### (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:
example_2.Test.main([Ljava/lang/String;)V: Graph of 67 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 15 rules for P and 5 rules for R.

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

Filtered ground terms:

172_0_divBy_LE(x1, x2, x3) → 172_0_divBy_LE(x2, x3)
194_0_main_Return(x1) → 194_0_main_Return

Filtered duplicate args:

172_0_divBy_LE(x1, x2) → 172_0_divBy_LE(x2)

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

Finished conversion. Obtained 1 rules for P and 1 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

The ITRS R consists of the following rules:
172_1_main_InvokeMethod(172_0_divBy_LE(0)) → 194_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0])) → COND_172_1_MAIN_INVOKEMETHOD(x0[0] > 1 && 0 <= x0[0] / 2, 172_0_divBy_LE(x0[0]))
(1): COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1])) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[1] / 2))

(0) -> (1), if ((x0[0] > 1 && 0 <= x0[0] / 2* TRUE)∧(172_0_divBy_LE(x0[0]) →* 172_0_divBy_LE(x0[1])))

(1) -> (0), if ((172_0_divBy_LE(x0[1] / 2) →* 172_0_divBy_LE(x0[0])))

The set Q consists of the following terms:
172_1_main_InvokeMethod(172_0_divBy_LE(0))

### (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 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0)) → COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), <=(0, /(x0, 2))), 172_0_divBy_LE(x0)) the following chains were created:
• We consider the chain 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0])) → COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0])), COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1])) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2))) which results in the following constraint:

(1)    (&&(>(x0[0], 1), <=(0, /(x0[0], 2)))=TRUE172_0_divBy_LE(x0[0])=172_0_divBy_LE(x0[1]) ⇒ 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0]))≥NonInfC∧172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0]))≥COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))∧(UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥))

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

(2)    (>(x0[0], 1)=TRUE<=(0, /(x0[0], 2))=TRUE172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0]))≥NonInfC∧172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0]))≥COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))∧(UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥))

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

(3)    (x0[0] + [-2] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(4)    (x0[0] + [-2] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(5)    (x0[0] + [-2] ≥ 0∧[2]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(6)    (x0[0] ≥ 0∧[4] + [2]x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥)∧[(-1)Bound*bni_16 + (2)bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(7)    (x0[0] ≥ 0∧[1] + x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥)∧[(-1)Bound*bni_16 + (2)bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0)) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0, 2))) the following chains were created:
• We consider the chain 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0])) → COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0])), COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1])) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2))) which results in the following constraint:

(8)    (&&(>(x0[0], 1), <=(0, /(x0[0], 2)))=TRUE172_0_divBy_LE(x0[0])=172_0_divBy_LE(x0[1]) ⇒ COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1]))≥NonInfC∧COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1]))≥172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))∧(UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥))

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

(9)    (>(x0[0], 1)=TRUE<=(0, /(x0[0], 2))=TRUECOND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[0]))≥NonInfC∧COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[0]))≥172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[0], 2)))∧(UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥))

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

(10)    (x0[0] + [-2] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_22] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

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

(11)    (x0[0] + [-2] ≥ 0∧max{x0[0], [-1]x0[0]} + [-1] ≥ 0 ⇒ (UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_22] + x0[0] + [-1]max{x0[0], [-1]x0[0]} ≥ 0)

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

(12)    (x0[0] + [-2] ≥ 0∧[2]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(13)    (x0[0] ≥ 0∧[4] + [2]x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_18 + (2)bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(14)    (x0[0] ≥ 0∧[1] + x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_18 + (2)bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0)) → COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0, 1), <=(0, /(x0, 2))), 172_0_divBy_LE(x0))
• (x0[0] ≥ 0∧[1] + x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))), ≥)∧[(-1)Bound*bni_16 + (2)bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0)) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0, 2)))
• (x0[0] ≥ 0∧[1] + x0[0] ≥ 0∧[2] + x0[0] ≥ 0 ⇒ (UIncreasing(172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))), ≥)∧[(-1)Bound*bni_18 + (2)bni_18] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_22] ≥ 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(172_1_main_InvokeMethod(x1)) = [-1] + [-1]x1
POL(172_0_divBy_LE(x1)) = [-1] + [-1]x1
POL(0) = 0
POL(194_0_main_Return) = [-1]
POL(172_1_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(COND_172_1_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(<=(x1, x2)) = [-1]
POL(2) = [2]

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

POL(/(x1, 2)1 @ {}) = max{x1, [-1]x1} + [-1]
POL(/(x1, 2)1 @ {172_1_MAIN_INVOKEMETHOD_1/0, 172_0_divBy_LE_1/0}) = max{x1, [-1]x1} + [-1]

The following pairs are in P>:

COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1])) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))

The following pairs are in Pbound:

172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0])) → COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))
COND_172_1_MAIN_INVOKEMETHOD(TRUE, 172_0_divBy_LE(x0[1])) → 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(/(x0[1], 2)))

The following pairs are in P:

172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0])) → COND_172_1_MAIN_INVOKEMETHOD(&&(>(x0[0], 1), <=(0, /(x0[0], 2))), 172_0_divBy_LE(x0[0]))

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

FALSE1&&(TRUE, FALSE)1
/1

### (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:
172_1_main_InvokeMethod(172_0_divBy_LE(0)) → 194_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 172_1_MAIN_INVOKEMETHOD(172_0_divBy_LE(x0[0])) → COND_172_1_MAIN_INVOKEMETHOD(x0[0] > 1 && 0 <= x0[0] / 2, 172_0_divBy_LE(x0[0]))

The set Q consists of the following terms:
172_1_main_InvokeMethod(172_0_divBy_LE(0))

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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