### (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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
Loop1.main([Ljava/lang/String;)V: Graph of 40 nodes with 1 SCC.

### (3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

### (4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: Loop1.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

### (5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 8 rules for P and 0 rules for R.

P rules:
207_0_main_Load(EOS(STATIC_207), java.lang.Object(ARRAY(i7)), i26, i26) → 209_0_main_ArrayLength(EOS(STATIC_209), java.lang.Object(ARRAY(i7)), i26, i26, java.lang.Object(ARRAY(i7)))
209_0_main_ArrayLength(EOS(STATIC_209), java.lang.Object(ARRAY(i7)), i26, i26, java.lang.Object(ARRAY(i7))) → 212_0_main_GE(EOS(STATIC_212), java.lang.Object(ARRAY(i7)), i26, i26, i7) | >=(i7, 0)
212_0_main_GE(EOS(STATIC_212), java.lang.Object(ARRAY(i7)), i26, i26, i7) → 214_0_main_GE(EOS(STATIC_214), java.lang.Object(ARRAY(i7)), i26, i26, i7)
214_0_main_GE(EOS(STATIC_214), java.lang.Object(ARRAY(i7)), i26, i26, i7) → 219_0_main_Inc(EOS(STATIC_219), java.lang.Object(ARRAY(i7)), i26) | <(i26, i7)
219_0_main_Inc(EOS(STATIC_219), java.lang.Object(ARRAY(i7)), i26) → 223_0_main_JMP(EOS(STATIC_223), java.lang.Object(ARRAY(i7)), +(i26, 1)) | >=(i26, 0)
223_0_main_JMP(EOS(STATIC_223), java.lang.Object(ARRAY(i7)), i30) → 239_0_main_Load(EOS(STATIC_239), java.lang.Object(ARRAY(i7)), i30)
R rules:

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

P rules:
207_0_main_Load(EOS(STATIC_207), java.lang.Object(ARRAY(x0)), x1, x1) → 207_0_main_Load(EOS(STATIC_207), java.lang.Object(ARRAY(x0)), +(x1, 1), +(x1, 1)) | &&(&&(>(+(x1, 1), 0), <(x1, x0)), >(+(x0, 1), 0))
R rules:

Filtered ground terms:

EOS(x1) → EOS

Filtered duplicate args:

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

P rules:
207_0_main_Load(java.lang.Object(ARRAY(x0)), x1) → 207_0_main_Load(java.lang.Object(ARRAY(x0)), +(x1, 1)) | &&(&&(>(x1, -1), <(x1, x0)), >(x0, -1))
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

P rules:
R rules:

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_207_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0] && x0[0] > -1, java.lang.Object(ARRAY(x0[0])), x1[0])

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

(1) -> (0), if (java.lang.Object(ARRAY(x0[1])) →* java.lang.Object(ARRAY(x0[0]))∧x1[1] + 1* x1[0])

The set Q is empty.

### (7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@6bfcc7a9 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_207_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1) the following chains were created:
• We consider the chain 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0]), COND_207_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:

(1)    (&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1))=TRUEjava.lang.Object(ARRAY(x0[0]))=java.lang.Object(ARRAY(x0[1]))∧x1[0]=x1[1]207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥NonInfC∧207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0])≥COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])∧(UIncreasing(COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(3)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(4)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(5)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 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] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_207_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0)), x1) → 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), +(x1, 1)) the following chains were created:
• We consider the chain COND_207_0_MAIN_LOAD(TRUE, java.lang.Object(ARRAY(x0[1])), x1[1]) → 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1)) which results in the following constraint:

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    ((UIncreasing(207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(9)    ((UIncreasing(207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(10)    ((UIncreasing(207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(11)    ((UIncreasing(207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0)), x1) → COND_207_0_MAIN_LOAD(&&(&&(>(x1, -1), <(x1, x0)), >(x0, -1)), java.lang.Object(ARRAY(x0)), x1)
• ([1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_207_0_MAIN_LOAD(&&(&&(>(x1[0], -1), <(x1[0], x0[0])), >(x0[0], -1)), java.lang.Object(ARRAY(x0[0])), x1[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• ((UIncreasing(207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[1])), +(x1[1], 1))), ≥)∧[bni_12] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(207_0_MAIN_LOAD(x1, x2)) = [-1] + [-1]x2 + x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1)) = x1
POL(COND_207_0_MAIN_LOAD(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (9) 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): 207_0_MAIN_LOAD(java.lang.Object(ARRAY(x0[0])), x1[0]) → COND_207_0_MAIN_LOAD(x1[0] > -1 && x1[0] < x0[0] && x0[0] > -1, java.lang.Object(ARRAY(x0[0])), x1[0])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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

R is empty.

The integer pair graph contains the following rules and edges: