### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: example_2/Test
`package example_2;public class Test {	public static int divBy(int x){		int r = 0;		int y;		while (x > 0) {			y = 2;			x = x/y;			r = r + x;		}		return r;	}	public static void main(String[] args) {		if (args.length > 0) {		        int x = args[0].length();			int r = divBy(x);			// System.out.println("Result: " + r);		}		// else System.out.println("Error: Incorrect call");	}}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 106 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:
Cond_Load460(TRUE, i52, i44) → Load460(i52 / 2, i44 + i52 / 2)
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:

Integer

The ITRS R consists of the following rules:
Cond_Load460(TRUE, i52, i44) → Load460(i52 / 2, i44 + i52 / 2)

The integer pair graph contains the following rules and edges:
(1): COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(i52[1] / 2, i44[1] + i52[1] / 2)

(0) -> (1), if ((i52[0]* i52[1])∧(i44[0]* i44[1])∧(i52[0] > 0* TRUE))

(1) -> (0), if ((i52[1] / 2* i52[0])∧(i44[1] + i52[1] / 2* i44[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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(i52[1] / 2, i44[1] + i52[1] / 2)

(0) -> (1), if ((i52[0]* i52[1])∧(i44[0]* i44[1])∧(i52[0] > 0* TRUE))

(1) -> (0), if ((i52[1] / 2* i52[0])∧(i44[1] + i52[1] / 2* i44[0]))

The set Q consists of the following terms:

### (9) 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 LOAD460(i52, i44) → COND_LOAD460(>(i52, 0), i52, i44) the following chains were created:
• We consider the chain LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0]), COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2))) which results in the following constraint:

We simplified constraint (1) using rule (IV) which results in the following new constraint:

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

(3)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(4)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(5)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(6)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

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

(7)    (i52[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair COND_LOAD460(TRUE, i52, i44) → LOAD460(/(i52, 2), +(i44, /(i52, 2))) the following chains were created:
• We consider the chain LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0]), COND_LOAD460(TRUE, i52[1], i44[1]) → LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2))), LOAD460(i52[0], i44[0]) → COND_LOAD460(>(i52[0], 0), i52[0], i44[0]) which results in the following constraint:

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:

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

(10)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧[1 + (-1)bso_20] + i52[0] + [-1]max{i52[0], [-1]i52[0]} ≥ 0)

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

(11)    (i52[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧[1 + (-1)bso_20] + i52[0] + [-1]max{i52[0], [-1]i52[0]} ≥ 0)

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

(12)    (i52[0] + [-1] ≥ 0∧[2]i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(13)    (i52[0] + [-1] ≥ 0∧[2]i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(14)    (i52[0] ≥ 0∧[2] + [2]i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_GCD) which results in the following new constraint:

(15)    (i52[0] ≥ 0∧[1] + i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i52[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD460(>(i52[0], 0), i52[0], i44[0])), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]i52[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• (i52[0] ≥ 0∧[1] + i52[0] ≥ 0 ⇒ (UIncreasing(LOAD460(/(i52[1], 2), +(i44[1], /(i52[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i52[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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) = [3]
POL(FALSE) = 0
POL(LOAD460(x1, x2)) = [2] + x1
POL(COND_LOAD460(x1, x2, x3)) = [2] + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(2) = [2]
POL(+(x1, x2)) = x1 + x2

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {LOAD460_2/0}) = max{x1, [-1]x1} + [-1]
POL(/(x1, 2)1 @ {LOAD460_2/1, +_2/1}) = [-1]max{x1, [-1]x1} + [1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

At least the following rules have been oriented under context sensitive arithmetic replacement:

/1

### (11) 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:

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) 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:

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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