Termination proof of /home/www/eval/JBC/AProVE-RTA2010/Int.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 DuplicateArgsRemoverProof (⇔)

↳6 ITRS

↳7 ITRStoIDPProof (⇔)

↳8 IDP

↳9 UsableRulesProof (⇔)

↳10 IDP

↳11 ItpfGraphProof (⇔)

↳12 IDP

↳13 IDPNonInfProof (⇐)

↳14 AND

    ↳15 IDP

        ↳16 IDependencyGraphProof (⇔)

        ↳17 TRUE

    ↳18 IDP

        ↳19 IDependencyGraphProof (⇔)

        ↳20 TRUE

(0) Obligation:

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

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

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

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

    // introduce sharing
    Int copy = orig;

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

  public static void main(String[] args) {
    Random.args = args;
    Int x = new Int();
    x.val = Random.random();
    Int y = new Int();
    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:
Graph of 216 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load529(java.lang.Object(Int(i9)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Cond_Load529(i9 < i15, java.lang.Object(Int(i9)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
Cond_Load529(TRUE, java.lang.Object(Int(i9)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Load529(java.lang.Object(Int(i9 + 1)), java.lang.Object(Int(i15)), java.lang.Object(Int(i9 + 1)))
The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)), java.lang.Object(Int(x0)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)), java.lang.Object(Int(x0)))

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Load529(x1, x2, x3) → Load529(x2, x3)
Cond_Load529(x1, x2, x3, x4) → Cond_Load529(x1, x3, x4)

(6) Obligation:

ITRS 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 TRS R consists of the following rules:
Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Cond_Load529(i9 < i15, java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
Cond_Load529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9 + 1)))
The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(8) 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

The ITRS R consists of the following rules:
Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Cond_Load529(i9 < i15, java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
Cond_Load529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → Load529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9 + 1)))

The integer pair graph contains the following rules and edges:
(0): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))
(1): COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

(0) -> (1), if ((java.lang.Object(Int(i15[0])) →^* java.lang.Object(Int(i15[1])))∧(i9[0] < i15[0] →^* TRUE)∧(java.lang.Object(Int(i9[0])) →^* java.lang.Object(Int(i9[1]))))

(1) -> (0), if ((java.lang.Object(Int(i15[1])) →^* java.lang.Object(Int(i15[0])))∧(java.lang.Object(Int(i9[1] + 1)) →^* java.lang.Object(Int(i9[0]))))

The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))

(9) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(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:
(0): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))
(1): COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

(0) -> (1), if ((java.lang.Object(Int(i15[0])) →^* java.lang.Object(Int(i15[1])))∧(i9[0] < i15[0] →^* TRUE)∧(java.lang.Object(Int(i9[0])) →^* java.lang.Object(Int(i9[1]))))

(1) -> (0), if ((java.lang.Object(Int(i15[1])) →^* java.lang.Object(Int(i15[0])))∧(java.lang.Object(Int(i9[1] + 1)) →^* java.lang.Object(Int(i9[0]))))

The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))

(11) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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

(0) -> (1), if (((i15[0] →^* i15[1]))∧(i9[0] < i15[0] →^* TRUE)∧((i9[0] →^* i9[1])))

(1) -> (0), if (((i15[1] →^* i15[0]))∧((i9[1] + 1 →^* i9[0])))

The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))

(13) 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 LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → COND_LOAD529(<(i9, i15), java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) the following chains were created:

We consider the chain LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))), COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1)))) which results in the following constraint:
(1)    (i15[0]=i15[1]∧<(i9[0], i15[0])=TRUE∧i9[0]=i9[1] ⇒ LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))≥NonInfC∧LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))≥COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))∧(U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (<(i9[0], i15[0])=TRUE ⇒ LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))≥NonInfC∧LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))≥COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))∧(U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i15[0] + [-1] + [-1]i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]i9[0] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i15[0] + [-1] + [-1]i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]i9[0] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i15[0] + [-1] + [-1]i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(-1)bni_8]i9[0] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (i15[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

(8)    (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_LOAD529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(+(i9, 1)))) the following chains were created:

We consider the chain COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1)))) which results in the following constraint:
(9)    (COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1])))≥NonInfC∧COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1])))≥LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))∧(U^Increasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    ((U^Increasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧[1 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    ((U^Increasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧[1 + (-1)bso_11] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    ((U^Increasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧[1 + (-1)bso_11] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    ((U^Increasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

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

LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → COND_LOAD529(<(i9, i15), java.lang.Object(Int(i15)), java.lang.Object(Int(i9)))
- (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)
- (i15[0] ≥ 0∧i9[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))), ≥)∧[(-1)Bound*bni_8] + [bni_8]i15[0] ≥ 0∧[(-1)bso_9] ≥ 0)
COND_LOAD529(TRUE, java.lang.Object(Int(i15)), java.lang.Object(Int(i9))) → LOAD529(java.lang.Object(Int(i15)), java.lang.Object(Int(+(i9, 1))))
- ((U^Increasing(LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))), ≥)∧0 = 0∧0 = 0∧[1 + (-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(LOAD529(x₁, x₂)) = [-1] + [-1]x₂ + x₁
POL(java.lang.Object(x₁)) = x₁
POL(Int(x₁)) = x₁
POL(COND_LOAD529(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(+(i9[1], 1))))

The following pairs are in P_bound:

LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))

The following pairs are in P_≥:

LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(<(i9[0], i15[0]), java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))

There are no usable rules.

(14) Complex Obligation (AND)

(15) 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): LOAD529(java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0]))) → COND_LOAD529(i9[0] < i15[0], java.lang.Object(Int(i15[0])), java.lang.Object(Int(i9[0])))

The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))

(16) IDependencyGraphProof (EQUIVALENT transformation)

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

(17) TRUE

(18) 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_LOAD529(TRUE, java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1]))) → LOAD529(java.lang.Object(Int(i15[1])), java.lang.Object(Int(i9[1] + 1)))

The set Q consists of the following terms:
Load529(java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))
Cond_Load529(TRUE, java.lang.Object(Int(x0)), java.lang.Object(Int(x1)))

(19) 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) DuplicateArgsRemoverProof (EQUIVALENT transformation)

(6) Obligation:

(7) ITRStoIDPProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) ItpfGraphProof (EQUIVALENT transformation)

(12) Obligation:

(13) IDPNonInfProof (SOUND transformation)

(14) Complex Obligation (AND)

(15) Obligation:

(16) IDependencyGraphProof (EQUIVALENT transformation)

(17) TRUE

(18) Obligation:

(19) IDependencyGraphProof (EQUIVALENT transformation)

(20) TRUE