Termination proof of /tmp/tmpCb2HWv/Loop1.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_16 (Sun Microsystems Inc.) Main-Class: Loop1

/**
 * A very simple loop over an array.
 *
 * All calls terminate.
 *
 * Julia + BinTerm prove that all calls terminate
 *
 * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A>
 */

public class Loop1 {
    public static void main(String[] args) {
	for (int i = 0; i < args.length; i++) {}
    }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Loop1.main([Ljava/lang/String;)V: Graph of 18 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 8 rules for P and 2 rules for R.

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

Filtered ground terms:

107_0_main_Load(x1, x2, x3, x4) → 107_0_main_Load(x2, x3, x4)
Cond_107_0_main_Load(x1, x2, x3, x4, x5) → Cond_107_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

107_0_main_Load(x1, x2, x3) → 107_0_main_Load(x1, x3)
Cond_107_0_main_Load(x1, x2, x3, x4) → Cond_107_0_main_Load(x1, x2, 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): 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_107_0_MAIN_LOAD(x2[0] >= 0 && x2[0] < x0[0], java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])
(1): COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), x2[1] + 1)

(0) -> (1), if ((x2[0] >= 0 && x2[0] < x0[0] →^* TRUE)∧(java.lang.Object(ARRAY(x0[0], x1[0])) →^* java.lang.Object(ARRAY(x0[1], x1[1])))∧(x2[0] →^* x2[1]))

(1) -> (0), if ((java.lang.Object(ARRAY(x0[1], x1[1])) →^* java.lang.Object(ARRAY(x0[0], x1[0])))∧(x2[1] + 1 →^* x2[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 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), x2) → COND_107_0_MAIN_LOAD(&&(>=(x2, 0), <(x2, x0)), java.lang.Object(ARRAY(x0, x1)), x2) the following chains were created:

We consider the chain 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]), COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1)) which results in the following constraint:
(1)    (&&(>=(x2[0], 0), <(x2[0], x0[0]))=TRUE∧java.lang.Object(ARRAY(x0[0], x1[0]))=java.lang.Object(ARRAY(x0[1], x1[1]))∧x2[0]=x2[1] ⇒ 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])≥NonInfC∧107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])≥COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])∧(U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])≥NonInfC∧107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])≥COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])∧(U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] + [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)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] + [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)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x2[0] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [(-1)bni_13]x2[0] + [bni_13]x0[0] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧0 = 0∧[(-1)Bound*bni_13 + bni_13] + [bni_13]x0[0] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

For Pair COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0, x1)), x2) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), +(x2, 1)) the following chains were created:

We consider the chain COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1)) which results in the following constraint:
(8)    (COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])≥NonInfC∧COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1])≥107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))∧(U^Increasing(107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

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

107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), x2) → COND_107_0_MAIN_LOAD(&&(>=(x2, 0), <(x2, x0)), java.lang.Object(ARRAY(x0, x1)), x2)
- (x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])), ≥)∧0 = 0∧[(-1)Bound*bni_13 + bni_13] + [bni_13]x0[0] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)
COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0, x1)), x2) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0, x1)), +(x2, 1))
- ((U^Increasing(107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-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(107_0_MAIN_LOAD(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(java.lang.Object(x₁)) = x₁
POL(ARRAY(x₁, x₂)) = [-1] + [-1]x₁
POL(COND_107_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), +(x2[1], 1))

The following pairs are in P_bound:

107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[0])

The following pairs are in P_≥:

107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0], x1[0])), x2[0]) → COND_107_0_MAIN_LOAD(&&(>=(x2[0], 0), <(x2[0], x0[0])), java.lang.Object(ARRAY(x0[0], x1[0])), x2[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_107_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1], x1[1])), x2[1]) → 107_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1], x1[1])), x2[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