### (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:

Filtered duplicate args:

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])

(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]))=TRUEjava.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])∧(UIncreasing(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:

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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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 ⇒ (UIncreasing(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:

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

(9)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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)    ((UIncreasing(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 ⇒ (UIncreasing(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)

• ((UIncreasing(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 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(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>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (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

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])

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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: