(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:
Graph of 58 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:
Load182(java.lang.Object(ARRAY(i11, a37data)), i7) → Cond_Load182(i7 >= 0 && i7 < i11 && i7 + 1 > 0, java.lang.Object(ARRAY(i11, a37data)), i7)
The set Q consists of the following terms:

(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:
Load182(java.lang.Object(ARRAY(i11, a37data)), i7) → Cond_Load182(i7 >= 0 && i7 < i11 && i7 + 1 > 0, java.lang.Object(ARRAY(i11, a37data)), i7)

The integer pair graph contains the following rules and edges:
(0): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

(0) -> (1), if ((i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0* TRUE)∧(java.lang.Object(ARRAY(i11[0], a37data[0])) →* java.lang.Object(ARRAY(i11[1], a37data[1])))∧(i7[0]* i7[1]))

(1) -> (0), if ((java.lang.Object(ARRAY(i11[1], a37data[1])) →* java.lang.Object(ARRAY(i11[0], a37data[0])))∧(i7[1] + 1* i7[0]))

The set Q consists of the following terms:

(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): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

(0) -> (1), if ((i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0* TRUE)∧(java.lang.Object(ARRAY(i11[0], a37data[0])) →* java.lang.Object(ARRAY(i11[1], a37data[1])))∧(i7[0]* i7[1]))

(1) -> (0), if ((java.lang.Object(ARRAY(i11[1], a37data[1])) →* java.lang.Object(ARRAY(i11[0], a37data[0])))∧(i7[1] + 1* i7[0]))

The set Q consists of the following terms:

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

(0) -> (1), if ((i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0* TRUE)∧((i11[0]* i11[1])∧(a37data[0]* a37data[1]))∧(i7[0]* i7[1]))

(1) -> (0), if (((i11[1]* i11[0])∧(a37data[1]* a37data[0]))∧(i7[1] + 1* i7[0]))

The set Q consists of the following terms:

(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 LOAD182(java.lang.Object(ARRAY(i11, a37data)), i7) → COND_LOAD182(&&(&&(>=(i7, 0), <(i7, i11)), >(+(i7, 1), 0)), java.lang.Object(ARRAY(i11, a37data)), i7) the following chains were created:
• We consider the chain LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]), COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1)) which results in the following constraint:

(1)    (&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0))=TRUEi11[0]=i11[1]a37data[0]=a37data[1]i7[0]=i7[1]LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥NonInfC∧LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])∧(UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥))

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

(2)    (>(+(i7[0], 1), 0)=TRUE>=(i7[0], 0)=TRUE<(i7[0], i11[0])=TRUELOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥NonInfC∧LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])≥COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])∧(UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥))

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

(3)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(6)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] + [-1] + [-1]i7[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i7[0] + [bni_14]i11[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(7)    (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i11[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11, a37data)), i7) → LOAD182(java.lang.Object(ARRAY(i11, a37data)), +(i7, 1)) the following chains were created:
• We consider the chain COND_LOAD182(TRUE, java.lang.Object(ARRAY(i11[1], a37data[1])), i7[1]) → LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[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(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(10)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(11)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(12)    ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD182(java.lang.Object(ARRAY(i11, a37data)), i7) → COND_LOAD182(&&(&&(>=(i7, 0), <(i7, i11)), >(+(i7, 1), 0)), java.lang.Object(ARRAY(i11, a37data)), i7)
• (i7[0] ≥ 0∧i7[0] ≥ 0∧i11[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]i11[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• ((UIncreasing(LOAD182(java.lang.Object(ARRAY(i11[1], a37data[1])), +(i7[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 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(LOAD182(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(java.lang.Object(x1)) = x1
POL(ARRAY(x1, x2)) = [-1]x1
POL(COND_LOAD182(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)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]

The following pairs are in P>:

The following pairs are in Pbound:

LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

The following pairs are in P:

LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(&&(&&(>=(i7[0], 0), <(i7[0], i11[0])), >(+(i7[0], 1), 0)), java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

There are no usable rules.

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD182(java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0]) → COND_LOAD182(i7[0] >= 0 && i7[0] < i11[0] && i7[0] + 1 > 0, java.lang.Object(ARRAY(i11[0], a37data[0])), i7[0])

The set Q consists of the following terms:

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges: