Termination proof of /tmp/tmpGfqQ1u/StupidArray.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 ITRStoIDPProof (⇔)

↳6 IDP

↳7 UsableRulesProof (⇔)

↳8 IDP

↳9 ItpfGraphProof (⇔)

↳10 IDP

↳11 IDPNonInfProof (⇐)

↳12 AND

    ↳13 IDP

        ↳14 IDependencyGraphProof (⇔)

        ↳15 TRUE

    ↳16 IDP

        ↳17 IDependencyGraphProof (⇔)

        ↳18 TRUE

(0) Obligation:

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

public class StupidArray {
  public static void main(String[] args) {
    int i = 0;
    while (true) {
      i = args.length + 1;
      args[i] = new String();
    }
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 143 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:
JMP261(java.lang.Object(ARRAY(i5, a7data))) → Cond_JMP261(i5 + 1 > 0 && i5 + 1 < i5, java.lang.Object(ARRAY(i5, a7data)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(i5, a7data))) → JMP261(java.lang.Object(ARRAY(i5, a7dataNew)))
The set Q consists of the following terms:
JMP261(java.lang.Object(ARRAY(x0, x1)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(x0, x1)))

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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

The ITRS R consists of the following rules:
JMP261(java.lang.Object(ARRAY(i5, a7data))) → Cond_JMP261(i5 + 1 > 0 && i5 + 1 < i5, java.lang.Object(ARRAY(i5, a7data)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(i5, a7data))) → JMP261(java.lang.Object(ARRAY(i5, a7dataNew)))

The integer pair graph contains the following rules and edges:
(0): JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0]))) → COND_JMP261(i5[0] + 1 > 0 && i5[0] + 1 < i5[0], java.lang.Object(ARRAY(i5[0], a7data[0])))
(1): COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1]))) → JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))

(0) -> (1), if ((java.lang.Object(ARRAY(i5[0], a7data[0])) →^* java.lang.Object(ARRAY(i5[1], a7data[1])))∧(i5[0] + 1 > 0 && i5[0] + 1 < i5[0] →^* TRUE))

(1) -> (0), if ((java.lang.Object(ARRAY(i5[1], a7dataNew[1])) →^* java.lang.Object(ARRAY(i5[0], a7data[0]))))

The set Q consists of the following terms:
JMP261(java.lang.Object(ARRAY(x0, x1)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(x0, x1)))

(7) 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.

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0]))) → COND_JMP261(i5[0] + 1 > 0 && i5[0] + 1 < i5[0], java.lang.Object(ARRAY(i5[0], a7data[0])))
(1): COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1]))) → JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))

(0) -> (1), if ((java.lang.Object(ARRAY(i5[0], a7data[0])) →^* java.lang.Object(ARRAY(i5[1], a7data[1])))∧(i5[0] + 1 > 0 && i5[0] + 1 < i5[0] →^* TRUE))

(1) -> (0), if ((java.lang.Object(ARRAY(i5[1], a7dataNew[1])) →^* java.lang.Object(ARRAY(i5[0], a7data[0]))))

The set Q consists of the following terms:
JMP261(java.lang.Object(ARRAY(x0, x1)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(x0, x1)))

(9) ItpfGraphProof (EQUIVALENT transformation)

Applied rule ItpfICap [ICap]

(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:

Boolean, Integer

(0) -> (1), if (((i5[0] →^* i5[1])∧(a7data[0] →^* a7data[1]))∧(i5[0] + 1 > 0 && i5[0] + 1 < i5[0] →^* TRUE))

(1) -> (0), if (((i5[1] →^* i5[0])∧(a7dataNew[1] →^* a7data[0])))

The set Q consists of the following terms:
JMP261(java.lang.Object(ARRAY(x0, x1)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(x0, x1)))

(11) 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 JMP261'(java.lang.Object(ARRAY(i5, a7data))) → COND_JMP261(&&(>(+(i5, 1), 0), <(+(i5, 1), i5)), java.lang.Object(ARRAY(i5, a7data))) the following chains were created:

We consider the chain JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0]))) → COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0]))), COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1]))) → JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1]))) which results in the following constraint:
(1)    (i5[0]=i5[1]∧a7data[0]=a7data[1]∧&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0]))=TRUE ⇒ JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0])))≥NonInfC∧JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0])))≥COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))∧(U^Increasing(COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(+(i5[0], 1), 0)=TRUE∧<(+(i5[0], 1), i5[0])=TRUE ⇒ JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0])))≥NonInfC∧JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0])))≥COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))∧(U^Increasing(COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i5[0] ≥ 0∧[-2] ≥ 0 ⇒ (U^Increasing(COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧[-2 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i5[0] ≥ 0∧[-2] ≥ 0 ⇒ (U^Increasing(COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧[-2 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i5[0] ≥ 0∧[-2] ≥ 0 ⇒ (U^Increasing(COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧[-2 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (i5[0] ≥ 0∧[-2] ≥ 0 ⇒ (U^Increasing(COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))), ≥)∧0 = 0∧[(-2)bni_13 + (-1)Bound*bni_13] ≥ 0∧0 = 0∧[-2 + (-1)bso_14] ≥ 0)

We solved constraint (6) using rule (IDP_SMT_SPLIT).

For Pair COND_JMP261(TRUE, java.lang.Object(ARRAY(i5, a7data))) → JMP261'(java.lang.Object(ARRAY(i5, a7dataNew))) the following chains were created:

We consider the chain COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1]))) → JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1]))) which results in the following constraint:
(7)    (COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1])))≥NonInfC∧COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1])))≥JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))∧(U^Increasing(JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))), ≥)∧[2 + (-1)bso_16] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))), ≥)∧[2 + (-1)bso_16] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))), ≥)∧[2 + (-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_16] ≥ 0)

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

JMP261'(java.lang.Object(ARRAY(i5, a7data))) → COND_JMP261(&&(>(+(i5, 1), 0), <(+(i5, 1), i5)), java.lang.Object(ARRAY(i5, a7data)))
COND_JMP261(TRUE, java.lang.Object(ARRAY(i5, a7data))) → JMP261'(java.lang.Object(ARRAY(i5, a7dataNew)))
- ((U^Increasing(JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_16] ≥ 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(JMP261'(x₁)) = [-1] + x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1]
POL(COND_JMP261(x₁, x₂)) = 0
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(0) = 0
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0]))) → COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))
COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1]))) → JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))

The following pairs are in P_bound:

JMP261'(java.lang.Object(ARRAY(i5[0], a7data[0]))) → COND_JMP261(&&(>(+(i5[0], 1), 0), <(+(i5[0], 1), i5[0])), java.lang.Object(ARRAY(i5[0], a7data[0])))

The following pairs are in P_≥:
none

There are no usable rules.

(12) Complex Obligation (AND)

(13) 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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
JMP261(java.lang.Object(ARRAY(x0, x1)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(x0, x1)))

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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:
none

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_JMP261(TRUE, java.lang.Object(ARRAY(i5[1], a7data[1]))) → JMP261'(java.lang.Object(ARRAY(i5[1], a7dataNew[1])))

The set Q consists of the following terms:
JMP261(java.lang.Object(ARRAY(x0, x1)))
Cond_JMP261(TRUE, java.lang.Object(ARRAY(x0, x1)))

(17) 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) ITRStoIDPProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) ItpfGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) IDPNonInfProof (SOUND transformation)

(12) Complex Obligation (AND)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE