### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: LeUserDefRec
`public class LeUserDefRec {	public static void main(String[] args) {		int x = args[0].length();		int y = args[1].length();		le(x, y);	}	public static boolean le(int x, int y) {		if (x > 0 && y > 0) {			return le(x-1, y-1);		} else {			return (x == 0);		}	}}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
LeUserDefRec.main([Ljava/lang/String;)V: Graph of 129 nodes with 0 SCCs.

LeUserDefRec.le(II)Z: Graph of 35 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 15 rules for P and 19 rules for R.

Combined rules. Obtained 1 rules for P and 4 rules for R.

Filtered ground terms:

415_0_le_LE(x1, x2, x3, x4) → 415_0_le_LE(x2, x3, x4)
Cond_415_0_le_LE(x1, x2, x3, x4, x5) → Cond_415_0_le_LE(x1, x3, x4, x5)
689_0_le_Return(x1, x2) → 689_0_le_Return(x2)
643_0_le_Return(x1, x2) → 643_0_le_Return
616_0_le_Return(x1) → 616_0_le_Return

Filtered duplicate args:

415_0_le_LE(x1, x2, x3) → 415_0_le_LE(x2, x3)
Cond_415_0_le_LE(x1, x2, x3, x4) → Cond_415_0_le_LE(x1, x3, x4)

Filtered unneeded arguments:

681_1_le_InvokeMethod(x1, x2, x3, x4) → 681_1_le_InvokeMethod(x1, x3, x4)

Combined rules. Obtained 1 rules for P and 4 rules for R.

Finished conversion. Obtained 1 rules for P and 4 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:
415_0_le_LE(x1, 0) → 616_0_le_Return
681_1_le_InvokeMethod(616_0_le_Return, 0, x2) → 689_0_le_Return(x0)
681_1_le_InvokeMethod(643_0_le_Return, x2, 0) → 689_0_le_Return(x1)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x2) → 689_0_le_Return(x1)

The integer pair graph contains the following rules and edges:
(0): 415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(x1[1] - 1, x0[1] - 1)

(0) -> (1), if ((x1[0] > 0 && x0[0] > 0* TRUE)∧(x1[0]* x1[1])∧(x0[0]* x0[1]))

(1) -> (0), if ((x1[1] - 1* x1[0])∧(x0[1] - 1* x0[0]))

The set Q consists of the following terms:
415_0_le_LE(x0, 0)
681_1_le_InvokeMethod(616_0_le_Return, 0, x0)
681_1_le_InvokeMethod(643_0_le_Return, x0, 0)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x1)

### (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 415_0_LE_LE(x1, x0) → COND_415_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:

(1)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]415_0_LE_LE(x1[0], x0[0])≥NonInfC∧415_0_LE_LE(x1[0], x0[0])≥COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

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

(2)    (>(x1[0], 0)=TRUE>(x0[0], 0)=TRUE415_0_LE_LE(x1[0], x0[0])≥NonInfC∧415_0_LE_LE(x1[0], x0[0])≥COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

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

(3)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_415_0_LE_LE(TRUE, x1, x0) → 415_0_LE_LE(-(x1, 1), -(x0, 1)) the following chains were created:
• We consider the chain COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(-(x1[1], 1), -(x0[1], 1)) which results in the following constraint:

(8)    (COND_415_0_LE_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_415_0_LE_LE(TRUE, x1[1], x0[1])≥415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))∧(UIncreasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥))

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

(9)    ((UIncreasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[4 + (-1)bso_17] ≥ 0)

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

(10)    ((UIncreasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[4 + (-1)bso_17] ≥ 0)

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

(11)    ((UIncreasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧[4 + (-1)bso_17] ≥ 0)

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

(12)    ((UIncreasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧0 = 0∧0 = 0∧[4 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 415_0_LE_LE(x1, x0) → COND_415_0_LE_LE(&&(>(x1, 0), >(x0, 0)), x1, x0)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(5)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• COND_415_0_LE_LE(TRUE, x1, x0) → 415_0_LE_LE(-(x1, 1), -(x0, 1))
• ((UIncreasing(415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))), ≥)∧0 = 0∧0 = 0∧[4 + (-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(415_0_le_LE(x1, x2)) = [-1]
POL(0) = 0
POL(616_0_le_Return) = [-1]
POL(681_1_le_InvokeMethod(x1, x2, x3)) = [-1]
POL(689_0_le_Return(x1)) = [-1]
POL(643_0_le_Return) = [-1]
POL(415_0_LE_LE(x1, x2)) = [1] + [2]x2 + [2]x1
POL(COND_415_0_LE_LE(x1, x2, x3)) = [1] + [2]x3 + [2]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]

The following pairs are in P>:

COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(-(x1[1], 1), -(x0[1], 1))

The following pairs are in Pbound:

415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

The following pairs are in P:

415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(&&(>(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])

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

The ITRS R consists of the following rules:
415_0_le_LE(x1, 0) → 616_0_le_Return
681_1_le_InvokeMethod(616_0_le_Return, 0, x2) → 689_0_le_Return(x0)
681_1_le_InvokeMethod(643_0_le_Return, x2, 0) → 689_0_le_Return(x1)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x2) → 689_0_le_Return(x1)

The integer pair graph contains the following rules and edges:
(0): 415_0_LE_LE(x1[0], x0[0]) → COND_415_0_LE_LE(x1[0] > 0 && x0[0] > 0, x1[0], x0[0])

The set Q consists of the following terms:
415_0_le_LE(x0, 0)
681_1_le_InvokeMethod(616_0_le_Return, 0, x0)
681_1_le_InvokeMethod(643_0_le_Return, x0, 0)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x1)

### (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

The ITRS R consists of the following rules:
415_0_le_LE(x1, 0) → 616_0_le_Return
681_1_le_InvokeMethod(616_0_le_Return, 0, x2) → 689_0_le_Return(x0)
681_1_le_InvokeMethod(643_0_le_Return, x2, 0) → 689_0_le_Return(x1)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x2) → 689_0_le_Return(x1)

The integer pair graph contains the following rules and edges:
(1): COND_415_0_LE_LE(TRUE, x1[1], x0[1]) → 415_0_LE_LE(x1[1] - 1, x0[1] - 1)

The set Q consists of the following terms:
415_0_le_LE(x0, 0)
681_1_le_InvokeMethod(616_0_le_Return, 0, x0)
681_1_le_InvokeMethod(643_0_le_Return, x0, 0)
681_1_le_InvokeMethod(689_0_le_Return(x0), x0, x1)

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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