.

IntRTA.count(LIntRTA;LIntRTA;)V: Graph of 24 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 14 rules for P and 2 rules for R.

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

Filtered ground terms:

880_0_count_FieldAccess(x1, x2, x3, x4, x5) → 880_0_count_FieldAccess(x2, x3, x4, x5)
IntRTA(x1, x2) → IntRTA(x2)
Cond_880_0_count_FieldAccess(x1, x2, x3, x4, x5, x6) → Cond_880_0_count_FieldAccess(x1, x3, x4, x5, x6)

Filtered duplicate args:

880_0_count_FieldAccess(x1, x2, x3, x4) → 880_0_count_FieldAccess(x2, x4)
Cond_880_0_count_FieldAccess(x1, x2, x3, x4, x5) → Cond_880_0_count_FieldAccess(x1, x3, x5)

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): 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_880_0_COUNT_FIELDACCESS(x1[0] > x0[0] && x0[0] >= 0, java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))
(1): COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1] + 1)))

(0) -> (1), if ((x1[0] > x0[0] && x0[0] >= 0 →^* TRUE)∧(java.lang.Object(IntRTA(x1[0])) →^* java.lang.Object(IntRTA(x1[1])))∧(java.lang.Object(IntRTA(x0[0])) →^* java.lang.Object(IntRTA(x0[1]))))

(1) -> (0), if ((java.lang.Object(IntRTA(x1[1])) →^* java.lang.Object(IntRTA(x1[0])))∧(java.lang.Object(IntRTA(x0[1] + 1)) →^* java.lang.Object(IntRTA(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 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → COND_880_0_COUNT_FIELDACCESS(&&(>(x1, x0), >=(x0, 0)), java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) the following chains were created:

We consider the chain 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))), COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1)))) which results in the following constraint:
(1)    (&&(>(x1[0], x0[0]), >=(x0[0], 0))=TRUE∧java.lang.Object(IntRTA(x1[0]))=java.lang.Object(IntRTA(x1[1]))∧java.lang.Object(IntRTA(x0[0]))=java.lang.Object(IntRTA(x0[1])) ⇒ 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥NonInfC∧880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))∧(U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], x0[0])=TRUE∧>=(x0[0], 0)=TRUE ⇒ 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥NonInfC∧880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))≥COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))∧(U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x0[0] + [(2)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x0[0] + [(2)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x0[0] + [(2)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(2)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(+(x0, 1)))) the following chains were created:

We consider the chain COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1)))) which results in the following constraint:
(7)    (COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1])))≥NonInfC∧COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1])))≥880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))∧(U^Increasing(880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

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

880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → COND_880_0_COUNT_FIELDACCESS(&&(>(x1, x0), >=(x0, 0)), java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0)))
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(2)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(x0))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1)), java.lang.Object(IntRTA(+(x0, 1))))
- ((U^Increasing(880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 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(880_0_COUNT_FIELDACCESS(x₁, x₂)) = [-1] + [-1]x₂ + [2]x₁
POL(java.lang.Object(x₁)) = x₁
POL(IntRTA(x₁)) = x₁
POL(COND_880_0_COUNT_FIELDACCESS(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(+(x0[1], 1))))

The following pairs are in P_bound:

880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))

The following pairs are in P_≥:

880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0]))) → COND_880_0_COUNT_FIELDACCESS(&&(>(x1[0], x0[0]), >=(x0[0], 0)), java.lang.Object(IntRTA(x1[0])), java.lang.Object(IntRTA(x0[0])))

There are no usable rules.

(6) Complex Obligation (AND)

(7) 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

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(10) 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_880_0_COUNT_FIELDACCESS(TRUE, java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1]))) → 880_0_COUNT_FIELDACCESS(java.lang.Object(IntRTA(x1[1])), java.lang.Object(IntRTA(x0[1] + 1)))

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE